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 23266

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	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		iitopon 23094	. . . . . . 7 ⊢ II ∈ (TopOn‘(0[,]1))
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
4		pi1xfr.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
5		cnf2 21465	. . . . . 6 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
6	3, 1, 4, 5	syl3anc 1439	. . . . 5 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
7		0elunit 12609	. . . . 5 ⊢ 0 ∈ (0[,]1)
8		ffvelrn 6623	. . . . 5 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
9	6, 7, 8	sylancl 580	. . . 4 ⊢ (𝜑 → (𝐹‘0) ∈ 𝑋)
10		pi1xfr.p	. . . . 5 ⊢ 𝑃 = (𝐽 π₁ (𝐹‘0))
11	10	pi1grp 23261	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘0) ∈ 𝑋) → 𝑃 ∈ Grp)
12	1, 9, 11	syl2anc 579	. . 3 ⊢ (𝜑 → 𝑃 ∈ Grp)
13		1elunit 12610	. . . . 5 ⊢ 1 ∈ (0[,]1)
14		ffvelrn 6623	. . . . 5 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
15	6, 13, 14	sylancl 580	. . . 4 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑋)
16		pi1xfr.q	. . . . 5 ⊢ 𝑄 = (𝐽 π₁ (𝐹‘1))
17	16	pi1grp 23261	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘1) ∈ 𝑋) → 𝑄 ∈ Grp)
18	1, 15, 17	syl2anc 579	. . 3 ⊢ (𝜑 → 𝑄 ∈ Grp)
19	12, 18	jca 507	. 2 ⊢ (𝜑 → (𝑃 ∈ Grp ∧ 𝑄 ∈ Grp))
20		pi1xfr.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
21		pi1xfr.g	. . . 4 ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃_ph‘𝐽), [(𝐼(_𝑝‘𝐽)(𝑔(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽)⟩)
22		pi1xfr.i	. . . . . . 7 ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))
23	22	pcorevcl 23236	. . . . . 6 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
24	4, 23	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
25	24	simp1d 1133	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
26	24	simp2d 1134	. . . . 5 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
27	26	eqcomd 2784	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))
28	24	simp3d 1135	. . . 4 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
29	10, 16, 20, 21, 1, 4, 25, 27, 28	pi1xfrf 23264	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄))
30	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
31	10, 1, 9, 30	pi1bas2 23252	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / ( ≃_ph‘𝐽)))
32	31	eleq2d 2845	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))))
33	32	biimpa 470	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
34		eqid 2778	. . . . . 6 ⊢ (∪ 𝐵 / ( ≃_ph‘𝐽)) = (∪ 𝐵 / ( ≃_ph‘𝐽))
35		fvoveq1 6947	. . . . . . . 8 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = (𝐺‘(𝑦(+_g‘𝑃)𝑧)))
36		fveq2 6448	. . . . . . . . 9 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃_ph‘𝐽)) = (𝐺‘𝑦))
37	36	oveq1d 6939	. . . . . . . 8 ⊢ ([𝑓]( ≃_ph‘𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
38	35, 37	eqeq12d 2793	. . . . . . 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))))
39	38	ralbidv 3168	. . . . . 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))))
40	31	eleq2d 2845	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))))
41	40	biimpa 470	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
42	41	adantlr 705	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽)))
43		oveq2 6932	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽)) = ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧))
44	43	fveq2d 6452	. . . . . . . . . 10 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽))) = (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)))
45		fveq2 6448	. . . . . . . . . . 11 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → (𝐺‘[ℎ]( ≃_ph‘𝐽)) = (𝐺‘𝑧))
46	45	oveq2d 6940	. . . . . . . . . 10 ⊢ ([ℎ]( ≃_ph‘𝐽) = 𝑧 → ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘[ℎ]( ≃_ph‘𝐽))) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)))
47	44, 46	eqeq12d 2793	. . . . . . . . 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))))
48		phtpcer 23206	. . . . . . . . . . . . . 14 ⊢ ( ≃_ph‘𝐽) Er (II Cn 𝐽)
49	48	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ( ≃_ph‘𝐽) Er (II Cn 𝐽))
50	10, 1, 9, 30	pi1eluni 23253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑓 ∈ ∪ 𝐵 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0))))
51	50	biimpa 470	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0)))
52	51	simp1d 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
53	52	3adant3 1123	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ (II Cn 𝐽))
