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Theorem pi1coghm 22669
 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]( ≃ph𝐽), [(𝐹𝑔)]( ≃ph𝐾)⟩)
pi1co.j (𝜑𝐽 ∈ (TopOn‘𝑋))
pi1co.f (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
pi1co.a (𝜑𝐴𝑋)
pi1co.b (𝜑 → (𝐹𝐴) = 𝐵)
Assertion
Ref Expression
pi1coghm (𝜑𝐺 ∈ (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 𝑓 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pi1co.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 pi1co.a . . . 4 (𝜑𝐴𝑋)
3 pi1co.p . . . . 5 𝑃 = (𝐽 π1 𝐴)
43pi1grp 22658 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑃 ∈ Grp)
51, 2, 4syl2anc 691 . . 3 (𝜑𝑃 ∈ Grp)
6 pi1co.f . . . . . 6 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
7 cntop2 20855 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
86, 7syl 17 . . . . 5 (𝜑𝐾 ∈ Top)
9 eqid 2610 . . . . . 6 𝐾 = 𝐾
109toptopon 20548 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
118, 10sylib 207 . . . 4 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
12 pi1co.b . . . . 5 (𝜑 → (𝐹𝐴) = 𝐵)
13 cnf2 20863 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋 𝐾)
141, 11, 6, 13syl3anc 1318 . . . . . 6 (𝜑𝐹:𝑋 𝐾)
1514, 2ffvelrnd 6268 . . . . 5 (𝜑 → (𝐹𝐴) ∈ 𝐾)
1612, 15eqeltrrd 2689 . . . 4 (𝜑𝐵 𝐾)
17 pi1co.q . . . . 5 𝑄 = (𝐾 π1 𝐵)
1817pi1grp 22658 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ 𝐵 𝐾) → 𝑄 ∈ Grp)
1911, 16, 18syl2anc 691 . . 3 (𝜑𝑄 ∈ Grp)
205, 19jca 553 . 2 (𝜑 → (𝑃 ∈ Grp ∧ 𝑄 ∈ Grp))
21 pi1co.v . . . 4 𝑉 = (Base‘𝑃)
22 pi1co.g . . . 4 𝐺 = ran (𝑔 𝑉 ↦ ⟨[𝑔]( ≃ph𝐽), [(𝐹𝑔)]( ≃ph𝐾)⟩)
233, 17, 21, 22, 1, 6, 2, 12pi1cof 22667 . . 3 (𝜑𝐺:𝑉⟶(Base‘𝑄))
2421a1i 11 . . . . . . . 8 (𝜑𝑉 = (Base‘𝑃))
253, 1, 2, 24pi1bas2 22649 . . . . . . 7 (𝜑𝑉 = ( 𝑉 / ( ≃ph𝐽)))
2625eleq2d 2673 . . . . . 6 (𝜑 → (𝑦𝑉𝑦 ∈ ( 𝑉 / ( ≃ph𝐽))))
2726biimpa 500 . . . . 5 ((𝜑𝑦𝑉) → 𝑦 ∈ ( 𝑉 / ( ≃ph𝐽)))
28 eqid 2610 . . . . . 6 ( 𝑉 / ( ≃ph𝐽)) = ( 𝑉 / ( ≃ph𝐽))
29 oveq1 6556 . . . . . . . . 9 ([𝑓]( ≃ph𝐽) = 𝑦 → ([𝑓]( ≃ph𝐽)(+g𝑃)𝑧) = (𝑦(+g𝑃)𝑧))
3029fveq2d 6107 . . . . . . . 8 ([𝑓]( ≃ph𝐽) = 𝑦 → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = (𝐺‘(𝑦(+g𝑃)𝑧)))
31 fveq2 6103 . . . . . . . . 9 ([𝑓]( ≃ph𝐽) = 𝑦 → (𝐺‘[𝑓]( ≃ph𝐽)) = (𝐺𝑦))
3231oveq1d 6564 . . . . . . . 8 ([𝑓]( ≃ph𝐽) = 𝑦 → ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))
3330, 32eqeq12d 2625 . . . . . . 7 ([𝑓]( ≃ph𝐽) = 𝑦 → ((𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)) ↔ (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧))))
3433ralbidv 2969 . . . . . 6 ([𝑓]( ≃ph𝐽) = 𝑦 → (∀𝑧𝑉 (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)) ↔ ∀𝑧𝑉 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧))))
35 oveq2 6557 . . . . . . . . . . 11 ([]( ≃ph𝐽) = 𝑧 → ([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽)) = ([𝑓]( ≃ph𝐽)(+g𝑃)𝑧))
3635fveq2d 6107 . . . . . . . . . 10 ([]( ≃ph𝐽) = 𝑧 → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)))
