Step
*
2
6
1
1
of Lemma
padic-ring_wf
1. p : {2...}
2. ℤ(p) ∈ RngSig
3. ∀[x,y,z:p-adics(p)]. (x + y + z = x + y + z ∈ p-adics(p))
4. ∀[x:p-adics(p)]. ((x + 0(p) = x ∈ p-adics(p)) ∧ (0(p) + x = x ∈ p-adics(p)))
5. ∀[x:p-adics(p)]. ((x + -(x) = 0(p) ∈ p-adics(p)) ∧ (-(x) + x = 0(p) ∈ p-adics(p)))
6. ∀[x,y,z:p-adics(p)]. (x * y * z = x * y * z ∈ p-adics(p))
7. ∀[x:p-adics(p)]. ((x * 1(p) = x ∈ p-adics(p)) ∧ (1(p) * x = x ∈ p-adics(p)))
8. ∀[a,x,y:p-adics(p)]. ((a * x + y = a * x + a * y ∈ p-adics(p)) ∧ (x + y * a = x * a + y * a ∈ p-adics(p)))
9. ∀[x,y:p-adics(p)]. (x * y = y * x ∈ p-adics(p))
10. ∀a,b:basic-padic(p). bpa-equiv(p;bpa-mul(p;a;b);pa-mul(p;a;b))
11. ∀a,b:basic-padic(p). bpa-equiv(p;bpa-add(p;a;b);pa-add(p;a;b))
12. IsMonoid(padic(p);λu,v. pa-mul(p;u;v);1(p))
13. a : padic(p)
14. x : padic(p)
15. y : padic(p)
16. pa-mul(p;a;pa-add(p;x;y)) = pa-add(p;pa-mul(p;a;x);pa-mul(p;a;y)) ∈ padic(p)
17. X1 : ℕ
18. X2 : p-adics(p)
19. Y1 : ℕ
20. Y2 : p-adics(p)
21. Z1 : ℕ
22. Z2 : p-adics(p)
23. M : ℤ
24. imax(X1;Y1) = M ∈ ℤ
⊢ ((X1 ≤ M) ∧ (Y1 ≤ M))
⇒ (p^((Z1 + M) + (M - X1))(p) * X2 * Z2 + p^((Z1 + M) + (M - Y1))(p) * Y2 * Z2
= p^((M + Z1) + ((Z1 + M) - X1 + Z1))(p) * X2 * Z2 + p^((M + Z1) + ((Z1 + M) - Y1 + Z1))(p) * Y2 * Z2
∈ p-adics(p))
BY
{ (Intro
THEN D -1
THEN (Subst' (M + Z1) + ((Z1 + M) - X1 + Z1) ~ (Z1 + M) + (M - X1) 0 THENA Auto)
THEN (Subst' (M + Z1) + ((Z1 + M) - Y1 + Z1) ~ (Z1 + M) + (M - Y1) 0 THENA Auto)
THEN GenConclTerms Auto [⌜M - X1⌝;⌜M - Y1⌝;⌜Z1 + M⌝]⋅) }
1
1. p : {2...}
2. ℤ(p) ∈ RngSig
3. ∀[x,y,z:p-adics(p)]. (x + y + z = x + y + z ∈ p-adics(p))
4. ∀[x:p-adics(p)]. ((x + 0(p) = x ∈ p-adics(p)) ∧ (0(p) + x = x ∈ p-adics(p)))
5. ∀[x:p-adics(p)]. ((x + -(x) = 0(p) ∈ p-adics(p)) ∧ (-(x) + x = 0(p) ∈ p-adics(p)))
6. ∀[x,y,z:p-adics(p)]. (x * y * z = x * y * z ∈ p-adics(p))
7. ∀[x:p-adics(p)]. ((x * 1(p) = x ∈ p-adics(p)) ∧ (1(p) * x = x ∈ p-adics(p)))
8. ∀[a,x,y:p-adics(p)]. ((a * x + y = a * x + a * y ∈ p-adics(p)) ∧ (x + y * a = x * a + y * a ∈ p-adics(p)))
9. ∀[x,y:p-adics(p)]. (x * y = y * x ∈ p-adics(p))
10. ∀a,b:basic-padic(p). bpa-equiv(p;bpa-mul(p;a;b);pa-mul(p;a;b))
11. ∀a,b:basic-padic(p). bpa-equiv(p;bpa-add(p;a;b);pa-add(p;a;b))
