Theorem funcringcsetclem9ALTV 48446

Description: Lemma 9 for funcringcsetcALTV 48447. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
funcringcsetcALTV.r	⊢ 𝑅 = (RingCatALTV‘𝑈)
funcringcsetcALTV.s	⊢ 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV.b	⊢ 𝐵 = (Base‘𝑅)
funcringcsetcALTV.c	⊢ 𝐶 = (Base‘𝑆)
funcringcsetcALTV.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcringcsetcALTV.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))

Assertion

Ref	Expression
funcringcsetclem9ALTV	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦   𝑥,𝑍,𝑦

Allowed substitution hints:   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem funcringcsetclem9ALTV

Step	Hyp	Ref	Expression
1		funcringcsetcALTV.r	. . . . . 6 ⊢ 𝑅 = (RingCatALTV‘𝑈)
2		funcringcsetcALTV.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
3		funcringcsetcALTV.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ WUni)
4	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑈 ∈ WUni)
5		eqid 2733	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
6		simpr1 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
7		simpr2 1196	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
8	1, 2, 4, 5, 6, 7	ringchomALTV 48427	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
9	8	eleq2d 2819	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ↔ 𝐻 ∈ (𝑋 RingHom 𝑌)))
10		simpr3 1197	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
11	1, 2, 4, 5, 7, 10	ringchomALTV 48427	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌(Hom ‘𝑅)𝑍) = (𝑌 RingHom 𝑍))
12	11	eleq2d 2819	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍) ↔ 𝐾 ∈ (𝑌 RingHom 𝑍)))
13	9, 12	anbi12d 632	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍)) ↔ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))))
14		rhmco 20418	. . . . . . . 8 ⊢ ((𝐾 ∈ (𝑌 RingHom 𝑍) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → (𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍))
15	14	ancoms 458	. . . . . . 7 ⊢ ((𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍)) → (𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍))
16	15	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍))
17		fvresi 7113	. . . . . 6 ⊢ ((𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍) → (( I ↾ (𝑋 RingHom 𝑍))‘(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
18	16, 17	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (( I ↾ (𝑋 RingHom 𝑍))‘(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
19		funcringcsetcALTV.s	. . . . . . . . 9 ⊢ 𝑆 = (SetCat‘𝑈)
20		funcringcsetcALTV.c	. . . . . . . . 9 ⊢ 𝐶 = (Base‘𝑆)
21		funcringcsetcALTV.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
22		funcringcsetcALTV.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
23	1, 19, 2, 20, 3, 21, 22	funcringcsetclem5ALTV 48442	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
24	23	3adantr2 1171	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
25	24	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
26	4	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑈 ∈ WUni)
27		eqid 2733	. . . . . . 7 ⊢ (comp‘𝑅) = (comp‘𝑅)
28	6	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑋 ∈ 𝐵)
29	7	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑌 ∈ 𝐵)
30	10	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑍 ∈ 𝐵)
31		simprl 770	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻 ∈ (𝑋 RingHom 𝑌))
32		simprr 772	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾 ∈ (𝑌 RingHom 𝑍))
33	1, 2, 26, 27, 28, 29, 30, 31, 32	ringccoALTV 48430	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻) = (𝐾 ∘ 𝐻))
34	25, 33	fveq12d 6835	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (( I ↾ (𝑋 RingHom 𝑍))‘(𝐾 ∘ 𝐻)))
35		eqid 2733	. . . . . . 7 ⊢ (comp‘𝑆) = (comp‘𝑆)
36	1, 19, 2, 20, 3, 21	funcringcsetclem2ALTV 48439	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
37	36	3ad2antr1 1189	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
38	37	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹‘𝑋) ∈ 𝑈)
39	1, 19, 2, 20, 3, 21	funcringcsetclem2ALTV 48439	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
40	39	3ad2antr2 1190	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
41	40	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹‘𝑌) ∈ 𝑈)
42	1, 19, 2, 20, 3, 21	funcringcsetclem2ALTV 48439	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) ∈ 𝑈)
43	42	3ad2antr3 1191	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) ∈ 𝑈)
44	43	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹‘𝑍) ∈ 𝑈)
45		eqid 2733	. . . . . . . . . . 11 ⊢ (Base‘𝑋) = (Base‘𝑋)
46		eqid 2733	. . . . . . . . . . 11 ⊢ (Base‘𝑌) = (Base‘𝑌)
47	45, 46	rhmf 20404	. . . . . . . . . 10 ⊢ (𝐻 ∈ (𝑋 RingHom 𝑌) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
48	47	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
49	1, 19, 2, 20, 3, 21	funcringcsetclem1ALTV 48438	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
50	49	3ad2antr1 1189	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
51	1, 19, 2, 20, 3, 21	funcringcsetclem1ALTV 48438	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
52	51	3ad2antr2 1190	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
53	50, 52	feq23d 6651	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
54	53	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
55	48, 54	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌))
56		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝜑)
57		3simpa 1148	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
58	57	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
59	1, 19, 2, 20, 3, 21, 22	funcringcsetclem6ALTV 48443	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
60	56, 58, 31, 59	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
61	60	feq1d 6638	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹‘𝑋)⟶(𝐹‘𝑌) ↔ 𝐻:(𝐹‘𝑋)⟶(𝐹‘𝑌)))
62	55, 61	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑌)‘𝐻):(𝐹‘𝑋)⟶(𝐹‘𝑌))
63		eqid 2733	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
64	46, 63	rhmf 20404	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝑌 RingHom 𝑍) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
65	64	ad2antll 729	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
66	1, 19, 2, 20, 3, 21	funcringcsetclem1ALTV 48438	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝐵) → (𝐹‘𝑍) = (Base‘𝑍))
67	66	3ad2antr3 1191	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐹‘𝑍) = (Base‘𝑍))
68	52, 67	feq23d 6651	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
69	68	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
70	65, 69	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍))
71		3simpc 1150	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵))
72	71	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵))
73	1, 19, 2, 20, 3, 21, 22	funcringcsetclem6ALTV 48443	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍)) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
74	56, 72, 32, 73	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
75	74	feq1d 6638	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹‘𝑌)⟶(𝐹‘𝑍) ↔ 𝐾:(𝐹‘𝑌)⟶(𝐹‘𝑍)))
76	70, 75	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑌𝐺𝑍)‘𝐾):(𝐹‘𝑌)⟶(𝐹‘𝑍))
77	19, 26, 35, 38, 41, 44, 62, 76	setcco 17992	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)) = (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)))
78	74, 60	coeq12d 5808	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾 ∘ 𝐻))
79	77, 78	eqtrd 2768	. . . . 5 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾 ∘ 𝐻))
80	18, 34, 79	3eqtr4d 2778	. . . 4 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
81	80	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))))
82	13, 81	sylbid 240	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))))
83	82	3impia 1117	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ⟨cop 4581   ↦ cmpt 5174   I cid 5513   ↾ cres 5621   ∘ ccom 5623  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  WUnicwun 10598  Basecbs 17122  Hom chom 17174  compcco 17175  SetCatcsetc 17984   RingHom crh 20389  RingCatALTVcringcALTV 48412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-wun 10600  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-plusg 17176  df-hom 17187  df-cco 17188  df-0g 17347  df-setc 17985  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-grp 18851  df-ghm 19127  df-mgp 20061  df-ur 20102  df-ring 20155  df-rhm 20392  df-ringcALTV 48413

This theorem is referenced by:  funcringcsetcALTV  48447