37 fveq2 6103 . . . . . . . . . . 11 ([]( ≃ph𝐽) = 𝑧 → (𝐺‘[]( ≃ph𝐽)) = (𝐺𝑧))
3837oveq2d 6565 . . . . . . . . . 10 ([]( ≃ph𝐽) = 𝑧 → ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)))
3936, 38eqeq12d 2625 . . . . . . . . 9 ([]( ≃ph𝐽) = 𝑧 → ((𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))) ↔ (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧))))
403, 1, 2, 24pi1eluni 22650 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑓 𝑉 ↔ (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐴)))
4140biimpa 500 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → (𝑓 ∈ (II Cn 𝐽) ∧ (𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐴))
4241simp1d 1066 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝑉) → 𝑓 ∈ (II Cn 𝐽))
4342adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝑓 ∈ (II Cn 𝐽))
441adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝑉) → 𝐽 ∈ (TopOn‘𝑋))
452adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝑉) → 𝐴𝑋)
4621a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝑉) → 𝑉 = (Base‘𝑃))
473, 44, 45, 46pi1eluni 22650 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → ( 𝑉 ↔ ( ∈ (II Cn 𝐽) ∧ (‘0) = 𝐴 ∧ (‘1) = 𝐴)))
4847biimpa 500 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → ( ∈ (II Cn 𝐽) ∧ (‘0) = 𝐴 ∧ (‘1) = 𝐴))
4948simp1d 1066 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → ∈ (II Cn 𝐽))
5041simp3d 1068 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → (𝑓‘1) = 𝐴)
5150adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝑓‘1) = 𝐴)
5248simp2d 1067 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → (‘0) = 𝐴)
5351, 52eqtr4d 2647 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝑓‘1) = (‘0))
546ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝐹 ∈ (𝐽 Cn 𝐾))
5543, 49, 53, 54copco 22626 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐹 ∘ (𝑓(*𝑝𝐽))) = ((𝐹𝑓)(*𝑝𝐾)(𝐹)))
5655eceq1d 7670 . . . . . . . . . . 11 (((𝜑𝑓 𝑉) ∧ 𝑉) → [(𝐹 ∘ (𝑓(*𝑝𝐽)))]( ≃ph𝐾) = [((𝐹𝑓)(*𝑝𝐾)(𝐹))]( ≃ph𝐾))
5743, 49, 53pcocn 22625 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝑓(*𝑝𝐽)) ∈ (II Cn 𝐽))
5843, 49pco0 22622 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝑓(*𝑝𝐽))‘0) = (𝑓‘0))
5941simp2d 1067 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → (𝑓‘0) = 𝐴)
6059adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝑓‘0) = 𝐴)
6158, 60eqtrd 2644 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝑓(*𝑝𝐽))‘0) = 𝐴)
6243, 49pco1 22623 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝑓(*𝑝𝐽))‘1) = (‘1))
6348simp3d 1068 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → (‘1) = 𝐴)
6462, 63eqtrd 2644 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝑓(*𝑝𝐽))‘1) = 𝐴)
651ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝐽 ∈ (TopOn‘𝑋))
662ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝐴𝑋)
6721a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝑉 = (Base‘𝑃))
