Theorem pi1coghm 23665

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 23654	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝑃 ∈ Grp)
5	1, 2, 4	syl2anc 586	. 2 ⊢ (𝜑 → 𝑃 ∈ Grp)
6		pi1co.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
7		cntop2 21849	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
9		toptopon2 21526	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 220	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		pi1co.b	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)
12		cnf2 21857	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶∪ 𝐾)
13	1, 10, 6, 12	syl3anc 1367	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐾)
14	13, 2	ffvelrnd 6852	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ∪ 𝐾)
15	11, 14	eqeltrrd 2914	. . 3 ⊢ (𝜑 → 𝐵 ∈ ∪ 𝐾)
16		pi1co.q	. . . 4 ⊢ 𝑄 = (𝐾 π1 𝐵)
17	16	pi1grp 23654	. . 3 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐵 ∈ ∪ 𝐾) → 𝑄 ∈ Grp)
18	10, 15, 17	syl2anc 586	. 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 23663	. . 3 ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑄))
22	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑉 = (Base‘𝑃))
23	3, 1, 2, 22	pi1bas2 23645	. . . . . . 7 ⊢ (𝜑 → 𝑉 = (∪ 𝑉 / ( ≃ph‘𝐽)))
24	23	eleq2d 2898	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝑉 ↔ 𝑦 ∈ (∪ 𝑉 / ( ≃ph‘𝐽))))
25	24	biimpa 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (∪ 𝑉 / ( ≃ph‘𝐽)))
26		eqid 2821	. . . . . 6 ⊢ (∪ 𝑉 / ( ≃ph‘𝐽)) = (∪ 𝑉 / ( ≃ph‘𝐽))
27		fvoveq1 7179	. . . . . . . 8 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = (𝐺‘(𝑦(+g‘𝑃)𝑧)))
28		fveq2 6670	. . . . . . . . 9 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃ph‘𝐽)) = (𝐺‘𝑦))
29	28	oveq1d 7171	. . . . . . . 8 ⊢ ([𝑓]( ≃ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
30	27, 29	eqeq12d 2837	. . . . . . 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 3197	. . . . . 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 7164	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧))
33	32	fveq2d 6674	. . . . . . . . . 10 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽))) = (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)))
34		fveq2 6670	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃ph‘𝐽)) = (𝐺‘𝑧))
35	34	oveq2d 7172	. . . . . . . . . 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 2837	. . . . . . . . 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 23646	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑓 ∈ ∪ 𝑉 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐴)))
38	37	biimpa 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐴))
39	38	simp1d 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝑓 ∈ (II Cn 𝐽))
40	39	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝑓 ∈ (II Cn 𝐽))
41	1	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐽 ∈ (TopOn‘𝑋))
42	2	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐴 ∈ 𝑋)
43	19	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝑉 = (Base‘𝑃))
44	3, 41, 42, 43	pi1eluni 23646	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (ℎ ∈ ∪ 𝑉 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = 𝐴 ∧ (ℎ‘1) = 𝐴)))
45	44	biimpa 479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = 𝐴 ∧ (ℎ‘1) = 𝐴))
46	45	simp1d 1138	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ℎ ∈ (II Cn 𝐽))
47	38	simp3d 1140	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝑓‘1) = 𝐴)
48	47	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓‘1) = 𝐴)
49	45	simp2d 1139	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (ℎ‘0) = 𝐴)
50	48, 49	eqtr4d 2859	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓‘1) = (ℎ‘0))
51	6	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐹 ∈ (𝐽 Cn 𝐾))
52	40, 46, 50, 51	copco 23622	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹 ∘ (𝑓(*𝑝‘𝐽)ℎ)) = ((𝐹 ∘ 𝑓)(*𝑝‘𝐾)(𝐹 ∘ ℎ)))
53	52	eceq1d 8328	. . . . . . . . . . 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 23621	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓(*𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽))
55	40, 46	pco0 23618	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = (𝑓‘0))
56	38	simp2d 1139	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝑓‘0) = 𝐴)
57	56	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓‘0) = 𝐴)
58	55, 57	eqtrd 2856	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ)‘0) = 𝐴)
59	40, 46	pco1 23619	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = (ℎ‘1))
