Theorem xpcval 17298

Description: Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)

Hypotheses

Ref	Expression
xpcval.t	⊢ 𝑇 = (𝐶 ×c 𝐷)
xpcval.x	⊢ 𝑋 = (Base‘𝐶)
xpcval.y	⊢ 𝑌 = (Base‘𝐷)
xpcval.h	⊢ 𝐻 = (Hom ‘𝐶)
xpcval.j	⊢ 𝐽 = (Hom ‘𝐷)
xpcval.o1	⊢ · = (comp‘𝐶)
xpcval.o2	⊢ ∙ = (comp‘𝐷)
xpcval.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
xpcval.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)
xpcval.b	⊢ (𝜑 → 𝐵 = (𝑋 × 𝑌))
xpcval.k	⊢ (𝜑 → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))))
xpcval.o	⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩)))

Assertion

Ref	Expression
xpcval	⊢ (𝜑 → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})

Distinct variable groups:   𝑓,𝑔,𝑢,𝑣,𝑥,𝑦,𝐵   𝜑,𝑓,𝑔,𝑢,𝑣,𝑥,𝑦   𝐶,𝑓,𝑔,𝑢,𝑣,𝑥,𝑦   𝐷,𝑓,𝑔,𝑢,𝑣,𝑥,𝑦   𝑓,𝐾,𝑔,𝑥,𝑦

Allowed substitution hints:   ∙ (𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑇(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   · (𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝐻(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝐽(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝐾(𝑣,𝑢)   𝑂(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑊(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑋(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)   𝑌(𝑥,𝑦,𝑣,𝑢,𝑓,𝑔)

Proof of Theorem xpcval

Dummy variables 𝑏 ℎ 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcval.t	. 2 ⊢ 𝑇 = (𝐶 ×c 𝐷)
2		df-xpc 17293	. . . 4 ⊢ ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩})
3	2	a1i 11	. . 3 ⊢ (𝜑 → ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩}))
4		fvex 6510	. . . . . 6 ⊢ (Base‘𝑟) ∈ V
5		fvex 6510	. . . . . 6 ⊢ (Base‘𝑠) ∈ V
6	4, 5	xpex 7292	. . . . 5 ⊢ ((Base‘𝑟) × (Base‘𝑠)) ∈ V
7	6	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) ∈ V)
8		simprl 759	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → 𝑟 = 𝐶)
9	8	fveq2d 6501	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → (Base‘𝑟) = (Base‘𝐶))
10		xpcval.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐶)
11	9, 10	syl6eqr 2827	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → (Base‘𝑟) = 𝑋)
12		simprr 761	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → 𝑠 = 𝐷)
13	12	fveq2d 6501	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → (Base‘𝑠) = (Base‘𝐷))
14		xpcval.y	. . . . . . 7 ⊢ 𝑌 = (Base‘𝐷)
15	13, 14	syl6eqr 2827	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → (Base‘𝑠) = 𝑌)
16	11, 15	xpeq12d 5435	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) = (𝑋 × 𝑌))
17		xpcval.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑋 × 𝑌))
18	17	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → 𝐵 = (𝑋 × 𝑌))
19	16, 18	eqtr4d 2812	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) = 𝐵)
20		vex 3413	. . . . . . 7 ⊢ 𝑏 ∈ V
21	20, 20	mpoex 7584	. . . . . 6 ⊢ (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) ∈ V
22	21	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) ∈ V)
23		simpr 477	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
24		simplrl 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑟 = 𝐶)
25	24	fveq2d 6501	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑟) = (Hom ‘𝐶))
26		xpcval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝐶)
27	25, 26	syl6eqr 2827	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑟) = 𝐻)
28	27	oveqd 6992	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → ((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) = ((1st ‘𝑢)𝐻(1st ‘𝑣)))
29		simplrr 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑠 = 𝐷)
30	29	fveq2d 6501	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = (Hom ‘𝐷))
31		xpcval.j	. . . . . . . . . 10 ⊢ 𝐽 = (Hom ‘𝐷)
32	30, 31	syl6eqr 2827	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = 𝐽)
33	32	oveqd 6992	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)) = ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))
34	28, 33	xpeq12d 5435	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣))) = (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))))
35	23, 23, 34	mpoeq123dv 7046	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))))
36		xpcval.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))))
37	36	ad2antrr 714	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))))
38	35, 37	eqtr4d 2812	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) = 𝐾)
39		simplr 757	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → 𝑏 = 𝐵)
40	39	opeq2d 4681	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
41		simpr 477	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ℎ = 𝐾)
42	41	opeq2d 4681	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ⟨(Hom ‘ndx), ℎ⟩ = ⟨(Hom ‘ndx), 𝐾⟩)
43	39, 39	xpeq12d 5435	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
44	41	oveqd 6992	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ((2nd ‘𝑥)ℎ𝑦) = ((2nd ‘𝑥)𝐾𝑦))
45	41	fveq1d 6499	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (ℎ‘𝑥) = (𝐾‘𝑥))
46	24	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → 𝑟 = 𝐶)
47	46	fveq2d 6501	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (comp‘𝑟) = (comp‘𝐶))
48		xpcval.o1	. . . . . . . . . . . . . 14 ⊢ · = (comp‘𝐶)
49	47, 48	syl6eqr 2827	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (comp‘𝑟) = · )
50	49	oveqd 6992	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦)) = (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦)))
51	50	oveqd 6992	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)) = ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)))
52	29	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → 𝑠 = 𝐷)
53	52	fveq2d 6501	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (comp‘𝑠) = (comp‘𝐷))
54		xpcval.o2	. . . . . . . . . . . . . 14 ⊢ ∙ = (comp‘𝐷)
55	53, 54	syl6eqr 2827	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (comp‘𝑠) = ∙ )
56	55	oveqd 6992	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦)) = (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦)))
57	56	oveqd 6992	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓)) = ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓)))
58	51, 57	opeq12d 4682	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩ = ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩)
59	44, 45, 58	mpoeq123dv 7046	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩) = (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩))
60	43, 39, 59	mpoeq123dv 7046	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩)))
61		xpcval.o	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩)))
62	61	ad3antrrr 718	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩)))
63	60, 62	eqtr4d 2812	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩)) = 𝑂)
64	63	opeq2d 4681	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩ = ⟨(comp‘ndx), 𝑂⟩)
65	40, 42, 64	tpeq123d 4555	. . . . 5 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
66	22, 38, 65	csbied2 3811	. . . 4 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → ⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
67	7, 19, 66	csbied2 3811	. . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
68		xpcval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
69	68	elexd 3430	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
70		xpcval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
71	70	elexd 3430	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
72		tpex 7286	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩} ∈ V
73	72	a1i 11	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩} ∈ V)
74	3, 67, 69, 71, 73	ovmpod 7117	. 2 ⊢ (𝜑 → (𝐶 ×c 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
75	1, 74	syl5eq 2821	1 ⊢ (𝜑 → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  Vcvv 3410  ⦋csb 3781  {ctp 4440  ⟨cop 4442   × cxp 5402  ‘cfv 6186  (class class class)co 6975   ∈ cmpo 6977  1^st c1st 7498  2^nd c2nd 7499  ndxcnx 16335  Basecbs 16338  Hom chom 16431  compcco 16432   ×_c cxpc 17289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-1st 7500  df-2nd 7501  df-xpc 17293

This theorem is referenced by:  xpcbas  17299  xpchomfval  17300  xpccofval  17303  catcxpccl  17328  xpcpropd  17329