683, 65, 66, 67pi1eluni 22650 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝑓(*𝑝𝐽)) ∈ 𝑉 ↔ ((𝑓(*𝑝𝐽)) ∈ (II Cn 𝐽) ∧ ((𝑓(*𝑝𝐽))‘0) = 𝐴 ∧ ((𝑓(*𝑝𝐽))‘1) = 𝐴)))
6957, 61, 64, 68mpbir3and 1238 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝑓(*𝑝𝐽)) ∈ 𝑉)
703, 17, 21, 22, 1, 6, 2, 12pi1coval 22668 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓(*𝑝𝐽)) ∈ 𝑉) → (𝐺‘[(𝑓(*𝑝𝐽))]( ≃ph𝐽)) = [(𝐹 ∘ (𝑓(*𝑝𝐽)))]( ≃ph𝐾))
7170adantlr 747 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ (𝑓(*𝑝𝐽)) ∈ 𝑉) → (𝐺‘[(𝑓(*𝑝𝐽))]( ≃ph𝐽)) = [(𝐹 ∘ (𝑓(*𝑝𝐽)))]( ≃ph𝐾))
7269, 71syldan 486 . . . . . . . . . . 11 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐺‘[(𝑓(*𝑝𝐽))]( ≃ph𝐽)) = [(𝐹 ∘ (𝑓(*𝑝𝐽)))]( ≃ph𝐾))
73 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑄) = (Base‘𝑄)
7411ad2antrr 758 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝐾 ∈ (TopOn‘ 𝐾))
7516ad2antrr 758 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝐵 𝐾)
76 eqid 2610 . . . . . . . . . . . 12 (+g𝑄) = (+g𝑄)
776adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → 𝐹 ∈ (𝐽 Cn 𝐾))
78 cnco 20880 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹𝑓) ∈ (II Cn 𝐾))
7942, 77, 78syl2anc 691 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝑉) → (𝐹𝑓) ∈ (II Cn 𝐾))
80 iitopon 22490 . . . . . . . . . . . . . . . . . 18 II ∈ (TopOn‘(0[,]1))
8180a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑓 𝑉) → II ∈ (TopOn‘(0[,]1)))
82 cnf2 20863 . . . . . . . . . . . . . . . . 17 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (II Cn 𝐽)) → 𝑓:(0[,]1)⟶𝑋)
8381, 44, 42, 82syl3anc 1318 . . . . . . . . . . . . . . . 16 ((𝜑𝑓 𝑉) → 𝑓:(0[,]1)⟶𝑋)
84 0elunit 12161 . . . . . . . . . . . . . . . 16 0 ∈ (0[,]1)
85 fvco3 6185 . . . . . . . . . . . . . . . 16 ((𝑓:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → ((𝐹𝑓)‘0) = (𝐹‘(𝑓‘0)))
8683, 84, 85sylancl 693 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → ((𝐹𝑓)‘0) = (𝐹‘(𝑓‘0)))
8759fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → (𝐹‘(𝑓‘0)) = (𝐹𝐴))
8812adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → (𝐹𝐴) = 𝐵)
8986, 87, 883eqtrd 2648 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝑉) → ((𝐹𝑓)‘0) = 𝐵)
90 1elunit 12162 . . . . . . . . . . . . . . . 16 1 ∈ (0[,]1)
91 fvco3 6185 . . . . . . . . . . . . . . . 16 ((𝑓:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → ((𝐹𝑓)‘1) = (𝐹‘(𝑓‘1)))
9283, 90, 91sylancl 693 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → ((𝐹𝑓)‘1) = (𝐹‘(𝑓‘1)))
9350fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → (𝐹‘(𝑓‘1)) = (𝐹𝐴))
9492, 93, 883eqtrd 2648 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝑉) → ((𝐹𝑓)‘1) = 𝐵)
9511adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → 𝐾 ∈ (TopOn‘ 𝐾))
9616adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → 𝐵 𝐾)
97 eqidd 2611 . . . . . . . . . . . . . . 15 ((𝜑𝑓 𝑉) → (Base‘𝑄) = (Base‘𝑄))
