Theorem pi1coghm 25028

Description: The mapping 𝐺 between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
pi1co.p	⊢ 𝑃 = (𝐽 π1 𝐴)
pi1co.q	⊢ 𝑄 = (𝐾 π1 𝐵)
pi1co.v	⊢ 𝑉 = (Base‘𝑃)
pi1co.g	⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝑉 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)
pi1co.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
pi1co.f	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
pi1co.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
pi1co.b	⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)

Assertion

Ref	Expression
pi1coghm	⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Distinct variable groups:   𝐴,𝑔   𝑔,𝐹   𝑔,𝐽   𝜑,𝑔   𝑔,𝐾   𝑃,𝑔   𝑄,𝑔   𝑔,𝑉

Allowed substitution hints:   𝐵(𝑔)   𝐺(𝑔)   𝑋(𝑔)

Proof of Theorem pi1coghm

Dummy variables ℎ 𝑓 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pi1co.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		pi1co.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
3		pi1co.p	. . . 4 ⊢ 𝑃 = (𝐽 π1 𝐴)
4	3	pi1grp 25017	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝑃 ∈ Grp)
5	1, 2, 4	syl2anc 585	. 2 ⊢ (𝜑 → 𝑃 ∈ Grp)
6		pi1co.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
7		cntop2 23206	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
9		toptopon2 22883	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 218	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		pi1co.b	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)
12		cnf2 23214	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶∪ 𝐾)
13	1, 10, 6, 12	syl3anc 1374	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
14	13, 2	ffvelcdmd 7037	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ∪ 𝐾)
15	11, 14	eqeltrrd 2837	. . 3 ⊢ (𝜑 → 𝐵 ∈ ∪ 𝐾)
16		pi1co.q	. . . 4 ⊢ 𝑄 = (𝐾 π1 𝐵)
17	16	pi1grp 25017	. . 3 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐵 ∈ ∪ 𝐾) → 𝑄 ∈ Grp)
18	10, 15, 17	syl2anc 585	. 2 ⊢ (𝜑 → 𝑄 ∈ Grp)
19		pi1co.v	. . . 4 ⊢ 𝑉 = (Base‘𝑃)
20		pi1co.g	. . . 4 ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝑉 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)
21	3, 16, 19, 20, 1, 6, 2, 11	pi1cof 25026	. . 3 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑄))
22	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑃))
23	3, 1, 2, 22	pi1bas2 25008	. . . . . . 7 ⊢ (𝜑 → 𝑉 = (∪ 𝑉 / ( ≃ph‘𝐽)))
24	23	eleq2d 2822	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (∪ 𝑉 / ( ≃ph‘𝐽))))
25	24	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (∪ 𝑉 / ( ≃ph‘𝐽)))
26		eqid 2736	. . . . . 6 ⊢ (∪ 𝑉 / ( ≃ph‘𝐽)) = (∪ 𝑉 / ( ≃ph‘𝐽))
27		fvoveq1 7390	. . . . . . . 8 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = (𝐺‘(𝑦(+g‘𝑃)𝑧)))
28		fveq2 6840	. . . . . . . . 9 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃ph‘𝐽)) = (𝐺‘𝑦))
29	28	oveq1d 7382	. . . . . . . 8 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
30	27, 29	eqeq12d 2752	. . . . . . 7 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → ((𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) ↔ (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))))
31	30	ralbidv 3160	. . . . . 6 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (∀𝑧 ∈ 𝑉 (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) ↔ ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))))
32		oveq2 7375	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧))
33	32	fveq2d 6844	. . . . . . . . . 10 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)))
34		fveq2 6840	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃ph‘𝐽)) = (𝐺‘𝑧))
35	34	oveq2d 7383	. . . . . . . . . 10 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
