Step
*
12
of Lemma
functor-curry_wf
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
⊢ a |→ b |→ cat-comp(C) (F <a, b>) (G <a, b>) (z <a, b>) (T <a, b>) (g <a, b>)
= a |→ b |→ T <a, b> o a |→ b |→ g <a, b>
∈ nat-trans(A;FUN(B;C);functor(ob(a) = functor(ob(b) = F <a, b>;
arrow(x@0,y,f) = F <a, x@0> <a, y> <cat-id(A) a, f>);
arrow(x@0,y,f) = b |→ F <x@0, b> <y, b> <f, cat-id(B) b>);functor(ob(a) = functor(ob(b) =\000C z <a, b>;
arrow(x,y,f) =
z <a, x> <a, y> <cat-id(\000CA) a, f>);
arrow(x,y,f) = b |→ z <x, b> <y, b>
<f, cat-id(B) b>))
BY
{ NatTransEq THEN Reduce 0 }
1
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
9. x@0 : cat-ob(A)
10. y : cat-ob(A)
11. f : cat-arrow(A) x@0 y
12. A1 : cat-ob(B)
13. B1 : cat-ob(B)
14. g1 : cat-arrow(B) A1 B1
⊢ (cat-comp(C) (F <x@0, A1>) (F <y, A1>) (F <y, B1>) (F <x@0, A1> <y, A1> <f, cat-id(B) A1>) (F <y, A1> <y, B1> <cat-id(\000CA) y, g1>))
= (cat-comp(C) (F <x@0, A1>) (F <x@0, B1>) (F <y, B1>) (F <x@0, A1> <x@0, B1> <cat-id(A) x@0, g1>) (F <x@0, B1> <y, B1> \000C<f, cat-id(B) B1>))
∈ (cat-arrow(C) (F <x@0, A1>) (F <y, B1>))
2
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
9. x@0 : cat-ob(A)
10. y : cat-ob(A)
11. z1 : cat-ob(A)
12. f : cat-arrow(A) x@0 y
13. g1 : cat-arrow(A) y z1
⊢ b |→ F <x@0, b> <z1, b> <cat-comp(A) x@0 y z1 f g1, cat-id(B) b>
= b |→ F <x@0, b> <y, b> <f, cat-id(B) b> o b |→ F <y, b> <z1, b> <g1, cat-id(B) b>
∈ nat-trans(B;C;functor(ob(b) = F <x@0, b>;
arrow(x1,y,f) = F <x@0, x1> <x@0, y> <cat-id(A) x@0, f>);functor(ob(b) = F <z1, b>;
arrow(x@0,y,f) = F <z1, x@0> <z1, y> <cat-id(A) z1, \000Cf>))
3
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
9. x@0 : cat-ob(A)
⊢ b |→ F <x@0, b> <x@0, b> <cat-id(A) x@0, cat-id(B) b>
= identity-trans(B;C;functor(ob(b) = F <x@0, b>;
arrow(x1,y,f) = F <x@0, x1> <x@0, y> <cat-id(A) x@0, f>))
∈ nat-trans(B;C;functor(ob(b) = F <x@0, b>;
arrow(x1,y,f) = F <x@0, x1> <x@0, y> <cat-id(A) x@0, f>);functor(ob(b) = F <x@0, b>;
arrow(x1,y,f) = F <x@0, x1> <x@0, y> <cat-id(A) x@0,\000C f>))
4
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
9. x : cat-ob(A)
10. y : cat-ob(A)
11. f : cat-arrow(A) x y
12. A1 : cat-ob(B)
13. B1 : cat-ob(B)
14. g1 : cat-arrow(B) A1 B1
⊢ (cat-comp(C) (z <x, A1>) (z <y, A1>) (z <y, B1>) (z <x, A1> <y, A1> <f, cat-id(B) A1>) (z <y, A1> <y, B1> <cat-id(A) y\000C, g1>))
= (cat-comp(C) (z <x, A1>) (z <x, B1>) (z <y, B1>) (z <x, A1> <x, B1> <cat-id(A) x, g1>) (z <x, B1> <y, B1> <f, cat-id(B\000C) B1>))
∈ (cat-arrow(C) (z <x, A1>) (z <y, B1>))
5
