Theorem funcringcsetclem9ALTV 48263

Description: Lemma 9 for funcringcsetcALTV 48264. (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 2736	. . . . . 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 48244	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
9	8	eleq2d 2821	. . . 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 48244	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌(Hom ‘𝑅)𝑍) = (𝑌 RingHom 𝑍))
12	11	eleq2d 2821	. . . 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 20466	. . . . . . . 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 7170	. . . . . 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 48259	. . . . . . . 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 2736	. . . . . . 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 48247	. . . . . 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 6888	. . . . 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 2736	. . . . . . 7 ⊢ (comp‘𝑆) = (comp‘𝑆)
36	1, 19, 2, 20, 3, 21	funcringcsetclem2ALTV 48256	. . . . . . . . 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 48256	. . . . . . . . 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 48256	. . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑋) = (Base‘𝑋)
46		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑌) = (Base‘𝑌)
47	45, 46	rhmf 20450	. . . . . . . . . 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 48255	. . . . . . . . . . . 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 48255	. . . . . . . . . . . 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 6706	. . . . . . . . . 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 48260	. . . . . . . . . 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 6695	. . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
64	46, 63	rhmf 20450	. . . . . . . . . 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 48255	. . . . . . . . . . . 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 6706	. . . . . . . . . 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 48260	. . . . . . . . . 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 6695	. . . . . . . 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 18101	. . . . . 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 5849	. . . . . 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 2771	. . . . 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 2781	. . . 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 1540   ∈ wcel 2109  ⟨cop 4612   ↦ cmpt 5206   I cid 5552   ↾ cres 5661   ∘ ccom 5663  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  WUnicwun 10719  Basecbs 17233  Hom chom 17287  compcco 17288  SetCatcsetc 18093   RingHom crh 20434  RingCatALTVcringcALTV 48229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-wun 10721  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-plusg 17289  df-hom 17300  df-cco 17301  df-0g 17460  df-setc 18094  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-grp 18924  df-ghm 19201  df-mgp 20106  df-ur 20147  df-ring 20200  df-rhm 20437  df-ringcALTV 48230

This theorem is referenced by:  funcringcsetcALTV  48264