36	33, 35	eqeq12d 2752	. . . . . . . . 9 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → ((𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))) ↔ (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))))
37	3, 1, 2, 22	pi1eluni 25009	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑓 ∈ ∪ 𝑉 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐴)))
38	37	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐴))
39	38	simp1d 1143	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝑓 ∈ (II Cn 𝐽))
40	39	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝑓 ∈ (II Cn 𝐽))
41	1	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐽 ∈ (TopOn‘𝑋))
42	2	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐴 ∈ 𝑋)
43	19	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝑉 = (Base‘𝑃))
44	3, 41, 42, 43	pi1eluni 25009	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (ℎ ∈ ∪ 𝑉 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = 𝐴 ∧ (ℎ‘1) = 𝐴)))
45	44	biimpa 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = 𝐴 ∧ (ℎ‘1) = 𝐴))
46	45	simp1d 1143	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ℎ ∈ (II Cn 𝐽))
47	38	simp3d 1145	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝑓‘1) = 𝐴)
48	47	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓‘1) = 𝐴)
49	45	simp2d 1144	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (ℎ‘0) = 𝐴)
50	48, 49	eqtr4d 2774	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓‘1) = (ℎ‘0))
51	6	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐹 ∈ (𝐽 Cn 𝐾))
52	40, 46, 50, 51	copco 24985	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹 ∘ (𝑓(*𝑝‘𝐽)ℎ)) = ((𝐹 ∘ 𝑓)(*𝑝‘𝐾)(𝐹 ∘ ℎ)))
53	52	eceq1d 8684	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → [(𝐹 ∘ (𝑓(*𝑝‘𝐽)ℎ))]( ≃ph‘𝐾) = [((𝐹 ∘ 𝑓)(*𝑝‘𝐾)(𝐹 ∘ ℎ))]( ≃ph‘𝐾))
54	40, 46, 50	pcocn 24984	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽))
55	40, 46	pco0 24981	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝑓‘0))
56	38	simp2d 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝑓‘0) = 𝐴)
57	56	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓‘0) = 𝐴)
58	55, 57	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = 𝐴)
59	40, 46	pco1 24982	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (ℎ‘1))
60	45	simp3d 1145	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (ℎ‘1) = 𝐴)
61	59, 60	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = 𝐴)
62	1	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐽 ∈ (TopOn‘𝑋))
63	2	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐴 ∈ 𝑋)
64	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝑉 = (Base‘𝑃))
65	3, 62, 63, 64	pi1eluni 25009	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝑉 ↔ ((𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽) ∧ ((𝑓(*𝑝‘𝐽)ℎ)‘0) = 𝐴 ∧ ((𝑓(*𝑝‘𝐽)ℎ)‘1) = 𝐴)))
66	54, 58, 61, 65	mpbir3and 1344	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝑉)
67	3, 16, 19, 20, 1, 6, 2, 11	pi1coval 25027	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝑉) → (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) = [(𝐹 ∘ (𝑓(*𝑝‘𝐽)ℎ))]( ≃ph‘𝐾))
68	67	adantlr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝑉) → (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) = [(𝐹 ∘ (𝑓(*𝑝‘𝐽)ℎ))]( ≃ph‘𝐾))
69	66, 68	syldan 592	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) = [(𝐹 ∘ (𝑓(*𝑝‘𝐽)ℎ))]( ≃ph‘𝐾))
70		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑄) = (Base‘𝑄)
71	10	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐾 ∈ (TopOn‘∪ 𝐾))
72	15	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐵 ∈ ∪ 𝐾)
73		eqid 2736	. . . . . . . . . . . 12 ⊢ (+g‘𝑄) = (+g‘𝑄)
74	6	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐹 ∈ (𝐽 Cn 𝐾))
75		cnco 23231	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ 𝑓) ∈ (II Cn 𝐾))
