Theorem funcringcsetclem9ALTV 48313

Description: Lemma 9 for funcringcsetcALTV 48314. (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 2730	. . . . . 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 48294	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
9	8	eleq2d 2815	. . . 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 48294	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌(Hom ‘𝑅)𝑍) = (𝑌 RingHom 𝑍))
12	11	eleq2d 2815	. . . 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 20417	. . . . . . . 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 7150	. . . . . 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 48309	. . . . . . . 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 2730	. . . . . . 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 48297	. . . . . 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 6868	. . . . 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 2730	. . . . . . 7 ⊢ (comp‘𝑆) = (comp‘𝑆)
36	1, 19, 2, 20, 3, 21	funcringcsetclem2ALTV 48306	. . . . . . . . 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 48306	. . . . . . . . 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 48306	. . . . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑋) = (Base‘𝑋)
46		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑌) = (Base‘𝑌)
47	45, 46	rhmf 20401	. . . . . . . . . 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 48305	. . . . . . . . . . . 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 48305	. . . . . . . . . . . 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 6686	. . . . . . . . . 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 48310	. . . . . . . . . 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 6673	. . . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑍) = (Base‘𝑍)
64	46, 63	rhmf 20401	. . . . . . . . . 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 48305	. . . . . . . . . . . 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 6686	. . . . . . . . . 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 48310	. . . . . . . . . 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 6673	. . . . . . . 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 18052	. . . . . 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 5831	. . . . . 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 2765	. . . . 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 2775	. . . 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 4598   ↦ cmpt 5191   I cid 5535   ↾ cres 5643   ∘ ccom 5645  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  WUnicwun 10660  Basecbs 17186  Hom chom 17238  compcco 17239  SetCatcsetc 18044   RingHom crh 20385  RingCatALTVcringcALTV 48279

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-wun 10662  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-plusg 17240  df-hom 17251  df-cco 17252  df-0g 17411  df-setc 18045  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-grp 18875  df-ghm 19152  df-mgp 20057  df-ur 20098  df-ring 20151  df-rhm 20388  df-ringcALTV 48280

This theorem is referenced by:  funcringcsetcALTV  48314