9817, 95, 96, 97pi1eluni 22650 . . . . . . . . . . . . . 14 ((𝜑𝑓 𝑉) → ((𝐹𝑓) ∈ (Base‘𝑄) ↔ ((𝐹𝑓) ∈ (II Cn 𝐾) ∧ ((𝐹𝑓)‘0) = 𝐵 ∧ ((𝐹𝑓)‘1) = 𝐵)))
9979, 89, 94, 98mpbir3and 1238 . . . . . . . . . . . . 13 ((𝜑𝑓 𝑉) → (𝐹𝑓) ∈ (Base‘𝑄))
10099adantr 480 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐹𝑓) ∈ (Base‘𝑄))
101 cnco 20880 . . . . . . . . . . . . . 14 (( ∈ (II Cn 𝐽) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹) ∈ (II Cn 𝐾))
10249, 54, 101syl2anc 691 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐹) ∈ (II Cn 𝐾))
10380a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑓 𝑉) ∧ 𝑉) → II ∈ (TopOn‘(0[,]1)))
104 cnf2 20863 . . . . . . . . . . . . . . . 16 ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐽 ∈ (TopOn‘𝑋) ∧ ∈ (II Cn 𝐽)) → :(0[,]1)⟶𝑋)
105103, 65, 49, 104syl3anc 1318 . . . . . . . . . . . . . . 15 (((𝜑𝑓 𝑉) ∧ 𝑉) → :(0[,]1)⟶𝑋)
106 fvco3 6185 . . . . . . . . . . . . . . 15 ((:(0[,]1)⟶𝑋 ∧ 0 ∈ (0[,]1)) → ((𝐹)‘0) = (𝐹‘(‘0)))
107105, 84, 106sylancl 693 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝐹)‘0) = (𝐹‘(‘0)))
10852fveq2d 6107 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐹‘(‘0)) = (𝐹𝐴))
10912ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐹𝐴) = 𝐵)
110107, 108, 1093eqtrd 2648 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝐹)‘0) = 𝐵)
111 fvco3 6185 . . . . . . . . . . . . . . 15 ((:(0[,]1)⟶𝑋 ∧ 1 ∈ (0[,]1)) → ((𝐹)‘1) = (𝐹‘(‘1)))
112105, 90, 111sylancl 693 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝐹)‘1) = (𝐹‘(‘1)))
11363fveq2d 6107 . . . . . . . . . . . . . 14 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐹‘(‘1)) = (𝐹𝐴))
114112, 113, 1093eqtrd 2648 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝐹)‘1) = 𝐵)
115 eqidd 2611 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘𝑄) = (Base‘𝑄))
11617, 11, 16, 115pi1eluni 22650 . . . . . . . . . . . . . 14 (𝜑 → ((𝐹) ∈ (Base‘𝑄) ↔ ((𝐹) ∈ (II Cn 𝐾) ∧ ((𝐹)‘0) = 𝐵 ∧ ((𝐹)‘1) = 𝐵)))
117116ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝐹) ∈ (Base‘𝑄) ↔ ((𝐹) ∈ (II Cn 𝐾) ∧ ((𝐹)‘0) = 𝐵 ∧ ((𝐹)‘1) = 𝐵)))
118102, 110, 114, 117mpbir3and 1238 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐹) ∈ (Base‘𝑄))
11917, 73, 74, 75, 76, 100, 118pi1addval 22656 . . . . . . . . . . 11 (((𝜑𝑓 𝑉) ∧ 𝑉) → ([(𝐹𝑓)]( ≃ph𝐾)(+g𝑄)[(𝐹)]( ≃ph𝐾)) = [((𝐹𝑓)(*𝑝𝐾)(𝐹))]( ≃ph𝐾))
12056, 72, 1193eqtr4d 2654 . . . . . . . . . 10 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐺‘[(𝑓(*𝑝𝐽))]( ≃ph𝐽)) = ([(𝐹𝑓)]( ≃ph𝐾)(+g𝑄)[(𝐹)]( ≃ph𝐾)))
121 eqid 2610 . . . . . . . . . . . 12 (+g𝑃) = (+g𝑃)
122 simplr 788 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝑓 𝑉)
123 simpr 476 . . . . . . . . . . . 12 (((𝜑𝑓 𝑉) ∧ 𝑉) → 𝑉)
1243, 21, 65, 66, 121, 122, 123pi1addval 22656 . . . . . . . . . . 11 (((𝜑𝑓 𝑉) ∧ 𝑉) → ([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽)) = [(𝑓(*𝑝𝐽))]( ≃ph𝐽))
125124fveq2d 6107 . . . . . . . . . 10 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = (𝐺‘[(𝑓(*𝑝𝐽))]( ≃ph𝐽)))
