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Theorem pi1xfr 22768
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]( ≃ph𝐽), [(𝐼(*𝑝𝐽)(𝑔(*𝑝𝐽)𝐹))]( ≃ph𝐽)⟩)
pi1xfr.j (𝜑𝐽 ∈ (TopOn‘𝑋))
pi1xfr.f (𝜑𝐹 ∈ (II Cn 𝐽))
pi1xfr.i 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))
Assertion
Ref Expression
pi1xfr (𝜑𝐺 ∈ (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 𝑢 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1xfr.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 iitopon 22595 . . . . . . 7 II ∈ (TopOn‘(0[,]1))
32a1i 11 . . . . . 6 (𝜑 → II ∈ (TopOn‘(0[,]1)))
4 pi1xfr.f . . . . . 6 (𝜑𝐹 ∈ (II Cn 𝐽))
5 cnf2 20966 . . . . . 6 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (II Cn 𝐽)) → 𝐹:(0[,]1)⟶𝑋)
63, 1, 4, 5syl3anc 1323 . . . . 5 (𝜑𝐹:(0[,]1)⟶𝑋)
7 0elunit 12235 . . . . 5 0 ∈ (0[,]1)
8 ffvelrn 6315 . . . . 5 ((𝐹:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ 𝑋)
96, 7, 8sylancl 693 . . . 4 (𝜑 → (𝐹‘0) ∈ 𝑋)
10 pi1xfr.p . . . . 5 𝑃 = (𝐽 π1 (𝐹‘0))
1110pi1grp 22763 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘0) ∈ 𝑋) → 𝑃 ∈ Grp)
121, 9, 11syl2anc 692 . . 3 (𝜑𝑃 ∈ Grp)
13 1elunit 12236 . . . . 5 1 ∈ (0[,]1)
14 ffvelrn 6315 . . . . 5 ((𝐹:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ 𝑋)
156, 13, 14sylancl 693 . . . 4 (𝜑 → (𝐹‘1) ∈ 𝑋)
16 pi1xfr.q . . . . 5 𝑄 = (𝐽 π1 (𝐹‘1))
1716pi1grp 22763 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘1) ∈ 𝑋) → 𝑄 ∈ Grp)
181, 15, 17syl2anc 692 . . 3 (𝜑𝑄 ∈ Grp)
1912, 18jca 554 . 2 (𝜑 → (𝑃 ∈ Grp ∧ 𝑄 ∈ Grp))
20 pi1xfr.b . . . 4 𝐵 = (Base‘𝑃)
21 pi1xfr.g . . . 4 𝐺 = ran (𝑔 𝐵 ↦ ⟨[𝑔]( ≃ph𝐽), [(𝐼(*𝑝𝐽)(𝑔(*𝑝𝐽)𝐹))]( ≃ph𝐽)⟩)
22 pi1xfr.i . . . . . . 7 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))
2322pcorevcl 22738 . . . . . 6 (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
244, 23syl 17 . . . . 5 (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0)))
2524simp1d 1071 . . . 4 (𝜑𝐼 ∈ (II Cn 𝐽))
2624simp2d 1072 . . . . 5 (𝜑 → (𝐼‘0) = (𝐹‘1))
2726eqcomd 2627 . . . 4 (𝜑 → (𝐹‘1) = (𝐼‘0))
2824simp3d 1073 . . . 4 (𝜑 → (𝐼‘1) = (𝐹‘0))
2910, 16, 20, 21, 1, 4, 25, 27, 28pi1xfrf 22766 . . 3 (𝜑𝐺:𝐵⟶(Base‘𝑄))
3020a1i 11 . . . . . . . 8 (𝜑𝐵 = (Base‘𝑃))
3110, 1, 9, 30pi1bas2 22754 . . . . . . 7 (𝜑𝐵 = ( 𝐵 / ( ≃ph𝐽)))
3231eleq2d 2684 . . . . . 6 (𝜑 → (𝑦𝐵𝑦 ∈ ( 𝐵 / ( ≃ph𝐽))))
3332biimpa 501 . . . . 5 ((𝜑𝑦𝐵) → 𝑦 ∈ ( 𝐵 / ( ≃ph𝐽)))
34 eqid 2621 . . . . . 6 ( 𝐵 / ( ≃ph𝐽)) = ( 𝐵 / ( ≃ph𝐽))
35 oveq1 6614 . . . . . . . . 9 ([𝑓]( ≃ph𝐽) = 𝑦 → ([𝑓]( ≃ph𝐽)(+g𝑃)𝑧) = (𝑦(+g𝑃)𝑧))
3635fveq2d 6154 . . . . . . . 8 ([𝑓]( ≃ph𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = (𝐺‘(𝑦(+g𝑃)𝑧)))
37 fveq2 6150 . . . . . . . . 9 ([𝑓]( ≃ph𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃ph𝐽)) = (𝐺𝑦))
3837oveq1d 6622 . . . . . . . 8 ([𝑓]( ≃ph𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))
3936, 38eqeq12d 2636 . . . . . . 7 ([𝑓]( ≃ph𝐽) = 𝑦 → ((𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)) ↔ (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧))))
