pi1xfr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pi1xfr		Structured version Visualization version GIF version

Theorem pi1xfr 25002

Description: Given a path 𝐹 and its inverse 𝐼 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.)

**Hypotheses**
Ref	Expression
pi1xfr.p	⊢ 𝑃 = (𝐽 π₁ (𝐹‘0))
pi1xfr.q	⊢ 𝑄 = (𝐽 π₁ (𝐹‘1))
pi1xfr.b	⊢ 𝐵 = (Base‘𝑃)
pi1xfr.g	⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃_ph‘𝐽), [(𝐼(_𝑝‘𝐽)(𝑔(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)⟩)
pi1xfr.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
pi1xfr.f	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
pi1xfr.i	⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))

**Assertion**
Ref	Expression
pi1xfr	⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Distinct variable groups: 𝑥,𝑔,𝐵 𝑔,𝐹,𝑥 𝑔,𝐼,𝑥 𝜑,𝑔,𝑥 𝑔,𝐽,𝑥 𝑃,𝑔,𝑥 𝑄,𝑔,𝑥 Allowed substitution hints: 𝐺(𝑥,𝑔) 𝑋(𝑥,𝑔)

Proof of Theorem pi1xfr Dummy variables 𝑓 ℎ 𝑢 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pi1xfr.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		iitopon 24819	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
3		pi1xfr.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
4		cnf2 23173	. . . . 5 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
5	2, 1, 3, 4	mp3an2i 1462	. . . 4 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
6		0elunit 13486	. . . 4 ⊢ 0 ∈ (0[,]1)
7		ffvelcdm 7096	. . . 4 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
8	5, 6, 7	sylancl 584	. . 3 ⊢ (𝜑 → (𝐹‘0) ∈ 𝑋)
9		pi1xfr.p	. . . 4 ⊢ 𝑃 = (𝐽 π₁ (𝐹‘0))
10	9	pi1grp 24997	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘0) ∈ 𝑋) → 𝑃 ∈ Grp)
11	1, 8, 10	syl2anc 582	. 2 ⊢ (𝜑 → 𝑃 ∈ Grp)
12		1elunit 13487	. . . 4 ⊢ 1 ∈ (0[,]1)
13		ffvelcdm 7096	. . . 4 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
14	5, 12, 13	sylancl 584	. . 3 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑋)
15		pi1xfr.q	. . . 4 ⊢ 𝑄 = (𝐽 π₁ (𝐹‘1))
16	15	pi1grp 24997	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘1) ∈ 𝑋) → 𝑄 ∈ Grp)
17	1, 14, 16	syl2anc 582	. 2 ⊢ (𝜑 → 𝑄 ∈ Grp)
18		pi1xfr.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
19		pi1xfr.g	. . . 4 ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃_ph‘𝐽), [(𝐼(_𝑝‘𝐽)(𝑔(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)⟩)
20		pi1xfr.i	. . . . . . 7 ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))
21	20	pcorevcl 24972	. . . . . 6 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
22	3, 21	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
23	22	simp1d 1139	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
24	22	simp2d 1140	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
25	24	eqcomd 2734	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))
26	22	simp3d 1141	. . . 4 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
27	9, 15, 18, 19, 1, 3, 23, 25, 26	pi1xfrf 25000	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄))
28	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
29	9, 1, 8, 28	pi1bas2 24988	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / ( ≃_ph‘𝐽)))
30	29	eleq2d 2815	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))))
31	30	biimpa 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
32		eqid 2728	. . . . . 6 ⊢ (∪ 𝐵 / ( ≃_ph‘𝐽)) = (∪ 𝐵 / ( ≃_ph‘𝐽))
33		fvoveq1 7449	. . . . . . . 8 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = (𝐺‘(𝑦(+_g‘𝑃)𝑧)))
34		fveq2 6902	. . . . . . . . 9 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃_ph‘𝐽)) = (𝐺‘𝑦))
35	34	oveq1d 7441	. . . . . . . 8 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
36	33, 35	eqeq12d 2744	. . . . . . 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))))
37	36	ralbidv 3175	. . . . . 6 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → (∀𝑧 ∈ 𝐵 (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧))))
38	29	eleq2d 2815	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))))
39	38	biimpa 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
40	39	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
41		oveq2 7434	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽)) = ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧))
42	41	fveq2d 6906	. . . . . . . . . 10 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽))) = (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)))
43		fveq2 6902	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃_ph‘𝐽)) = (𝐺‘𝑧))
44	43	oveq2d 7442	. . . . . . . . . 10 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘[ℎ]( ≃_ph‘𝐽))) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)))
45	42, 44	eqeq12d 2744	. . . . . . . . 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))))
