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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 3967 | . . 3 ⊢ (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd)) | |
| 2 | 1 | anbi1i 624 | . 2 ⊢ ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
| 3 | fvexd 6921 | . . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V) | |
| 4 | simpl 482 | . . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺) | |
| 5 | 4 | fveq2d 6910 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = (invg‘𝐺)) |
| 6 | istgp.2 | . . . . . 6 ⊢ 𝐼 = (invg‘𝐺) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = 𝐼) |
| 8 | id 22 | . . . . . . 7 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓)) | |
| 9 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺)) | |
| 10 | istgp.1 | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺) | |
| 11 | 9, 10 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽) |
| 12 | 8, 11 | sylan9eqr 2799 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽) |
| 13 | 12, 12 | oveq12d 7449 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽)) |
| 14 | 7, 13 | eleq12d 2835 | . . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽))) |
| 15 | 3, 14 | sbcied 3832 | . . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽))) |
| 16 | df-tgp 24081 | . . 3 ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)} | |
| 17 | 15, 16 | elrab2 3695 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
| 18 | df-3an 1089 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) | |
| 19 | 2, 17, 18 | 3bitr4i 303 | 1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 ∩ cin 3950 ‘cfv 6561 (class class class)co 7431 TopOpenctopn 17466 Grpcgrp 18951 invgcminusg 18952 Cn ccn 23232 TopMndctmd 24078 TopGrpctgp 24079 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-tgp 24081 |
| This theorem is referenced by: tgpgrp 24086 tgptmd 24087 tgpinv 24093 istgp2 24099 oppgtgp 24106 subgtgp 24113 symgtgp 24114 prdstgpd 24133 tlmtgp 24204 nrgtdrg 24714 |
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