4039ralbidv 2980 . . . . . 6 ([𝑓]( ≃ph𝐽) = 𝑦 → (∀𝑧𝐵 (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)) ↔ ∀𝑧𝐵 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧))))
4131eleq2d 2684 . . . . . . . . . 10 (𝜑 → (𝑧𝐵𝑧 ∈ ( 𝐵 / ( ≃ph𝐽))))
4241biimpa 501 . . . . . . . . 9 ((𝜑𝑧𝐵) → 𝑧 ∈ ( 𝐵 / ( ≃ph𝐽)))
4342adantlr 750 . . . . . . . 8 (((𝜑𝑓 𝐵) ∧ 𝑧𝐵) → 𝑧 ∈ ( 𝐵 / ( ≃ph𝐽)))
44 oveq2 6615 . . . . . . . . . . 11 ([]( ≃ph𝐽) = 𝑧 → ([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽)) = ([𝑓]( ≃ph𝐽)(+g𝑃)𝑧))
4544fveq2d 6154 . . . . . . . . . 10 ([]( ≃ph𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)))
46 fveq2 6150 . . . . . . . . . . 11 ([]( ≃ph𝐽) = 𝑧 → (𝐺‘[]( ≃ph𝐽)) = (𝐺𝑧))
4746oveq2d 6623 . . . . . . . . . 10 ([]( ≃ph𝐽) = 𝑧 → ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)))
4845, 47eqeq12d 2636 . . . . . . . . 9 ([]( ≃ph𝐽) = 𝑧 → ((𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))) ↔ (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧))))
49 phtpcer 22707 . . . . . . . . . . . . . 14 ( ≃ph𝐽) Er (II Cn 𝐽)
5049a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → ( ≃ph𝐽) Er (II Cn 𝐽))
5110, 1, 9, 30pi1eluni 22755 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑓 𝐵 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0))))
5251biimpa 501 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓 𝐵) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = (𝐹‘0) ∧ (𝑓‘1) = (𝐹‘0)))
5352simp1d 1071 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓 𝐵) → 𝑓 ∈ (II Cn 𝐽))
54533adant3 1079 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵 𝐵) → 𝑓 ∈ (II Cn 𝐽))
5510, 1, 9, 30pi1eluni 22755 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ( 𝐵 ↔ ( ∈ (II Cn 𝐽) ∧ (‘0) = (𝐹‘0) ∧ (‘1) = (𝐹‘0))))
5655adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓 𝐵) → ( 𝐵 ↔ ( ∈ (II Cn 𝐽) ∧ (‘0) = (𝐹‘0) ∧ (‘1) = (𝐹‘0))))
5756biimp3a 1429 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓 𝐵 𝐵) → ( ∈ (II Cn 𝐽) ∧ (‘0) = (𝐹‘0) ∧ (‘1) = (𝐹‘0)))
5857simp1d 1071 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵 𝐵) → ∈ (II Cn 𝐽))
5954, 58pco0 22727 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽))‘0) = (𝑓‘0))
6052simp2d 1072 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵) → (𝑓‘0) = (𝐹‘0))
61603adant3 1079 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵 𝐵) → (𝑓‘0) = (𝐹‘0))
6259, 61eqtrd 2655 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽))‘0) = (𝐹‘0))
6352simp3d 1073 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓 𝐵) → (𝑓‘1) = (𝐹‘0))
64633adant3 1079 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓 𝐵 𝐵) → (𝑓‘1) = (𝐹‘0))
6557simp2d 1072 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓 𝐵 𝐵) → (‘0) = (𝐹‘0))
6664, 65eqtr4d 2658 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵 𝐵) → (𝑓‘1) = (‘0))
6754, 58, 66pcocn 22730 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵 𝐵) → (𝑓(*𝑝𝐽)) ∈ (II Cn 𝐽))
6843ad2ant1 1080 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵 𝐵) → 𝐹 ∈ (II Cn 𝐽))
6967, 68pco0 22727 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵 𝐵) → (((𝑓(*𝑝𝐽))(*𝑝𝐽)𝐹)‘0) = ((𝑓(*𝑝𝐽))‘0))
