Step
*
7
of Lemma
fps-deriv-mul
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
⊢ d(<b>*g)/dx = ((<b>*dg/dx)+(d<b>/dx*g)) ∈ PowerSeries(X;r)
BY
{ (InstLemma `fps-linear-ucont-equal` [⌜X⌝;⌜eq⌝;⌜r⌝;⌜λ2g.d(<b>*g)/dx⌝;⌜λ2g.((<b>*dg/dx)+(d<b>/dx*g))⌝]⋅ THENA Auto) }
1
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
⊢ fps-ucont(X;eq;r;f.d(<b>*f)/dx)
2
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
15. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
⊢ fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
3
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
15. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
16. fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
17. f1 : PowerSeries(X;r)
18. g1 : PowerSeries(X;r)
⊢ d(<b>*(f1+g1))/dx = (d(<b>*f1)/dx+d(<b>*g1)/dx) ∈ PowerSeries(X;r)
4
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
15. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
16. fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
17. ∀f,g:PowerSeries(X;r). (d(<b>*(f+g))/dx = (d(<b>*f)/dx+d(<b>*g)/dx) ∈ PowerSeries(X;r))
18. f1 : PowerSeries(X;r)
19. g1 : PowerSeries(X;r)
⊢ ((<b>*d(f1+g1)/dx)+(d<b>/dx*(f1+g1))) = (((<b>*df1/dx)+(d<b>/dx*f1))+((<b>*dg1/dx)+(d<b>/dx*g1))) ∈ PowerSeries(X;r)
5
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
15. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
16. fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
17. ∀f,g:PowerSeries(X;r). (d(<b>*(f+g))/dx = (d(<b>*f)/dx+d(<b>*g)/dx) ∈ PowerSeries(X;r))
18. ∀f,g:PowerSeries(X;r).
(((<b>*d(f+g)/dx)+(d<b>/dx*(f+g))) = (((<b>*df/dx)+(d<b>/dx*f))+((<b>*dg/dx)+(d<b>/dx*g))) ∈ PowerSeries(X;r))
19. c : |r|
20. f1 : PowerSeries(X;r)
⊢ d(<b>*(c)*f1)/dx = (c)*d(<b>*f1)/dx ∈ PowerSeries(X;r)
6
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
15. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
16. fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
17. ∀f,g:PowerSeries(X;r). (d(<b>*(f+g))/dx = (d(<b>*f)/dx+d(<b>*g)/dx) ∈ PowerSeries(X;r))
18. ∀f,g:PowerSeries(X;r).
(((<b>*d(f+g)/dx)+(d<b>/dx*(f+g))) = (((<b>*df/dx)+(d<b>/dx*f))+((<b>*dg/dx)+(d<b>/dx*g))) ∈ PowerSeries(X;r))
19. ∀c:|r|. ∀f:PowerSeries(X;r). (d(<b>*(c)*f)/dx = (c)*d(<b>*f)/dx ∈ PowerSeries(X;r))
20. c : |r|
21. f1 : PowerSeries(X;r)
⊢ ((<b>*d(c)*f1/dx)+(d<b>/dx*(c)*f1)) = (c)*((<b>*df1/dx)+(d<b>/dx*f1)) ∈ PowerSeries(X;r)
7
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
15. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
16. fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
17. ∀f,g:PowerSeries(X;r). (d(<b>*(f+g))/dx = (d(<b>*f)/dx+d(<b>*g)/dx) ∈ PowerSeries(X;r))
18. ∀f,g:PowerSeries(X;r).
(((<b>*d(f+g)/dx)+(d<b>/dx*(f+g))) = (((<b>*df/dx)+(d<b>/dx*f))+((<b>*dg/dx)+(d<b>/dx*g))) ∈ PowerSeries(X;r))
19. ∀c:|r|. ∀f:PowerSeries(X;r). (d(<b>*(c)*f)/dx = (c)*d(<b>*f)/dx ∈ PowerSeries(X;r))
20. ∀c:|r|. ∀f:PowerSeries(X;r). (((<b>*d(c)*f/dx)+(d<b>/dx*(c)*f)) = (c)*((<b>*df/dx)+(d<b>/dx*f)) ∈ PowerSeries(X;r))
21. b@0 : bag(X)
⊢ d(<b>*<b@0>)/dx = ((<b>*d<b@0>/dx)+(d<b>/dx*<b@0>)) ∈ PowerSeries(X;r)
8
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))
12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))
13. ∀c:|r|. ∀f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
14. b : bag(X)
15. λ2g.d(<b>*g)/dx = λ2g.((<b>*dg/dx)+(d<b>/dx*g)) ∈ (PowerSeries(X;r) ⟶ PowerSeries(X;r))
⊢ d(<b>*g)/dx = ((<b>*dg/dx)+(d<b>/dx*g)) ∈ PowerSeries(X;r)
Latex:
Latex:
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. \mforall{}f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx))
11. \mforall{}f,g@0:PowerSeries(X;r).
((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))))
12. \mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx)
13. \mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)))
14. b : bag(X)
\mvdash{} d(<b>*g)/dx = ((<b>*dg/dx)+(d<b>/dx*g))
By
Latex:
(InstLemma `fps-linear-ucont-equal` [\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}eq\mkleeneclose{};\mkleeneopen{}r\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}g.d(<b>*g)/dx\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}g.((<b>*dg/dx)+(d<b>/dx*g))\mkleeneclose{}]
\mcdot{}
THENA Auto
)
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