46		phtpcer 24941	. . . . . . . . . . . . . 14 ⊢ ( ≃_ph‘𝐽) Er (II Cn 𝐽)
47	46	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ( ≃_ph‘𝐽) Er (II Cn 𝐽))
48	9, 1, 8, 28	pi1eluni 24989	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑓 ∈ ∪ 𝐵 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0))))
49	48	biimpa 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0)))
50	49	simp1d 1139	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
51	50	3adant3 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
52	9, 1, 8, 28	pi1eluni 24989	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))))
53	52	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))))
54	53	biimp3a 1465	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))
55	54	simp1d 1139	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ (II Cn 𝐽))
56	51, 55	pco0 24961	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘0) = (𝑓‘0))
57	49	simp2d 1140	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
58	57	3adant3 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
59	56, 58	eqtrd 2768	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘0) = (𝐹‘0))
60	49	simp3d 1141	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
61	60	3adant3 1129	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
62	54	simp2d 1140	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘0) = (𝐹‘0))
63	61, 62	eqtr4d 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (ℎ‘0))
64	51, 55, 63	pcocn 24964	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽))
65	3	3ad2ant1 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
66	64, 65	pco0 24961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)‘0) = ((𝑓(*_𝑝‘𝐽)ℎ)‘0))
67	26	3ad2ant1 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
68	59, 66, 67	3eqtr4rd 2779	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)‘0))
69	23	3ad2ant1 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
70	47, 69	erref 8751	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼( ≃_ph‘𝐽)𝐼)
71	54	simp3d 1141	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘1) = (𝐹‘0))
72		eqid 2728	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (0[,]1) ↦ if(𝑢 ≤ (1 / 2), if(𝑢 ≤ (1 / 4), (2 · 𝑢), (𝑢 + (1 / 4))), ((𝑢 / 2) + (1 / 2)))) = (𝑢 ∈ (0[,]1) ↦ if(𝑢 ≤ (1 / 2), if(𝑢 ≤ (1 / 4), (2 · 𝑢), (𝑢 + (1 / 4))), ((𝑢 / 2) + (1 / 2))))
73	51, 55, 65, 63, 71, 72	pcoass 24971	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)(𝑓(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))
74	55, 65	pco0 24961	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘0) = (ℎ‘0))
75	63, 74	eqtr4d 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘0))
76	47, 51	erref 8751	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓( ≃_ph‘𝐽)𝑓)
77	65, 69	pco1 24962	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*_𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
78	62, 74, 67	3eqtr4rd 2779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘0))
79	77, 78	eqtrd 2768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(_𝑝‘𝐽)𝐼)‘1) = ((ℎ(_𝑝‘𝐽)𝐹)‘0))
80		eqid 2728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
81	20, 80	pcorev2 24975	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*_𝑝‘𝐽)𝐼)( ≃_ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
82	65, 81	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹(*_𝑝‘𝐽)𝐼)( ≃_ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
83	55, 65, 71	pcocn 24964	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
84	47, 83	erref 8751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))
85	79, 82, 84	pcohtpy 24967	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(_𝑝‘𝐽)𝐼)(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(((0[,]1) × {(𝐹‘0)})(_𝑝‘𝐽)(ℎ(*_𝑝‘𝐽)𝐹)))
86	74, 62	eqtrd 2768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘0) = (𝐹‘0))
87	80	pcopt 24969	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℎ(_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽) ∧ ((ℎ(_𝑝‘𝐽)𝐹)‘0) = (𝐹‘0)) → (((0[,]1) × {(𝐹‘0)})(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(ℎ(*_𝑝‘𝐽)𝐹))
88	83, 86, 87	syl2anc 582	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((0[,]1) × {(𝐹‘0)})(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(ℎ(*_𝑝‘𝐽)𝐹))
89	47, 85, 88	ertrd 8747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(_𝑝‘𝐽)𝐼)(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))
90	24	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
91	90	eqcomd 2734	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
92	65, 69, 83, 91, 78, 72	pcoass 24971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(_𝑝‘𝐽)𝐼)(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(𝐹(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))))