60	45	simp3d 1140	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (ℎ‘1) = 𝐴)
61	59, 60	eqtrd 2856	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝑓(*𝑝‘𝐽)ℎ)‘1) = 𝐴)
62	1	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐽 ∈ (TopOn‘𝑋))
63	2	ad2antrr 724	. . . . . . . . . . . . . 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 23646	. . . . . . . . . . . . 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 1338	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝑉)
67	3, 16, 19, 20, 1, 6, 2, 11	pi1coval 23664	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓(*𝑝‘𝐽)ℎ) ∈ ∪ 𝑉) → (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) = [(𝐹 ∘ (𝑓(*𝑝‘𝐽)ℎ))]( ≃ph‘𝐾))
68	67	adantlr 713	. . . . . . . . . . . 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 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽)) = [(𝐹 ∘ (𝑓(*𝑝‘𝐽)ℎ))]( ≃ph‘𝐾))
70		eqid 2821	. . . . . . . . . . . 12 ⊢ (Base‘𝑄) = (Base‘𝑄)
71	10	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐾 ∈ (TopOn‘∪ 𝐾))
72	15	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝐵 ∈ ∪ 𝐾)
73		eqid 2821	. . . . . . . . . . . 12 ⊢ (+g‘𝑄) = (+g‘𝑄)
74	6	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐹 ∈ (𝐽 Cn 𝐾))
75		cnco 21874	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ 𝑓) ∈ (II Cn 𝐾))
76	39, 74, 75	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹 ∘ 𝑓) ∈ (II Cn 𝐾))
77		iitopon 23487	. . . . . . . . . . . . . . . . 17 ⊢ II ∈ (TopOn‘(0[,]1))
78		cnf2 21857	. . . . . . . . . . . . . . . . 17 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (II Cn 𝐽)) → 𝑓:(0[,]1)⟶𝑋)
79	77, 41, 39, 78	mp3an2i 1462	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝑓:(0[,]1)⟶𝑋)
80		0elunit 12856	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ (0[,]1)
81		fvco3 6760	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝑓)‘0) = (𝐹‘(𝑓‘0)))
82	79, 80, 81	sylancl 588	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓)‘0) = (𝐹‘(𝑓‘0)))
83	56	fveq2d 6674	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹‘(𝑓‘0)) = (𝐹‘𝐴))
84	11	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹‘𝐴) = 𝐵)
85	82, 83, 84	3eqtrd 2860	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓)‘0) = 𝐵)
86		1elunit 12857	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ (0[,]1)
87		fvco3 6760	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝑓)‘1) = (𝐹‘(𝑓‘1)))
88	79, 86, 87	sylancl 588	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓)‘1) = (𝐹‘(𝑓‘1)))
89	47	fveq2d 6674	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹‘(𝑓‘1)) = (𝐹‘𝐴))
90	88, 89, 84	3eqtrd 2860	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ((𝐹 ∘ 𝑓)‘1) = 𝐵)
91	10	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐾 ∈ (TopOn‘∪ 𝐾))
92	15	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝐵 ∈ ∪ 𝐾)
93		eqidd 2822	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (Base‘𝑄) = (Base‘𝑄))
94	16, 91, 92, 93	pi1eluni 23646	. . . . . . . . . . . . . 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 1338	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐹 ∘ 𝑓) ∈ ∪ (Base‘𝑄))
96	95	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹 ∘ 𝑓) ∈ ∪ (Base‘𝑄))
97		cnco 21874	. . . . . . . . . . . . . 14 ⊢ ((ℎ ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹 ∘ ℎ) ∈ (II Cn 𝐾))
98	46, 51, 97	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹 ∘ ℎ) ∈ (II Cn 𝐾))
99		cnf2 21857	. . . . . . . . . . . . . . . 16 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ ℎ ∈ (II Cn 𝐽)) → ℎ:(0[,]1)⟶𝑋)
100	77, 62, 46, 99	mp3an2i 1462	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ℎ:(0[,]1)⟶𝑋)
101		fvco3 6760	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ ℎ)‘0) = (𝐹‘(ℎ‘0)))
102	100, 80, 101	sylancl 588	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ)‘0) = (𝐹‘(ℎ‘0)))
103	49	fveq2d 6674	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹‘(ℎ‘0)) = (𝐹‘𝐴))
104	11	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹‘𝐴) = 𝐵)
105	102, 103, 104	3eqtrd 2860	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ)‘0) = 𝐵)
106		fvco3 6760	. . . . . . . . . . . . . . 15 ⊢ ((ℎ:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ ℎ)‘1) = (𝐹‘(ℎ‘1)))
107	100, 86, 106	sylancl 588	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ)‘1) = (𝐹‘(ℎ‘1)))
108	60	fveq2d 6674	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹‘(ℎ‘1)) = (𝐹‘𝐴))
109	107, 108, 104	3eqtrd 2860	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ((𝐹 ∘ ℎ)‘1) = 𝐵)