70283ad2ant1 1080 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵 𝐵) → (𝐼‘1) = (𝐹‘0))
7162, 69, 703eqtr4rd 2666 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → (𝐼‘1) = (((𝑓(*𝑝𝐽))(*𝑝𝐽)𝐹)‘0))
72253ad2ant1 1080 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵 𝐵) → 𝐼 ∈ (II Cn 𝐽))
7350, 72erref 7710 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → 𝐼( ≃ph𝐽)𝐼)
7457simp3d 1073 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵 𝐵) → (‘1) = (𝐹‘0))
75 eqid 2621 . . . . . . . . . . . . . . . . 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))))
7654, 58, 68, 66, 74, 75pcoass 22737 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽))(*𝑝𝐽)𝐹)( ≃ph𝐽)(𝑓(*𝑝𝐽)((*𝑝𝐽)𝐹)))
7758, 68pco0 22727 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓 𝐵 𝐵) → (((*𝑝𝐽)𝐹)‘0) = (‘0))
7866, 77eqtr4d 2658 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵 𝐵) → (𝑓‘1) = (((*𝑝𝐽)𝐹)‘0))
7950, 54erref 7710 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵 𝐵) → 𝑓( ≃ph𝐽)𝑓)
8068, 72pco1 22728 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑓 𝐵 𝐵) → ((𝐹(*𝑝𝐽)𝐼)‘1) = (𝐼‘1))
8165, 77, 703eqtr4rd 2666 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑓 𝐵 𝐵) → (𝐼‘1) = (((*𝑝𝐽)𝐹)‘0))
8280, 81eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓 𝐵 𝐵) → ((𝐹(*𝑝𝐽)𝐼)‘1) = (((*𝑝𝐽)𝐹)‘0))
83 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . 23 ((0[,]1) × {(𝐹‘0)}) = ((0[,]1) × {(𝐹‘0)})
8422, 83pcorev2 22741 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝𝐽)𝐼)( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))
8568, 84syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓 𝐵 𝐵) → (𝐹(*𝑝𝐽)𝐼)( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))
8658, 68, 74pcocn 22730 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑓 𝐵 𝐵) → ((*𝑝𝐽)𝐹) ∈ (II Cn 𝐽))
8750, 86erref 7710 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓 𝐵 𝐵) → ((*𝑝𝐽)𝐹)( ≃ph𝐽)((*𝑝𝐽)𝐹))
8882, 85, 87pcohtpy 22733 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓 𝐵 𝐵) → ((𝐹(*𝑝𝐽)𝐼)(*𝑝𝐽)((*𝑝𝐽)𝐹))( ≃ph𝐽)(((0[,]1) × {(𝐹‘0)})(*𝑝𝐽)((*𝑝𝐽)𝐹)))
8977, 65eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓 𝐵 𝐵) → (((*𝑝𝐽)𝐹)‘0) = (𝐹‘0))
9083pcopt 22735 . . . . . . . . . . . . . . . . . . . . 21 ((((*𝑝𝐽)𝐹) ∈ (II Cn 𝐽) ∧ (((*𝑝𝐽)𝐹)‘0) = (𝐹‘0)) → (((0[,]1) × {(𝐹‘0)})(*𝑝𝐽)((*𝑝𝐽)𝐹))( ≃ph𝐽)((*𝑝𝐽)𝐹))
9186, 89, 90syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓 𝐵 𝐵) → (((0[,]1) × {(𝐹‘0)})(*𝑝𝐽)((*𝑝𝐽)𝐹))( ≃ph𝐽)((*𝑝𝐽)𝐹))
9250, 88, 91ertrd 7706 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓 𝐵 𝐵) → ((𝐹(*𝑝𝐽)𝐼)(*𝑝𝐽)((*𝑝𝐽)𝐹))( ≃ph𝐽)((*𝑝𝐽)𝐹))
93263ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑓 𝐵 𝐵) → (𝐼‘0) = (𝐹‘1))
9493eqcomd 2627 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓 𝐵 𝐵) → (𝐹‘1) = (𝐼‘0))
9568, 72, 86, 94, 81, 75pcoass 22737 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓 𝐵 𝐵) → ((𝐹(*𝑝𝐽)𝐼)(*𝑝𝐽)((*𝑝𝐽)𝐹))( ≃ph𝐽)(𝐹(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))))
9650, 92, 95ertr3d 7708 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵 𝐵) → ((*𝑝𝐽)𝐹)( ≃ph𝐽)(𝐹(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))))
