Theorem pi1xfrcnvlem 24963

Description: Given a path 𝐹 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

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)))
pi1xfrcnv.h	⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)

Assertion

Ref	Expression
pi1xfrcnvlem	⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)

Distinct variable groups:   𝑔,ℎ,𝑥,𝐵   𝑔,𝐹,ℎ,𝑥   𝑔,𝐼,ℎ,𝑥   ℎ,𝐺   𝜑,𝑔,ℎ,𝑥   𝑔,𝐽,ℎ,𝑥   𝑃,𝑔,ℎ,𝑥   𝑄,𝑔,ℎ,𝑥

Allowed substitution hints:   𝐺(𝑥,𝑔)   𝐻(𝑥,𝑔,ℎ)   𝑋(𝑥,𝑔,ℎ)

Proof of Theorem pi1xfrcnvlem

Step	Hyp	Ref	Expression
1		pi1xfr.g	. . . 4 ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)
2		fvex 6874	. . . . 5 ⊢ ( ≃ph‘𝐽) ∈ V
3		ecexg 8678	. . . . 5 ⊢ (( ≃ph‘𝐽) ∈ V → [𝑔]( ≃ph‘𝐽) ∈ V)
4	2, 3	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [𝑔]( ≃ph‘𝐽) ∈ V)
5		ecexg 8678	. . . . 5 ⊢ (( ≃ph‘𝐽) ∈ V → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ V)
6	2, 5	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽) ∈ V)
7	1, 4, 6	fliftcnv 7289	. . 3 ⊢ (𝜑 → ◡𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩))
8		pi1xfr.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
9		pi1xfr.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))
10	9	pcorevcl 24932	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
11	8, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
12	11	simp1d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼 ∈ (II Cn 𝐽))
14		pi1xfr.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝐽 π1 (𝐹‘0))
15		pi1xfr.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
16		iitopon 24779	. . . . . . . . . . . . . 14 ⊢ II ∈ (TopOn‘(0[,]1))
17		cnf2 23143	. . . . . . . . . . . . . 14 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
18	16, 15, 8, 17	mp3an2i 1468	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:(0[,]1)⟶𝑋)
19		0elunit 13437	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
20		ffvelcdm 7056	. . . . . . . . . . . . 13 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
21	18, 19, 20	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘0) ∈ 𝑋)
22		pi1xfr.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑃)
23	22	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = (Base‘𝑃))
24	14, 15, 21, 23	pi1eluni 24949	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↔ (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0))))
25	24	biimpa 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0) ∧ (𝑔‘1) = (𝐹‘0)))
26	25	simp1d 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔 ∈ (II Cn 𝐽))
27	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹 ∈ (II Cn 𝐽))
28	25	simp3d 1144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = (𝐹‘0))
29	26, 27, 28	pcocn 24924	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)𝐹) ∈ (II Cn 𝐽))
30	11	simp3d 1144	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))
31	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝐹‘0))
32	25	simp2d 1143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘0) = (𝐹‘0))
33	31, 32	eqtr4d 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = (𝑔‘0))
34	26, 27	pco0 24921	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘0) = (𝑔‘0))
35	33, 34	eqtr4d 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘0))
36	13, 29, 35	pcocn 24924	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽))
37	13, 29	pco0 24921	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐼‘0))
38	11	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1))
39	38	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼‘0) = (𝐹‘1))
40	37, 39	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1))
41	13, 29	pco1 24922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = ((𝑔(*𝑝‘𝐽)𝐹)‘1))
42	26, 27	pco1 24922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐹‘1))
43	41, 42	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))
44		pi1xfr.q	. . . . . . . . 9 ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))
45		1elunit 13438	. . . . . . . . . 10 ⊢ 1 ∈ (0[,]1)
46		ffvelcdm 7056	. . . . . . . . . 10 ⊢ ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
47	18, 45, 46	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑋)
48		eqidd 2731	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑄) = (Base‘𝑄))
49	44, 15, 47, 48	pi1eluni 24949	. . . . . . . 8 ⊢ (𝜑 → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄) ↔ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))))
50	49	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄) ↔ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))‘1) = (𝐹‘1))))
51	36, 40, 43, 50	mpbir3and 1343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) ∈ ∪ (Base‘𝑄))
52		eqidd 2731	. . . . . 6 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))) = (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))))
53		eqidd 2731	. . . . . 6 ⊢ (𝜑 → (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) = (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩))
54		eceq1 8713	. . . . . . 7 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → [ℎ]( ≃ph‘𝐽) = [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
55		oveq1 7397	. . . . . . . . 9 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → (ℎ(*𝑝‘𝐽)𝐼) = ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))
56	55	oveq2d 7406	. . . . . . . 8 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → (𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼)) = (𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)))
57	56	eceq1d 8714	. . . . . . 7 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽) = [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽))
58	54, 57	opeq12d 4848	. . . . . 6 ⊢ (ℎ = (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)) → ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩ = ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)
59	51, 52, 53, 58	fmptco 7104	. . . . 5 ⊢ (𝜑 → ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩))
60		phtpcer 24901	. . . . . . . . 9 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽)
61	60	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ( ≃ph‘𝐽) Er (II Cn 𝐽))
62	13, 26	pco0 24921	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)𝑔)‘0) = (𝐼‘0))
63	62, 39	eqtr2d 2766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = ((𝐼(*𝑝‘𝐽)𝑔)‘0))
64	61, 27	erref 8694	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐹( ≃ph‘𝐽)𝐹)