76	39, 74, 75	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹 ∘ 𝑓) ∈ (II Cn 𝐾))
77		iitopon 24846	. . . . . . . . . . . . . . . . 17 ⊢ II ∈ (TopOn‘(0[,]1))
78		cnf2 23214	. . . . . . . . . . . . . . . . 17 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (II Cn 𝐽)) → 𝑓:(0[,]1)⟶𝑋)
79	77, 41, 39, 78	mp3an2i 1469	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝑓:(0[,]1)⟶𝑋)
80		0elunit 13422	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ (0[,]1)
81		fvco3 6939	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝑓)‘0) = (𝐹‘(𝑓‘0)))
82	79, 80, 81	sylancl 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓)‘0) = (𝐹‘(𝑓‘0)))
83	56	fveq2d 6844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹‘(𝑓‘0)) = (𝐹‘𝐴))
84	11	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹‘𝐴) = 𝐵)
85	82, 83, 84	3eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓)‘0) = 𝐵)
86		1elunit 13423	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0[,]1)
87		fvco3 6939	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑓)‘1) = (𝐹‘(𝑓‘1)))
88	79, 86, 87	sylancl 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓)‘1) = (𝐹‘(𝑓‘1)))
89	47	fveq2d 6844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹‘(𝑓‘1)) = (𝐹‘𝐴))
90	88, 89, 84	3eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓)‘1) = 𝐵)
91	10	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐾 ∈ (TopOn‘∪ 𝐾))
92	15	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐵 ∈ ∪ 𝐾)
93		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (Base‘𝑄) = (Base‘𝑄))
94	16, 91, 92, 93	pi1eluni 25009	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓) ∈ ∪ (Base‘𝑄) ↔ ((𝐹 ∘ 𝑓) ∈ (II Cn 𝐾) ∧ ((𝐹 ∘ 𝑓)‘0) = 𝐵 ∧ ((𝐹 ∘ 𝑓)‘1) = 𝐵)))
95	76, 85, 90, 94	mpbir3and 1344	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹 ∘ 𝑓) ∈ ∪ (Base‘𝑄))
96	95	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹 ∘ 𝑓) ∈ ∪ (Base‘𝑄))
97		cnco 23231	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ ℎ) ∈ (II Cn 𝐾))
98	46, 51, 97	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹 ∘ ℎ) ∈ (II Cn 𝐾))
99		cnf2 23214	. . . . . . . . . . . . . . . 16 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ ℎ ∈ (II Cn 𝐽)) → ℎ:(0[,]1)⟶𝑋)
100	77, 62, 46, 99	mp3an2i 1469	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ℎ:(0[,]1)⟶𝑋)
101		fvco3 6939	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ ℎ)‘0) = (𝐹‘(ℎ‘0)))
102	100, 80, 101	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ)‘0) = (𝐹‘(ℎ‘0)))
103	49	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹‘(ℎ‘0)) = (𝐹‘𝐴))
104	11	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹‘𝐴) = 𝐵)
105	102, 103, 104	3eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ)‘0) = 𝐵)
106		fvco3 6939	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ ℎ)‘1) = (𝐹‘(ℎ‘1)))
107	100, 86, 106	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ)‘1) = (𝐹‘(ℎ‘1)))
108	60	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹‘(ℎ‘1)) = (𝐹‘𝐴))
109	107, 108, 104	3eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ)‘1) = 𝐵)
110		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑄) = (Base‘𝑄))
111	16, 10, 15, 110	pi1eluni 25009	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐹 ∘ ℎ) ∈ ∪ (Base‘𝑄) ↔ ((𝐹 ∘ ℎ) ∈ (II Cn 𝐾) ∧ ((𝐹 ∘ ℎ)‘0) = 𝐵 ∧ ((𝐹 ∘ ℎ)‘1) = 𝐵)))
112	111	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ) ∈ ∪ (Base‘𝑄) ↔ ((𝐹 ∘ ℎ) ∈ (II Cn 𝐾) ∧ ((𝐹 ∘ ℎ)‘0) = 𝐵 ∧ ((𝐹 ∘ ℎ)‘1) = 𝐵)))
113	98, 105, 109, 112	mpbir3and 1344	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹 ∘ ℎ) ∈ ∪ (Base‘𝑄))
114	16, 70, 71, 72, 73, 96, 113	pi1addval 25015	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ([(𝐹 ∘ 𝑓)]( ≃ph‘𝐾)(+g‘𝑄)[(𝐹 ∘ ℎ)]( ≃ph‘𝐾)) = [((𝐹 ∘ 𝑓)(*𝑝‘𝐾)(𝐹 ∘ ℎ))]( ≃ph‘𝐾))