9778, 79, 96pcohtpy 22733 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵 𝐵) → (𝑓(*𝑝𝐽)((*𝑝𝐽)𝐹))( ≃ph𝐽)(𝑓(*𝑝𝐽)(𝐹(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)))))
9872, 86, 81pcocn 22730 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵 𝐵) → (𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)) ∈ (II Cn 𝐽))
9972, 86pco0 22727 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))‘0) = (𝐼‘0))
10099, 93eqtrd 2655 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))‘0) = (𝐹‘1))
101100eqcomd 2627 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑓 𝐵 𝐵) → (𝐹‘1) = ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))‘0))
10254, 68, 98, 64, 101, 75pcoass 22737 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽)𝐹)(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)))( ≃ph𝐽)(𝑓(*𝑝𝐽)(𝐹(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)))))
10350, 97, 102ertr4d 7709 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵 𝐵) → (𝑓(*𝑝𝐽)((*𝑝𝐽)𝐹))( ≃ph𝐽)((𝑓(*𝑝𝐽)𝐹)(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))))
10450, 76, 103ertrd 7706 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽))(*𝑝𝐽)𝐹)( ≃ph𝐽)((𝑓(*𝑝𝐽)𝐹)(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))))
10571, 73, 104pcohtpy 22733 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → (𝐼(*𝑝𝐽)((𝑓(*𝑝𝐽))(*𝑝𝐽)𝐹))( ≃ph𝐽)(𝐼(*𝑝𝐽)((𝑓(*𝑝𝐽)𝐹)(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)))))
1064adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵) → 𝐹 ∈ (II Cn 𝐽))
10753, 106, 63pcocn 22730 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵) → (𝑓(*𝑝𝐽)𝐹) ∈ (II Cn 𝐽))
1081073adant3 1079 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → (𝑓(*𝑝𝐽)𝐹) ∈ (II Cn 𝐽))
10953, 106pco0 22727 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵) → ((𝑓(*𝑝𝐽)𝐹)‘0) = (𝑓‘0))
11028adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝐵) → (𝐼‘1) = (𝐹‘0))
11160, 109, 1103eqtr4rd 2666 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵) → (𝐼‘1) = ((𝑓(*𝑝𝐽)𝐹)‘0))
1121113adant3 1079 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → (𝐼‘1) = ((𝑓(*𝑝𝐽)𝐹)‘0))
11354, 68pco1 22728 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽)𝐹)‘1) = (𝐹‘1))
114113, 100eqtr4d 2658 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽)𝐹)‘1) = ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))‘0))
11572, 108, 98, 112, 114, 75pcoass 22737 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)))( ≃ph𝐽)(𝐼(*𝑝𝐽)((𝑓(*𝑝𝐽)𝐹)(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)))))
11650, 105, 115ertr4d 7709 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → (𝐼(*𝑝𝐽)((𝑓(*𝑝𝐽))(*𝑝𝐽)𝐹))( ≃ph𝐽)((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))))
11750, 116erthi 7741 . . . . . . . . . . . 12 ((𝜑𝑓 𝐵 𝐵) → [(𝐼(*𝑝𝐽)((𝑓(*𝑝𝐽))(*𝑝𝐽)𝐹))]( ≃ph𝐽) = [((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)))]( ≃ph𝐽))
11813ad2ant1 1080 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
11954, 58pco1 22728 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽))‘1) = (‘1))