65	61, 13	erref 8694	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝐼( ≃ph‘𝐽)𝐼)
66		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
67	66	pcopt2 24930	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘1) = (𝐹‘0)) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)𝑔)
68	26, 28, 67	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)𝑔)
69	39	eqcomd 2736	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹‘1) = (𝐼‘0))
70		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), if(𝑥 ≤ (1 / 4), (2 · 𝑥), (𝑥 + (1 / 4))), ((𝑥 / 2) + (1 / 2)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), if(𝑥 ≤ (1 / 4), (2 · 𝑥), (𝑥 + (1 / 4))), ((𝑥 / 2) + (1 / 2))))
71	26, 27, 13, 28, 69, 70	pcoass 24931	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼)))
72	27, 13	pco0 24921	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘0) = (𝐹‘0))
73	28, 72	eqtr4d 2768	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔‘1) = ((𝐹(*𝑝‘𝐽)𝐼)‘0))
74	61, 26	erref 8694	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)𝑔)
75	9, 66	pcorev2 24935	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
76	27, 75	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))
77	73, 74, 76	pcohtpy 24927	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)(𝐹(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)(𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)})))
78	61, 71, 77	ertr2d 8691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝑔(*𝑝‘𝐽)((0[,]1) × {(𝐹‘0)}))( ≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))
79	61, 68, 78	ertr3d 8692	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → 𝑔( ≃ph‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼))
80	33, 65, 79	pcohtpy 24927	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)))
81	42, 39	eqtr4d 2768	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝑔(*𝑝‘𝐽)𝐹)‘1) = (𝐼‘0))
82	13, 29, 13, 35, 81, 70	pcoass 24931	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)( ≃ph‘𝐽)(𝐼(*𝑝‘𝐽)((𝑔(*𝑝‘𝐽)𝐹)(*𝑝‘𝐽)𝐼)))
83	61, 80, 82	ertr4d 8693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐼(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))
84	63, 64, 83	pcohtpy 24927	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼)))
85	27, 13, 26, 69, 33, 70	pcoass 24931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔)))
86	27, 13	pco1 24922	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝐼‘1))
87	86, 33	eqtrd 2765	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)‘1) = (𝑔‘0))
88	87, 76, 74	pcohtpy 24927	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)(((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔))
89	66	pcopt 24929	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ (II Cn 𝐽) ∧ (𝑔‘0) = (𝐹‘0)) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
90	26, 32, 89	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (((0[,]1) × {(𝐹‘0)})(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
91	61, 88, 90	ertrd 8690	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ((𝐹(*𝑝‘𝐽)𝐼)(*𝑝‘𝐽)𝑔)( ≃ph‘𝐽)𝑔)
92	61, 85, 91	ertr3d 8692	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)(𝐼(*𝑝‘𝐽)𝑔))( ≃ph‘𝐽)𝑔)
93	61, 84, 92	ertr3d 8692	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → (𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))( ≃ph‘𝐽)𝑔)
94	61, 93	erthi 8730	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽) = [𝑔]( ≃ph‘𝐽))
95	94	opeq2d 4847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ ∪ 𝐵) → ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩ = ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩)
96	95	mpteq2dva 5203	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)((𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) = (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩))
97	59, 96	eqtrd 2765	. . . 4 ⊢ (𝜑 → ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩))
98	97	rneqd 5905	. . 3 ⊢ (𝜑 → ran ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽), [𝑔]( ≃ph‘𝐽)⟩))
99	7, 98	eqtr4d 2768	. 2 ⊢ (𝜑 → ◡𝐺 = ran ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))))
100		rncoss 5942	. . 3 ⊢ ran ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) ⊆ ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)
101		pi1xfrcnv.h	. . 3 ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)
102	100, 101	sseqtrri 3999	. 2 ⊢ ran ((ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩) ∘ (𝑔 ∈ ∪ 𝐵 ↦ (𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹)))) ⊆ 𝐻
103	99, 102	eqsstrdi 3994	1 ⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3917  ifcif 4491  {csn 4592  ⟨cop 4598  ∪ cuni 4874   class class class wbr 5110   ↦ cmpt 5191   × cxp 5639  ^◡ccnv 5640  ran crn 5642   ∘ ccom 5645  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   Er wer 8671  [cec 8672  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080   ≤ cle 11216   − cmin 11412   / cdiv 11842  2c2 12248  4c4 12250  [,]cicc 13316  Basecbs 17186  TopOnctopon 22804   Cn ccn 23118  IIcii 24775   ≃_phcphtpc 24875  *_𝑝cpco 24907   π₁ cpi1 24910

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  ax-pre-sup 11153  ax-addf 11154

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-int 4914  df-iun 4960  df-iin 4961  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-se 5595  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-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-ec 8676  df-qs 8680  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-fi 9369  df-sup 9400  df-inf 9401  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  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-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ioo 13317  df-icc 13320  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-rest 17392  df-topn 17393  df-0g 17411  df-gsum 17412  df-topgen 17413  df-pt 17414  df-prds 17417  df-xrs 17472  df-qtop 17477  df-imas 17478  df-qus 17479  df-xps 17480  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-mulg 19007  df-cntz 19256  df-cmn 19719  df-psmet 21263  df-xmet 21264  df-met 21265  df-bl 21266  df-mopn 21267  df-cnfld 21272  df-top 22788  df-topon 22805  df-topsp 22827  df-bases 22840  df-cld 22913  df-cn 23121  df-cnp 23122  df-tx 23456  df-hmeo 23649  df-xms 24215  df-ms 24216  df-tms 24217  df-ii 24777  df-htpy 24876  df-phtpy 24877  df-phtpc 24898  df-pco 24912  df-om1 24913  df-pi1 24915

This theorem is referenced by:  pi1xfrcnv  24964