54	10, 1, 9, 30	pi1eluni 23253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))))
55	54	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (ℎ ∈ ∪ 𝐵 ↔ (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0))))
56	55	biimp3a 1542	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ ∈ (II Cn 𝐽) ∧ (ℎ‘0) = (𝐹‘0) ∧ (ℎ‘1) = (𝐹‘0)))
57	56	simp1d 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ (II Cn 𝐽))
58	53, 57	pco0 23225	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘0) = (𝑓‘0))
59	51	simp2d 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
60	59	3adant3 1123	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘0) = (𝐹‘0))
61	58, 60	eqtrd 2814	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘0) = (𝐹‘0))
62	51	simp3d 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
63	62	3adant3 1123	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (𝐹‘0))
64	56	simp2d 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘0) = (𝐹‘0))
65	63, 64	eqtr4d 2817	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = (ℎ‘0))
66	53, 57, 65	pcocn 23228	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)ℎ) ∈ (II Cn 𝐽))
67	4	3ad2ant1 1124	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
68	66, 67	pco0 23225	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)‘0) = ((𝑓(*_𝑝‘𝐽)ℎ)‘0))
69	28	3ad2ant1 1124	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
70	61, 68, 69	3eqtr4rd 2825	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = (((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)‘0))
71	25	3ad2ant1 1124	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
72	49, 71	erref 8048	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐼( ≃_ph‘𝐽)𝐼)
73	56	simp3d 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ‘1) = (𝐹‘0))
74		eqid 2778	. . . . . . . . . . . . . . . . 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))))
75	53, 57, 67, 65, 73, 74	pcoass 23235	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(_𝑝‘𝐽)ℎ)(_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)(𝑓(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)))
76	57, 67	pco0 23225	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘0) = (ℎ‘0))
77	65, 76	eqtr4d 2817	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘0))
78	49, 53	erref 8048	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓( ≃_ph‘𝐽)𝑓)
79	67, 71	pco1 23226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(*_𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
80	64, 76, 69	3eqtr4rd 2825	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘0))
81	79, 80	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(_𝑝‘𝐽)𝐼)‘1) = ((ℎ(_𝑝‘𝐽)𝐹)‘0))
82		eqid 2778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
83	22, 82	pcorev2 23239	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*_𝑝‘𝐽)𝐼)( ≃_ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
84	67, 83	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹(*_𝑝‘𝐽)𝐼)( ≃_ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
85	57, 67, 73	pcocn 23228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
86	49, 85	erref 8048	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))
87	81, 84, 86	pcohtpy 23231	. . . . . . . . . . . . . . . . . . . 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)))
88	76, 64	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘0) = (𝐹‘0))
89	82	pcopt 23233	. . . . . . . . . . . . . . . . . . . . 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))
90	85, 88, 89	syl2anc 579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (((0[,]1) × {(𝐹‘0)})(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(ℎ(*_𝑝‘𝐽)𝐹))
91	49, 87, 90	ertrd 8044	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐹(_𝑝‘𝐽)𝐼)(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))( ≃_ph‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))
92	26	3ad2ant1 1124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
93	92	eqcomd 2784	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
94	67, 71, 85, 93, 80, 74	pcoass 23235	. . . . . . . . . . . . . . . . . . 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))))
95	49, 91, 94	ertr3d 8046	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (ℎ(_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)(𝐹(_𝑝‘𝐽)(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))))
96	77, 78, 95	pcohtpy 23231	. . . . . . . . . . . . . . . . 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)))))