120119, 74eqtrd 2655 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽))‘1) = (𝐹‘0))
12110, 1, 9, 30pi1eluni 22755 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑓(*𝑝𝐽)) ∈ 𝐵 ↔ ((𝑓(*𝑝𝐽)) ∈ (II Cn 𝐽) ∧ ((𝑓(*𝑝𝐽))‘0) = (𝐹‘0) ∧ ((𝑓(*𝑝𝐽))‘1) = (𝐹‘0))))
1221213ad2ant1 1080 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → ((𝑓(*𝑝𝐽)) ∈ 𝐵 ↔ ((𝑓(*𝑝𝐽)) ∈ (II Cn 𝐽) ∧ ((𝑓(*𝑝𝐽))‘0) = (𝐹‘0) ∧ ((𝑓(*𝑝𝐽))‘1) = (𝐹‘0))))
12367, 62, 120, 122mpbir3and 1243 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → (𝑓(*𝑝𝐽)) ∈ 𝐵)
12410, 16, 20, 21, 118, 68, 72, 94, 70, 123pi1xfrval 22767 . . . . . . . . . . . 12 ((𝜑𝑓 𝐵 𝐵) → (𝐺‘[(𝑓(*𝑝𝐽))]( ≃ph𝐽)) = [(𝐼(*𝑝𝐽)((𝑓(*𝑝𝐽))(*𝑝𝐽)𝐹))]( ≃ph𝐽))
125 eqid 2621 . . . . . . . . . . . . 13 (Base‘𝑄) = (Base‘𝑄)
126153ad2ant1 1080 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → (𝐹‘1) ∈ 𝑋)
127 eqid 2621 . . . . . . . . . . . . 13 (+g𝑄) = (+g𝑄)
12825adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵) → 𝐼 ∈ (II Cn 𝐽))
129128, 107, 111pcocn 22730 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵) → (𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹)) ∈ (II Cn 𝐽))
1301293adant3 1079 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → (𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹)) ∈ (II Cn 𝐽))
131128, 107pco0 22727 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵) → ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))‘0) = (𝐼‘0))
13226adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵) → (𝐼‘0) = (𝐹‘1))
133131, 132eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵) → ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))‘0) = (𝐹‘1))
1341333adant3 1079 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))‘0) = (𝐹‘1))
135128, 107pco1 22728 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵) → ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))‘1) = ((𝑓(*𝑝𝐽)𝐹)‘1))
13653, 106pco1 22728 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝐵) → ((𝑓(*𝑝𝐽)𝐹)‘1) = (𝐹‘1))
137135, 136eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵) → ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))‘1) = (𝐹‘1))
1381373adant3 1079 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))‘1) = (𝐹‘1))
139 eqidd 2622 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → (Base‘𝑄) = (Base‘𝑄))
14016, 118, 126, 139pi1eluni 22755 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹)) ∈ (Base‘𝑄) ↔ ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))‘1) = (𝐹‘1))))
141130, 134, 138, 140mpbir3and 1243 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → (𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹)) ∈ (Base‘𝑄))
14272, 86pco1 22728 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))‘1) = (((*𝑝𝐽)𝐹)‘1))
14358, 68pco1 22728 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝐵 𝐵) → (((*𝑝𝐽)𝐹)‘1) = (𝐹‘1))
144142, 143eqtrd 2655 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))‘1) = (𝐹‘1))