110		eqidd 2822	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Base‘𝑄) = (Base‘𝑄))
111	16, 10, 15, 110	pi1eluni 23646	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐹 ∘ ℎ) ∈ ∪ (Base‘𝑄) ↔ ((𝐹 ∘ ℎ) ∈ (II Cn 𝐾) ∧ ((𝐹 ∘ ℎ)‘0) = 𝐵 ∧ ((𝐹 ∘ ℎ)‘1) = 𝐵)))
112	111	ad2antrr 724	. . . . . . . . . . . . 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 1338	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐹 ∘ ℎ) ∈ ∪ (Base‘𝑄))
114	16, 70, 71, 72, 73, 96, 113	pi1addval 23652	. . . . . . . . . . 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 2866	. . . . . . . . . 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 2821	. . . . . . . . . . . 12 ⊢ (+g‘𝑃) = (+g‘𝑃)
117		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → 𝑓 ∈ ∪ 𝑉)
118		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ℎ ∈ ∪ 𝑉)
119	3, 19, 62, 63, 116, 117, 118	pi1addval 23652	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → ([𝑓]( ≃ph‘𝐽)(+g‘𝑃)[ℎ]( ≃ph‘𝐽)) = [(𝑓(*𝑝‘𝐽)ℎ)]( ≃ph‘𝐽))
120	119	fveq2d 6674	. . . . . . . . . 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 23664	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐹 ∘ 𝑓)]( ≃ph‘𝐾))
122	121	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[𝑓]( ≃ph‘𝐽)) = [(𝐹 ∘ 𝑓)]( ≃ph‘𝐾))
123	3, 16, 19, 20, 1, 6, 2, 11	pi1coval 23664	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[ℎ]( ≃ph‘𝐽)) = [(𝐹 ∘ ℎ)]( ≃ph‘𝐾))
124	123	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) ∧ ℎ ∈ ∪ 𝑉) → (𝐺‘[ℎ]( ≃ph‘𝐽)) = [(𝐹 ∘ ℎ)]( ≃ph‘𝐾))
125	122, 124	oveq12d 7174	. . . . . . . . . 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 2866	. . . . . . . . 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 8364	. . . . . . . 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 3182	. . . . . . 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 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → 𝑉 = (∪ 𝑉 / ( ≃ph‘𝐽)))
130	129	raleqdv 3415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → (∀𝑧 ∈ 𝑉 (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)) ↔ ∀𝑧 ∈ (∪ 𝑉 / ( ≃ph‘𝐽))(𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧))))
131	128, 130	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝑉) → ∀𝑧 ∈ 𝑉 (𝐺‘([𝑓]( ≃ph‘𝐽)(+g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph‘𝐽))(+g‘𝑄)(𝐺‘𝑧)))
132	26, 31, 131	ectocld 8364	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝑉 / ( ≃ph‘𝐽))) → ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
133	25, 132	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
134	133	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧)))
135	21, 134	jca 514	. 2 ⊢ (𝜑 → (𝐺:𝑉⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝑉 ∀𝑧 ∈ 𝑉 (𝐺‘(𝑦(+g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+g‘𝑄)(𝐺‘𝑧))))
136	19, 70, 116, 73	isghm 18358	. 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)))))
137	5, 18, 135, 136	syl21anbrc 1340	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ⟨cop 4573  ∪ cuni 4838   ↦ cmpt 5146  ran crn 5556   ∘ ccom 5559  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  [cec 8287   / cqs 8288  0cc0 10537  1c1 10538  [,]cicc 12742  Basecbs 16483  +_gcplusg 16565  Grpcgrp 18103   GrpHom cghm 18355  Topctop 21501  TopOnctopon 21518   Cn ccn 21832  IIcii 23483   ≃_phcphtpc 23573  *_𝑝cpco 23604   π₁ cpi1 23607

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616  ax-mulf 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-ec 8291  df-qs 8295  df-map 8408  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-fi 8875  df-sup 8906  df-inf 8907  df-oi 8974  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-q 12350  df-rp 12391  df-xneg 12508  df-xadd 12509  df-xmul 12510  df-ioo 12743  df-icc 12746  df-fz 12894  df-fzo 13035  df-seq 13371  df-exp 13431  df-hash 13692  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-hom 16589  df-cco 16590  df-rest 16696  df-topn 16697  df-0g 16715  df-gsum 16716  df-topgen 16717  df-pt 16718  df-prds 16721  df-xrs 16775  df-qtop 16780  df-imas 16781  df-qus 16782  df-xps 16783  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-grp 18106  df-mulg 18225  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-cnfld 20546  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cld 21627  df-cn 21835  df-cnp 21836  df-tx 22170  df-hmeo 22363  df-xms 22930  df-ms 22931  df-tms 22932  df-ii 23485  df-htpy 23574  df-phtpy 23575  df-phtpc 23596  df-pco 23609  df-om1 23610  df-pi1 23612

This theorem is referenced by: (None)