97	71, 85, 80	pcocn 23228	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
98	71, 85	pco0 23225	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
99	98, 92	eqtrd 2814	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
100	99	eqcomd 2784	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘0))
101	53, 67, 97, 63, 100, 74	pcoass 23235	. . . . . . . . . . . . . . . . 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)))))
102	49, 96, 101	ertr4d 8047	. . . . . . . . . . . . . . . 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))))
103	49, 75, 102	ertrd 8044	. . . . . . . . . . . . . . 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))))
104	70, 72, 103	pcohtpy 23231	. . . . . . . . . . . . . 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)))))
105	4	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
106	52, 105, 62	pcocn 23228	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
107	106	3adant3 1123	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
108	52, 105	pco0 23225	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)𝐹)‘0) = (𝑓‘0))
109	28	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
110	59, 108, 109	3eqtr4rd 2825	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘0))
111	110	3adant3 1123	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘0))
112	53, 67	pco1 23226	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
113	112, 99	eqtr4d 2817	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(_𝑝‘𝐽)𝐹)‘1) = ((𝐼(_𝑝‘𝐽)(ℎ(*_𝑝‘𝐽)𝐹))‘0))
114	71, 107, 97, 111, 113, 74	pcoass 23235	. . . . . . . . . . . . . 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)))))
115	49, 104, 114	ertr4d 8047	. . . . . . . . . . . . 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))))
116	49, 115	erthi 8077	. . . . . . . . . . . 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))
117	1	3ad2ant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
118	53, 57	pco1 23226	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘1) = (ℎ‘1))
119	118, 73	eqtrd 2814	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)ℎ)‘1) = (𝐹‘0))
120	10, 1, 9, 30	pi1eluni 23253	. . . . . . . . . . . . . . 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))))
121	120	3ad2ant1 1124	. . . . . . . . . . . . . 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))))
122	66, 61, 119, 121	mpbir3and 1399	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝑓(*_𝑝‘𝐽)ℎ) ∈ ∪ 𝐵)
123	10, 16, 20, 21, 117, 67, 71, 93, 69, 122	pi1xfrval 23265	. . . . . . . . . . . 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))
124		eqid 2778	. . . . . . . . . . . . 13 ⊢ (Base‘𝑄) = (Base‘𝑄)
125	15	3ad2ant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘1) ∈ 𝑋)
126		eqid 2778	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑄) = (+_g‘𝑄)
127	25	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
128	127, 106, 110	pcocn 23228	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
129	128	3adant3 1123	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
130	127, 106	pco0 23225	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
131	26	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
132	130, 131	eqtrd 2814	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
133	132	3adant3 1123	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
134	127, 106	pco1 23226	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = ((𝑓(*_𝑝‘𝐽)𝐹)‘1))
135	52, 105	pco1 23226	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝑓(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
136	134, 135	eqtrd 2814	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
137	136	3adant3 1123	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
138		eqidd 2779	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (Base‘𝑄) = (Base‘𝑄))
139	16, 117, 125, 138	pi1eluni 23253	. . . . . . . . . . . . . 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))))
140	129, 133, 137, 139	mpbir3and 1399	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
141	71, 85	pco1 23226	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘1) = ((ℎ(*_𝑝‘𝐽)𝐹)‘1))
142	57, 67	pco1 23226	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((ℎ(*_𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
143	141, 142	eqtrd 2814	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ((𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