14516, 118, 126, 139pi1eluni 22755 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵 𝐵) → ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)) ∈ (Base‘𝑄) ↔ ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)) ∈ (II Cn 𝐽) ∧ ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))‘0) = (𝐹‘1) ∧ ((𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))‘1) = (𝐹‘1))))
14698, 100, 144, 145mpbir3and 1243 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → (𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)) ∈ (Base‘𝑄))
14716, 125, 118, 126, 127, 141, 146pi1addval 22761 . . . . . . . . . . . 12 ((𝜑𝑓 𝐵 𝐵) → ([(𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))]( ≃ph𝐽)(+g𝑄)[(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))]( ≃ph𝐽)) = [((𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))(*𝑝𝐽)(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹)))]( ≃ph𝐽))
148117, 124, 1473eqtr4d 2665 . . . . . . . . . . 11 ((𝜑𝑓 𝐵 𝐵) → (𝐺‘[(𝑓(*𝑝𝐽))]( ≃ph𝐽)) = ([(𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))]( ≃ph𝐽)(+g𝑄)[(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))]( ≃ph𝐽)))
14993ad2ant1 1080 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → (𝐹‘0) ∈ 𝑋)
150 eqid 2621 . . . . . . . . . . . . 13 (+g𝑃) = (+g𝑃)
151 simp2 1060 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → 𝑓 𝐵)
152 simp3 1061 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵 𝐵) → 𝐵)
15310, 20, 118, 149, 150, 151, 152pi1addval 22761 . . . . . . . . . . . 12 ((𝜑𝑓 𝐵 𝐵) → ([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽)) = [(𝑓(*𝑝𝐽))]( ≃ph𝐽))
154153fveq2d 6154 . . . . . . . . . . 11 ((𝜑𝑓 𝐵 𝐵) → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = (𝐺‘[(𝑓(*𝑝𝐽))]( ≃ph𝐽)))
1551adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵) → 𝐽 ∈ (TopOn‘𝑋))
15627adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵) → (𝐹‘1) = (𝐼‘0))
157 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝐵) → 𝑓 𝐵)
15810, 16, 20, 21, 155, 106, 128, 156, 110, 157pi1xfrval 22767 . . . . . . . . . . . . 13 ((𝜑𝑓 𝐵) → (𝐺‘[𝑓]( ≃ph𝐽)) = [(𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))]( ≃ph𝐽))
1591583adant3 1079 . . . . . . . . . . . 12 ((𝜑𝑓 𝐵 𝐵) → (𝐺‘[𝑓]( ≃ph𝐽)) = [(𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))]( ≃ph𝐽))
16010, 16, 20, 21, 118, 68, 72, 94, 70, 152pi1xfrval 22767 . . . . . . . . . . . 12 ((𝜑𝑓 𝐵 𝐵) → (𝐺‘[]( ≃ph𝐽)) = [(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))]( ≃ph𝐽))
161159, 160oveq12d 6625 . . . . . . . . . . 11 ((𝜑𝑓 𝐵 𝐵) → ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))) = ([(𝐼(*𝑝𝐽)(𝑓(*𝑝𝐽)𝐹))]( ≃ph𝐽)(+g𝑄)[(𝐼(*𝑝𝐽)((*𝑝𝐽)𝐹))]( ≃ph𝐽)))
162148, 154, 1613eqtr4d 2665 . . . . . . . . . 10 ((𝜑𝑓 𝐵 𝐵) → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))))
1631623expa 1262 . . . . . . . . 9 (((𝜑𝑓 𝐵) ∧ 𝐵) → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))))
16434, 48, 163ectocld 7762 . . . . . . . 8 (((𝜑𝑓 𝐵) ∧ 𝑧 ∈ ( 𝐵 / ( ≃ph𝐽))) → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)))
16543, 164syldan 487 . . . . . . 7 (((𝜑𝑓 𝐵) ∧ 𝑧𝐵) → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)))
166165ralrimiva 2960 . . . . . 6 ((𝜑𝑓 𝐵) → ∀𝑧𝐵 (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)))
