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Theorem xpcval 18168

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 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))))
xpcval.o	⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘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 18163	. . . 4 ⊢ ×_c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
3	2	a1i 11	. . 3 ⊢ (𝜑 → ×_c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}))
4		fvex 6910	. . . . . 6 ⊢ (Base‘𝑟) ∈ V
5		fvex 6910	. . . . . 6 ⊢ (Base‘𝑠) ∈ V
6	4, 5	xpex 7755	. . . . 5 ⊢ ((Base‘𝑟) × (Base‘𝑠)) ∈ V
7	6	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) ∈ V)
8		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → 𝑟 = 𝐶)
9	8	fveq2d 6901	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → (Base‘𝑟) = (Base‘𝐶))
10		xpcval.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐶)
11	9, 10	eqtr4di 2786	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → (Base‘𝑟) = 𝑋)
12		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → 𝑠 = 𝐷)
13	12	fveq2d 6901	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → (Base‘𝑠) = (Base‘𝐷))
14		xpcval.y	. . . . . . 7 ⊢ 𝑌 = (Base‘𝐷)
15	13, 14	eqtr4di 2786	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → (Base‘𝑠) = 𝑌)
16	11, 15	xpeq12d 5709	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) = (𝑋 × 𝑌))
17		xpcval.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑋 × 𝑌))
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → 𝐵 = (𝑋 × 𝑌))
19	16, 18	eqtr4d 2771	. . . 4 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → ((Base‘𝑟) × (Base‘𝑠)) = 𝐵)
20		vex 3475	. . . . . . 7 ⊢ 𝑏 ∈ V
21	20, 20	mpoex 8084	. . . . . 6 ⊢ (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) ∈ V
22	21	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) ∈ V)
23		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
24		simplrl 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑟 = 𝐶)
25	24	fveq2d 6901	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑟) = (Hom ‘𝐶))
26		xpcval.h	. . . . . . . . . 10 ⊢ 𝐻 = (Hom ‘𝐶)
27	25, 26	eqtr4di 2786	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑟) = 𝐻)
28	27	oveqd 7437	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → ((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) = ((1^st ‘𝑢)𝐻(1^st ‘𝑣)))
29		simplrr 777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝑠 = 𝐷)
30	29	fveq2d 6901	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = (Hom ‘𝐷))
31		xpcval.j	. . . . . . . . . 10 ⊢ 𝐽 = (Hom ‘𝐷)
32	30, 31	eqtr4di 2786	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (Hom ‘𝑠) = 𝐽)
33	32	oveqd 7437	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)) = ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))
34	28, 33	xpeq12d 5709	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣))) = (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))
35	23, 23, 34	mpoeq123dv 7495	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))))
36		xpcval.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))))
37	36	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))))
38	35, 37	eqtr4d 2771	. . . . 5 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → (𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) = 𝐾)
39		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → 𝑏 = 𝐵)
40	39	opeq2d 4881	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
41		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ℎ = 𝐾)
42	41	opeq2d 4881	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ⟨(Hom ‘ndx), ℎ⟩ = ⟨(Hom ‘ndx), 𝐾⟩)
43	39, 39	xpeq12d 5709	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
44	41	oveqd 7437	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ((2^nd ‘𝑥)ℎ𝑦) = ((2^nd ‘𝑥)𝐾𝑦))
45	41	fveq1d 6899	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (ℎ‘𝑥) = (𝐾‘𝑥))
46	24	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → 𝑟 = 𝐶)
47	46	fveq2d 6901	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (comp‘𝑟) = (comp‘𝐶))
48		xpcval.o1	. . . . . . . . . . . . . 14 ⊢ · = (comp‘𝐶)
49	47, 48	eqtr4di 2786	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (comp‘𝑟) = · )
50	49	oveqd 7437	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦)) = (⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦)))
51	50	oveqd 7437	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)) = ((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)))
52	29	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → 𝑠 = 𝐷)
53	52	fveq2d 6901	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (comp‘𝑠) = (comp‘𝐷))
54		xpcval.o2	. . . . . . . . . . . . . 14 ⊢ ∙ = (comp‘𝐷)
55	53, 54	eqtr4di 2786	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (comp‘𝑠) = ∙ )
56	55	oveqd 7437	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦)) = (⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦)))
57	56	oveqd 7437	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓)) = ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓)))
58	51, 57	opeq12d 4882	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩ = ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)
59	44, 45, 58	mpoeq123dv 7495	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩) = (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩))
60	43, 39, 59	mpoeq123dv 7495	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
61		xpcval.o	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
62	61	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩ · (1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩ ∙ (2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
63	60, 62	eqtr4d 2771	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = 𝑂)
64	63	opeq2d 4881	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩ = ⟨(comp‘ndx), 𝑂⟩)
65	40, 42, 64	tpeq123d 4753	. . . . 5 ⊢ ((((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐾) → {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
66	22, 38, 65	csbied2 3932	. . . 4 ⊢ (((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) ∧ 𝑏 = 𝐵) → ⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
67	7, 19, 66	csbied2 3932	. . 3 ⊢ ((𝜑 ∧ (𝑟 = 𝐶 ∧ 𝑠 = 𝐷)) → ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1^st ‘𝑢)(Hom ‘𝑟)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝑠)(2^nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝑟)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝑠)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
68		xpcval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
69	68	elexd 3492	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
70		xpcval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
71	70	elexd 3492	. . 3 ⊢ (𝜑 → 𝐷 ∈ V)
72		tpex 7749	. . . 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 7573	. 2 ⊢ (𝜑 → (𝐶 ×_c 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})
75	1, 74	eqtrid 2780	1 ⊢ (𝜑 → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐾⟩, ⟨(comp‘ndx), 𝑂⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⦋csb 3892 {ctp 4633 ⟨cop 4635 × cxp 5676 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7991 2^nd c2nd 7992 ndxcnx 17162 Basecbs 17180 Hom chom 17244 compcco 17245 ×_c cxpc 18159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-xpc 18163

This theorem is referenced by: xpcbas 18169 xpchomfval 18170 xpccofval 18173 catcxpccl 18198 catcxpcclOLD 18199 xpcpropd 18200

W3C validator