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Mirrors > Home > MPE Home > Th. List > istgp | Structured version Visualization version GIF version |
Description: The predicate "is a topological group". Definition 1 of [BourbakiTop1] p. III.1. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
istgp.1 | ⊢ 𝐽 = (TopOpen‘𝐺) |
istgp.2 | ⊢ 𝐼 = (invg‘𝐺) |
Ref | Expression |
---|---|
istgp | ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4172 | . . 3 ⊢ (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd)) | |
2 | 1 | anbi1i 625 | . 2 ⊢ ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
3 | fvexd 6688 | . . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V) | |
4 | simpl 485 | . . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺) | |
5 | 4 | fveq2d 6677 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = (invg‘𝐺)) |
6 | istgp.2 | . . . . . 6 ⊢ 𝐼 = (invg‘𝐺) | |
7 | 5, 6 | syl6eqr 2877 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = 𝐼) |
8 | id 22 | . . . . . . 7 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓)) | |
9 | fveq2 6673 | . . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺)) | |
10 | istgp.1 | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺) | |
11 | 9, 10 | syl6eqr 2877 | . . . . . . 7 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽) |
12 | 8, 11 | sylan9eqr 2881 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽) |
13 | 12, 12 | oveq12d 7177 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽)) |
14 | 7, 13 | eleq12d 2910 | . . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽))) |
15 | 3, 14 | sbcied 3817 | . . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽))) |
16 | df-tgp 22684 | . . 3 ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)} | |
17 | 15, 16 | elrab2 3686 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
18 | df-3an 1085 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) | |
19 | 2, 17, 18 | 3bitr4i 305 | 1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 Vcvv 3497 [wsbc 3775 ∩ cin 3938 ‘cfv 6358 (class class class)co 7159 TopOpenctopn 16698 Grpcgrp 18106 invgcminusg 18107 Cn ccn 21835 TopMndctmd 22681 TopGrpctgp 22682 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-tgp 22684 |
This theorem is referenced by: tgpgrp 22689 tgptmd 22690 tgpinv 22696 istgp2 22702 oppgtgp 22709 subgtgp 22716 symgtgp 22717 prdstgpd 22736 tlmtgp 22807 nrgtdrg 23305 |
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