16734, 40, 166ectocld 7762 . . . . 5 ((𝜑𝑦 ∈ ( 𝐵 / ( ≃ph𝐽))) → ∀𝑧𝐵 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))
16833, 167syldan 487 . . . 4 ((𝜑𝑦𝐵) → ∀𝑧𝐵 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))
169168ralrimiva 2960 . . 3 (𝜑 → ∀𝑦𝐵𝑧𝐵 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))
17029, 169jca 554 . 2 (𝜑 → (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦𝐵𝑧𝐵 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧))))
17120, 125, 150, 127isghm 17584 . 2 (𝐺 ∈ (𝑃 GrpHom 𝑄) ↔ ((𝑃 ∈ Grp ∧ 𝑄 ∈ Grp) ∧ (𝐺:𝐵⟶(Base‘𝑄) ∧ ∀𝑦𝐵𝑧𝐵 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))))
17219, 170, 171sylanbrc 697 1 (𝜑𝐺 ∈ (𝑃 GrpHom 𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  ifcif 4060  {csn 4150  cop 4156   cuni 4404   class class class wbr 4615  cmpt 4675   × cxp 5074  ran crn 5077  wf 5845  cfv 5849  (class class class)co 6607   Er wer 7687  [cec 7688   / cqs 7689  0cc0 9883  1c1 9884   + caddc 9886   · cmul 9888  cle 10022  cmin 10213   / cdiv 10631  2c2 11017  4c4 11019  [,]cicc 12123  Basecbs 15784  +gcplusg 15865  Grpcgrp 17346   GrpHom cghm 17581  TopOnctopon 20637   Cn ccn 20941  IIcii 22591  phcphtpc 22681  *𝑝cpco 22713   π1 cpi1 22716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961  ax-addf 9962  ax-mulf 9963
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-of 6853  df-om 7016  df-1st 7116  df-2nd 7117  df-supp 7244  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-ec 7692  df-qs 7696  df-map 7807  df-ixp 7856  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-fsupp 8223  df-fi 8264  df-sup 8295  df-inf 8296  df-oi 8362  df-card 8712  df-cda 8937  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-7 11031  df-8 11032  df-9 11033  df-n0 11240  df-z 11325  df-dec 11441  df-uz 11635  df-q 11736  df-rp 11780  df-xneg 11893  df-xadd 11894  df-xmul 11895  df-ioo 12124  df-icc 12127  df-fz 12272  df-fzo 12410  df-seq 12745  df-exp 12804  df-hash 13061  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-mulr 15879  df-starv 15880  df-sca 15881  df-vsca 15882  df-ip 15883  df-tset 15884  df-ple 15885  df-ds 15888  df-unif 15889  df-hom 15890  df-cco 15891  df-rest 16007  df-topn 16008  df-0g 16026  df-gsum 16027  df-topgen 16028  df-pt 16029  df-prds 16032  df-xrs 16086  df-qtop 16091  df-imas 16092  df-qus 16093  df-xps 16094  df-mre 16170  df-mrc 16171  df-acs 16173  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-submnd 17260  df-grp 17349  df-mulg 17465  df-ghm 17582  df-cntz 17674  df-cmn 18119  df-psmet 19660  df-xmet 19661  df-met 19662  df-bl 19663  df-mopn 19664  df-cnfld 19669  df-top 20621  df-topon 20638  df-topsp 20651  df-bases 20664  df-cld 20736  df-cn 20944  df-cnp 20945  df-tx 21278  df-hmeo 21471  df-xms 22038  df-ms 22039  df-tms 22040  df-ii 22593  df-htpy 22682  df-phtpy 22683  df-phtpc 22704  df-pco 22718  df-om1 22719  df-pi1 22721
This theorem is referenced by:  pi1xfrcnv  22770  pi1xfrgim  22771
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