93	47, 89, 92	ertr3d 8749	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)(𝐹(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))))
94	75, 76, 93	pcohtpy 24967	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(𝑓(_𝑝‘𝐽)(𝐹(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))))
95	69, 83, 78	pcocn 24964	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
96	69, 83	pco0 24961	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
97	96, 90	eqtrd 2768	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
98	97	eqcomd 2734	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0))
99	51, 65, 95, 61, 98, 72	pcoass 24971	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(_𝑝‘𝐽)𝐹)(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))( ≃_ph‘𝐽)(𝑓(_𝑝‘𝐽)(𝐹(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))))
100	47, 94, 99	ertr4d 8750	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)((𝑓(_𝑝‘𝐽)𝐹)(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))))
101	47, 73, 100	ertrd 8747	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)((𝑓(_𝑝‘𝐽)𝐹)(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))))
102	68, 70, 101	pcohtpy 24967	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(𝐼(_𝑝‘𝐽)((𝑓(_𝑝‘𝐽)𝐹)(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))))
103	3	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
104	50, 103, 60	pcocn 24964	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
105	104	3adant3 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
106	50, 103	pco0 24961	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)𝐹)‘0) = (𝑓‘0))
107	26	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
108	57, 106, 107	3eqtr4rd 2779	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘0))
109	108	3adant3 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘0))
110	51, 65	pco1 24962	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
111	110, 97	eqtr4d 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(_𝑝‘𝐽)𝐹)‘1) = ((𝐼(_𝑝‘𝐽)(ℎ(*_𝑝‘𝐽)𝐹))‘0))
112	69, 105, 95, 109, 111, 72	pcoass 24971	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))( ≃_ph‘𝐽)(𝐼(_𝑝‘𝐽)((𝑓(_𝑝‘𝐽)𝐹)(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))))
113	47, 102, 112	ertr4d 8750	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))))
114	47, 113	erthi 8783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → [(𝐼(_𝑝‘𝐽)((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽) = [((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))]( ≃_ph‘𝐽))
115	1	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
116	51, 55	pco1 24962	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘1) = (ℎ‘1))
117	116, 71	eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘1) = (𝐹‘0))
118	9, 1, 8, 28	pi1eluni 24989	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑓(_𝑝‘𝐽)ℎ) ∈ ∪ 𝐵 ↔ ((𝑓(_𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽) ∧ ((𝑓(_𝑝‘𝐽)ℎ)‘0) = (𝐹‘0) ∧ ((𝑓(_𝑝‘𝐽)ℎ)‘1) = (𝐹‘0))))
119	118	3ad2ant1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(_𝑝‘𝐽)ℎ) ∈ ∪ 𝐵 ↔ ((𝑓(_𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽) ∧ ((𝑓(_𝑝‘𝐽)ℎ)‘0) = (𝐹‘0) ∧ ((𝑓(_𝑝‘𝐽)ℎ)‘1) = (𝐹‘0))))
120	64, 59, 117, 119	mpbir3and 1339	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)ℎ) ∈ ∪ 𝐵)
121	9, 15, 18, 19, 115, 65, 69, 91, 67, 120	pi1xfrval 25001	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[(𝑓(_𝑝‘𝐽)ℎ)]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
122		eqid 2728	. . . . . . . . . . . . 13 ⊢ (Base‘𝑄) = (Base‘𝑄)
123	14	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋)
124		eqid 2728	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑄) = (+_g‘𝑄)
125	23	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
126	125, 104, 108	pcocn 24964	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
127	126	3adant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
128	125, 104	pco0 24961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
129	24	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
130	128, 129	eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
131	130	3adant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
132	125, 104	pco1 24962	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘1))
133	50, 103	pco1 24962	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
134	132, 133	eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
135	134	3adant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
136		eqidd 2729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (Base‘𝑄) = (Base‘𝑄))