115	53, 69, 114	3eqtr4d 2781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) = ([(𝐹 ∘ 𝑓)]( ≃ph‘𝐾)(+g‘𝑄)[(𝐹 ∘ ℎ)]( ≃ph‘𝐾)))
116		eqid 2736	. . . . . . . . . . . 12 ⊢ (+g‘𝑃) = (+g‘𝑃)
117		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝑓 ∈ ∪ 𝑉)
118		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ℎ ∈ ∪ 𝑉)
119	3, 19, 62, 63, 116, 117, 118	pi1addval 25015	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = [(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽))
120	119	fveq2d 6844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)))
121	3, 16, 19, 20, 1, 6, 2, 11	pi1coval 25027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐹 ∘ 𝑓)]( ≃ph‘𝐾))
122	121	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐹 ∘ 𝑓)]( ≃ph‘𝐾))
123	3, 16, 19, 20, 1, 6, 2, 11	pi1coval 25027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[ℎ]( ≃ph‘𝐽)) = [(𝐹 ∘ ℎ)]( ≃ph‘𝐾))
124	123	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[ℎ]( ≃ph‘𝐽)) = [(𝐹 ∘ ℎ)]( ≃ph‘𝐾))
125	122, 124	oveq12d 7385	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))) = ([(𝐹 ∘ 𝑓)]( ≃ph‘𝐾)(+g‘𝑄)[(𝐹 ∘ ℎ)]( ≃ph‘𝐾)))
126	115, 120, 125	3eqtr4d 2781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘[ℎ]( ≃ph‘𝐽))))
127	26, 36, 126	ectocld 8729	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ 𝑧 ∈ (∪ 𝑉 / ( ≃ph‘𝐽))) → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
128	127	ralrimiva 3129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ∀𝑧 ∈ (∪ 𝑉 / ( ≃ph‘𝐽))(𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
129	23	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝑉 = (∪ 𝑉 / ( ≃ph‘𝐽)))
130	128, 129	raleqtrrdv 3299	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ∀𝑧 ∈ 𝑉 (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
131	26, 31, 130	ectocld 8729	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝑉 / ( ≃ph‘𝐽))) → ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
132	25, 131	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
133	132	ralrimiva 3129	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
134	21, 133	jca 511	. 2 ⊢ (𝜑 → (𝐺:𝑉⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))))
135	19, 70, 116, 73	isghm 19190	. 2 ⊢ (𝐺 ∈ (𝑃 GrpHom 𝑄) ↔ ((𝑃 ∈ Grp ∧ 𝑄 ∈ Grp) ∧ (𝐺:𝑉⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))))
136	5, 18, 134, 135	syl21anbrc 1346	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ⟨cop 4573  ∪ cuni 4850   ↦ cmpt 5166  ran crn 5632   ∘ ccom 5635  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  [cec 8641   / cqs 8642  0cc0 11038  1c1 11039  [,]cicc 13301  Basecbs 17179  +_gcplusg 17220  Grpcgrp 18909   GrpHom cghm 19187  Topctop 22858  TopOnctopon 22875   Cn ccn 23189  IIcii 24842   ≃_phcphtpc 24936  *_𝑝cpco 24967   π₁ cpi1 24970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-ec 8645  df-qs 8649  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-fi 9324  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-ioo 13302  df-icc 13305  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-rest 17385  df-topn 17386  df-0g 17404  df-gsum 17405  df-topgen 17406  df-pt 17407  df-prds 17410  df-xrs 17466  df-qtop 17471  df-imas 17472  df-qus 17473  df-xps 17474  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-grp 18912  df-mulg 19044  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-cnfld 21353  df-top 22859  df-topon 22876  df-topsp 22898  df-bases 22911  df-cld 22984  df-cn 23192  df-cnp 23193  df-tx 23527  df-hmeo 23720  df-xms 24285  df-ms 24286  df-tms 24287  df-ii 24844  df-htpy 24937  df-phtpy 24938  df-phtpc 24959  df-pco 24972  df-om1 24973  df-pi1 24975

This theorem is referenced by: (None)