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
9. x : cat-ob(A)
10. y : cat-ob(A)
11. z1 : cat-ob(A)
12. f : cat-arrow(A) x y
13. g1 : cat-arrow(A) y z1
⊢ b |→ z <x, b> <z1, b> <cat-comp(A) x y z1 f g1, cat-id(B) b>
= b |→ z <x, b> <y, b> <f, cat-id(B) b> o b |→ z <y, b> <z1, b> <g1, cat-id(B) b>
∈ nat-trans(B;C;functor(ob(b) = z <x, b>;
arrow(x@0,y,f) = z <x, x@0> <x, y> <cat-id(A) x, f>);functor(ob(b) = z <z1, b>;
arrow(x,y,f) = z <z1, x> <z1, y> <cat-id(A) z1, f>))
6
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
9. x : cat-ob(A)
⊢ b |→ z <x, b> <x, b> <cat-id(A) x, cat-id(B) b>
= identity-trans(B;C;functor(ob(b) = z <x, b>;
arrow(x@0,y,f) = z <x, x@0> <x, y> <cat-id(A) x, f>))
∈ nat-trans(B;C;functor(ob(b) = z <x, b>;
arrow(x@0,y,f) = z <x, x@0> <x, y> <cat-id(A) x, f>);functor(ob(b) = z <x, b>;
arrow(x@0,y,f) = z <x, x@0> <x, y> <cat-id(A) x, f>))
7
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
9. a : cat-ob(A)
⊢ b |→ cat-comp(C) (F <a, b>) (G <a, b>) (z <a, b>) (T <a, b>) (g <a, b>)
= b |→ T <a, b> o b |→ g <a, b>
∈ nat-trans(B;C;functor(ob(b) = F <a, b>;
arrow(x@0,y,f) = F <a, x@0> <a, y> <cat-id(A) a, f>);functor(ob(b) = z <a, b>;
arrow(x,y,f) = z <a, x> <a, y> <cat-id(A) a, f>))
8
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A × B;C))
5. G : cat-ob(FUN(A × B;C))
6. z : cat-ob(FUN(A × B;C))
7. T : cat-arrow(FUN(A × B;C)) F G
8. g : cat-arrow(FUN(A × B;C)) G z
9. A@0 : cat-ob(A)
10. B@0 : cat-ob(A)
11. g1 : cat-arrow(A) A@0 B@0
⊢ b |→ cat-comp(C) (F <A@0, b>) (G <A@0, b>) (z <A@0, b>) (T <A@0, b>) (g <A@0, b>) o b |→ z <A@0, b> <B@0, b> <g1, cat-\000Cid(B) b>
= b |→ F <A@0, b> <B@0, b> <g1, cat-id(B) b> o b |→ cat-comp(C) (F <B@0, b>) (G <B@0, b>) (z <B@0, b>) (T <B@0, b>) (g <\000CB@0, b>)
∈ nat-trans(B;C;functor(ob(b) = F <A@0, b>;
arrow(x@0,y,f) = F <A@0, x@0> <A@0, y> <cat-id(A) A@0, f>);functor(ob(b) = z <B@0, b>;
arrow(x,y,f) = z <B@0, x> <B@0, y> <cat-id(A) B@0, \000Cf>))
Latex:
Latex:
1. A : SmallCategory
2. B : SmallCategory
3. C : SmallCategory
4. F : cat-ob(FUN(A \mtimes{} B;C))
5. G : cat-ob(FUN(A \mtimes{} B;C))
6. z : cat-ob(FUN(A \mtimes{} B;C))
7. T : cat-arrow(FUN(A \mtimes{} B;C)) F G
8. g : cat-arrow(FUN(A \mtimes{} B;C)) G z
\mvdash{} a |\mrightarrow{} b |\mrightarrow{} cat-comp(C) (F <a, b>) (G <a, b>) (z <a, b>) (T <a, b>) (g <a, b>)
= a |\mrightarrow{} b |\mrightarrow{} T <a, b> o a |\mrightarrow{} b |\mrightarrow{} g <a, b>
By
Latex:
NatTransEq THEN Reduce 0
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