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