1263, 17, 21, 22, 1, 6, 2, 12pi1coval 22668 . . . . . . . . . . . 12 ((𝜑𝑓 𝑉) → (𝐺‘[𝑓]( ≃ph𝐽)) = [(𝐹𝑓)]( ≃ph𝐾))
127126adantr 480 . . . . . . . . . . 11 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐺‘[𝑓]( ≃ph𝐽)) = [(𝐹𝑓)]( ≃ph𝐾))
1283, 17, 21, 22, 1, 6, 2, 12pi1coval 22668 . . . . . . . . . . . 12 ((𝜑 𝑉) → (𝐺‘[]( ≃ph𝐽)) = [(𝐹)]( ≃ph𝐾))
129128adantlr 747 . . . . . . . . . . 11 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐺‘[]( ≃ph𝐽)) = [(𝐹)]( ≃ph𝐾))
130127, 129oveq12d 6567 . . . . . . . . . 10 (((𝜑𝑓 𝑉) ∧ 𝑉) → ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))) = ([(𝐹𝑓)]( ≃ph𝐾)(+g𝑄)[(𝐹)]( ≃ph𝐾)))
131120, 125, 1303eqtr4d 2654 . . . . . . . . 9 (((𝜑𝑓 𝑉) ∧ 𝑉) → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)[]( ≃ph𝐽))) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺‘[]( ≃ph𝐽))))
13228, 39, 131ectocld 7701 . . . . . . . 8 (((𝜑𝑓 𝑉) ∧ 𝑧 ∈ ( 𝑉 / ( ≃ph𝐽))) → (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)))
133132ralrimiva 2949 . . . . . . 7 ((𝜑𝑓 𝑉) → ∀𝑧 ∈ ( 𝑉 / ( ≃ph𝐽))(𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)))
13425adantr 480 . . . . . . . 8 ((𝜑𝑓 𝑉) → 𝑉 = ( 𝑉 / ( ≃ph𝐽)))
135134raleqdv 3121 . . . . . . 7 ((𝜑𝑓 𝑉) → (∀𝑧𝑉 (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)) ↔ ∀𝑧 ∈ ( 𝑉 / ( ≃ph𝐽))(𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧))))
136133, 135mpbird 246 . . . . . 6 ((𝜑𝑓 𝑉) → ∀𝑧𝑉 (𝐺‘([𝑓]( ≃ph𝐽)(+g𝑃)𝑧)) = ((𝐺‘[𝑓]( ≃ph𝐽))(+g𝑄)(𝐺𝑧)))
13728, 34, 136ectocld 7701 . . . . 5 ((𝜑𝑦 ∈ ( 𝑉 / ( ≃ph𝐽))) → ∀𝑧𝑉 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))
13827, 137syldan 486 . . . 4 ((𝜑𝑦𝑉) → ∀𝑧𝑉 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))
139138ralrimiva 2949 . . 3 (𝜑 → ∀𝑦𝑉𝑧𝑉 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))
14023, 139jca 553 . 2 (𝜑 → (𝐺:𝑉⟶(Base‘𝑄) ∧ ∀𝑦𝑉𝑧𝑉 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧))))
14121, 73, 121, 76isghm 17483 . 2 (𝐺 ∈ (𝑃 GrpHom 𝑄) ↔ ((𝑃 ∈ Grp ∧ 𝑄 ∈ Grp) ∧ (𝐺:𝑉⟶(Base‘𝑄) ∧ ∀𝑦𝑉𝑧𝑉 (𝐺‘(𝑦(+g𝑃)𝑧)) = ((𝐺𝑦)(+g𝑄)(𝐺𝑧)))))
14220, 140, 141sylanbrc 695 1 (𝜑𝐺 ∈ (𝑃 GrpHom 𝑄))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643  ran crn 5039   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  [cec 7627   / cqs 7628  0cc0 9815  1c1 9816  [,]cicc 12049  Basecbs 15695  +gcplusg 15768  Grpcgrp 17245   GrpHom cghm 17480  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  IIcii 22486   ≃phcphtpc 22576  *𝑝cpco 22608   π1 cpi1 22611 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-qus 15992  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-mulg 17364  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-cn 20841  df-cnp 20842  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937  df-ii 22488  df-htpy 22577  df-phtpy 22578  df-phtpc 22599  df-pco 22613  df-om1 22614  df-pi1 22616 This theorem is referenced by: (None)
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