12. IsMonoid(padic(p);λu,v. pa-mul(p;u;v);1(p))
13. a : padic(p)
14. x : padic(p)
15. y : padic(p)
16. pa-mul(p;a;pa-add(p;x;y)) = pa-add(p;pa-mul(p;a;x);pa-mul(p;a;y)) ∈ padic(p)
17. X1 : ℕ
18. X2 : p-adics(p)
19. Y1 : ℕ
20. Y2 : p-adics(p)
21. Z1 : ℕ
22. Z2 : p-adics(p)
23. M : ℤ
24. imax(X1;Y1) = M ∈ ℤ
25. X1 ≤ M
26. Y1 ≤ M
27. v : ℤ
28. (M - X1) = v ∈ ℤ
29. v1 : ℤ
30. (M - Y1) = v1 ∈ ℤ
31. v2 : ℤ
32. (Z1 + M) = v2 ∈ ℤ
⊢ p^(v2 + v)(p) * X2 * Z2 + p^(v2 + v1)(p) * Y2 * Z2 = p^(v2 + v)(p) * X2 * Z2 + p^(v2 + v1)(p) * Y2 * Z2 ∈ p-adics(p)
Latex:
Latex:
1. p : \{2...\}
2. \mBbbZ{}(p) \mmember{} RngSig
3. \mforall{}[x,y,z:p-adics(p)]. (x + y + z = x + y + z)
4. \mforall{}[x:p-adics(p)]. ((x + 0(p) = x) \mwedge{} (0(p) + x = x))
5. \mforall{}[x:p-adics(p)]. ((x + -(x) = 0(p)) \mwedge{} (-(x) + x = 0(p)))
6. \mforall{}[x,y,z:p-adics(p)]. (x * y * z = x * y * z)
7. \mforall{}[x:p-adics(p)]. ((x * 1(p) = x) \mwedge{} (1(p) * x = x))
8. \mforall{}[a,x,y:p-adics(p)]. ((a * x + y = a * x + a * y) \mwedge{} (x + y * a = x * a + y * a))
9. \mforall{}[x,y:p-adics(p)]. (x * y = y * x)
10. \mforall{}a,b:basic-padic(p). bpa-equiv(p;bpa-mul(p;a;b);pa-mul(p;a;b))
11. \mforall{}a,b:basic-padic(p). bpa-equiv(p;bpa-add(p;a;b);pa-add(p;a;b))
12. IsMonoid(padic(p);\mlambda{}u,v. pa-mul(p;u;v);1(p))
13. a : padic(p)
14. x : padic(p)
15. y : padic(p)
16. pa-mul(p;a;pa-add(p;x;y)) = pa-add(p;pa-mul(p;a;x);pa-mul(p;a;y))
17. X1 : \mBbbN{}
18. X2 : p-adics(p)
19. Y1 : \mBbbN{}
20. Y2 : p-adics(p)
21. Z1 : \mBbbN{}
22. Z2 : p-adics(p)
23. M : \mBbbZ{}
24. imax(X1;Y1) = M
\mvdash{} ((X1 \mleq{} M) \mwedge{} (Y1 \mleq{} M))
{}\mRightarrow{} (p\^{}((Z1 + M) + (M - X1))(p) * X2 * Z2 + p\^{}((Z1 + M) + (M - Y1))(p) * Y2 * Z2
= p\^{}((M + Z1) + ((Z1 + M) - X1 + Z1))(p) * X2 * Z2 + p\^{}((M + Z1)
+ ((Z1 + M) - Y1 + Z1))(p) * Y2 * Z2)
By
Latex:
(Intro
THEN D -1
THEN (Subst' (M + Z1) + ((Z1 + M) - X1 + Z1) \msim{} (Z1 + M) + (M - X1) 0 THENA Auto)
THEN (Subst' (M + Z1) + ((Z1 + M) - Y1 + Z1) \msim{} (Z1 + M) + (M - Y1) 0 THENA Auto)
THEN GenConclTerms Auto [\mkleeneopen{}M - X1\mkleeneclose{};\mkleeneopen{}M - Y1\mkleeneclose{};\mkleeneopen{}Z1 + M\mkleeneclose{}]\mcdot{})
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