137	15, 115, 123, 136	pi1eluni 24989	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄) ↔ ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))))
138	127, 131, 135, 137	mpbir3and 1339	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
139	69, 83	pco1 24962	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘1))
140	55, 65	pco1 24962	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
141	139, 140	eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
142	15, 115, 123, 136	pi1eluni 24989	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄) ↔ ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))))
143	95, 97, 141, 142	mpbir3and 1339	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
144	15, 122, 115, 123, 124, 138, 143	pi1addval 24995	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([(𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)(+_g‘𝑄)[(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)) = [((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(*_𝑝‘𝐽)𝐹)))]( ≃_ph‘𝐽))
145	114, 121, 144	3eqtr4d 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[(𝑓(_𝑝‘𝐽)ℎ)]( ≃_ph‘𝐽)) = ([(𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)(+_g‘𝑄)[(𝐼(_𝑝‘𝐽)(ℎ(*_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)))
146	8	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋)
147		eqid 2728	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
148		simp2 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
149		simp3 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ ∪ 𝐵)
150	9, 18, 115, 146, 147, 148, 149	pi1addval 24995	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽)) = [(𝑓(*_𝑝‘𝐽)ℎ)]( ≃_ph‘𝐽))
151	150	fveq2d 6906	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽))) = (𝐺‘[(𝑓(*_𝑝‘𝐽)ℎ)]( ≃_ph‘𝐽)))
152	1	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
153	25	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
154		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
155	9, 15, 18, 19, 152, 103, 125, 153, 107, 154	pi1xfrval 25001	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
156	155	3adant3 1129	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
157	9, 15, 18, 19, 115, 65, 69, 91, 67, 149	pi1xfrval 25001	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[ℎ]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
158	156, 157	oveq12d 7444	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘[ℎ]( ≃_ph‘𝐽))) = ([(𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)(+_g‘𝑄)[(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)))
159	145, 151, 158	3eqtr4d 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽))) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘[ℎ]( ≃_ph‘𝐽))))
160	159	3expa 1115	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽))) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘[ℎ]( ≃_ph‘𝐽))))
161	32, 45, 160	ectocld 8809	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))) → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)))
162	40, 161	syldan 589	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)))
163	162	ralrimiva 3143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)))
164	32, 37, 163	ectocld 8809	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
165	31, 164	syldan 589	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
166	165	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
167	27, 166	jca 510	. 2 ⊢ (𝜑 → (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧))))
168	18, 122, 147, 124	isghm 19177	. 2 ⊢ (𝐺 ∈ (𝑃 GrpHom 𝑄) ↔ ((𝑃 ∈ Grp ∧ 𝑄 ∈ Grp) ∧ (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))))
169	11, 17, 167, 168	syl21anbrc 1341	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ifcif 4532 {csn 4632 ⟨cop 4638 ∪ cuni 4912 class class class wbr 5152 ↦ cmpt 5235 × cxp 5680 ran crn 5683 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Er wer 8728 [cec 8729 / cqs 8730 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 ≤ cle 11287 − cmin 11482 / cdiv 11909 2c2 12305 4c4 12307 [,]cicc 13367 Basecbs 17187 +_gcplusg 17240 Grpcgrp 18897 GrpHom cghm 19174 TopOnctopon 22832 Cn ccn 23148 IIcii 24815 ≃_phcphtpc 24915 *_𝑝cpco 24947 π₁ cpi1 24950

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-ec 8733 df-qs 8737 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-icc 13371 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-qus 17498 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-mulg 19031 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-cn 23151 df-cnp 23152 df-tx 23486 df-hmeo 23679 df-xms 24246 df-ms 24247 df-tms 24248 df-ii 24817 df-htpy 24916 df-phtpy 24917 df-phtpc 24938 df-pco 24952 df-om1 24953 df-pi1 24955

This theorem is referenced by: pi1xfrcnv 25004 pi1xfrgim 25005

W3C validator