144	16, 117, 125, 138	pi1eluni 23253	. . . . . . . . . . . . . 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))))
145	97, 99, 143, 144	mpbir3and 1399	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
146	16, 124, 117, 125, 126, 140, 145	pi1addval 23259	. . . . . . . . . . . 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))
147	116, 123, 146	3eqtr4d 2824	. . . . . . . . . . 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)))
148	9	3ad2ant1 1124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐹‘0) ∈ 𝑋)
149		eqid 2778	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
150		simp2 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
151		simp3 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ℎ ∈ ∪ 𝐵)
152	10, 20, 117, 148, 149, 150, 151	pi1addval 23259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → ([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽)) = [(𝑓(*_𝑝‘𝐽)ℎ)]( ≃_ph‘𝐽))
153	152	fveq2d 6452	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)[ℎ]( ≃_ph‘𝐽))) = (𝐺‘[(𝑓(*_𝑝‘𝐽)ℎ)]( ≃_ph‘𝐽)))
154	1	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
155	27	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
156		simpr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → 𝑓 ∈ ∪ 𝐵)
157	10, 16, 20, 21, 154, 105, 127, 155, 109, 156	pi1xfrval 23265	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
158	157	3adant3 1123	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[𝑓]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)(𝑓(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
159	10, 16, 20, 21, 117, 67, 71, 93, 69, 151	pi1xfrval 23265	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵 ∧ ℎ ∈ ∪ 𝐵) → (𝐺‘[ℎ]( ≃_ph‘𝐽)) = [(𝐼(_𝑝‘𝐽)(ℎ(_𝑝‘𝐽)𝐹))]( ≃_ph‘𝐽))
160	158, 159	oveq12d 6942	. . . . . . . . . . 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)))
161	147, 153, 160	3eqtr4d 2824	. . . . . . . . . 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))))
162	161	3expa 1108	. . . . . . . . 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))))
163	34, 47, 162	ectocld 8099	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))) → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)))
164	42, 163	syldan 585	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) ∧ 𝑧 ∈ 𝐵) → (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)))
165	164	ralrimiva 3148	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ∪ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘([𝑓]( ≃_ph‘𝐽)(+_g‘𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃_ph‘𝐽))(+_g‘𝑄)(𝐺‘𝑧)))
166	34, 39, 165	ectocld 8099	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (∪ 𝐵 / ( ≃_ph‘𝐽))) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
167	33, 166	syldan 585	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
168	167	ralrimiva 3148	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧)))
169	29, 168	jca 507	. 2 ⊢ (𝜑 → (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐺‘(𝑦(+_g‘𝑃)𝑧)) = ((𝐺‘𝑦)(+_g‘𝑄)(𝐺‘𝑧))))
170	20, 124, 149, 126	isghm 18048	. 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)))))
171	19, 169, 170	sylanbrc 578	1 ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ifcif 4307 {csn 4398 ⟨cop 4404 ∪ cuni 4673 class class class wbr 4888 ↦ cmpt 4967 × cxp 5355 ran crn 5358 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 Er wer 8025 [cec 8026 / cqs 8027 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 ≤ cle 10414 − cmin 10608 / cdiv 11034 2c2 11434 4c4 11436 [,]cicc 12494 Basecbs 16259 +_gcplusg 16342 Grpcgrp 17813 GrpHom cghm 18045 TopOnctopon 21126 Cn ccn 21440 IIcii 23090 ≃_phcphtpc 23180 *_𝑝cpco 23211 π₁ cpi1 23214

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-ec 8030 df-qs 8034 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioo 12495 df-icc 12498 df-fz 12648 df-fzo 12789 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-hom 16366 df-cco 16367 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-pt 16495 df-prds 16498 df-xrs 16552 df-qtop 16557 df-imas 16558 df-qus 16559 df-xps 16560 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-grp 17816 df-mulg 17932 df-ghm 18046 df-cntz 18137 df-cmn 18585 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-cn 21443 df-cnp 21444 df-tx 21778 df-hmeo 21971 df-xms 22537 df-ms 22538 df-tms 22539 df-ii 23092 df-htpy 23181 df-phtpy 23182 df-phtpc 23203 df-pco 23216 df-om1 23217 df-pi1 23219

This theorem is referenced by: pi1xfrcnv 23268 pi1xfrgim 23269

W3C validator