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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 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 3758 | . . 3 ⊢ (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd)) | |
2 | 1 | anbi1i 727 | . 2 ⊢ ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
3 | fvex 6113 | . . . . 5 ⊢ (TopOpen‘𝑓) ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V) |
5 | simpl 472 | . . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺) | |
6 | 5 | fveq2d 6107 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = (invg‘𝐺)) |
7 | istgp.2 | . . . . . 6 ⊢ 𝐼 = (invg‘𝐺) | |
8 | 6, 7 | syl6eqr 2662 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = 𝐼) |
9 | id 22 | . . . . . . 7 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓)) | |
10 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺)) | |
11 | istgp.1 | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺) | |
12 | 10, 11 | syl6eqr 2662 | . . . . . . 7 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽) |
13 | 9, 12 | sylan9eqr 2666 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽) |
14 | 13, 13 | oveq12d 6567 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽)) |
15 | 8, 14 | eleq12d 2682 | . . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽))) |
16 | 4, 15 | sbcied 3439 | . . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽))) |
17 | df-tgp 21687 | . . 3 ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)} | |
18 | 16, 17 | elrab2 3333 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
19 | df-3an 1033 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) | |
20 | 2, 18, 19 | 3bitr4i 291 | 1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 ∩ cin 3539 ‘cfv 5804 (class class class)co 6549 TopOpenctopn 15905 Grpcgrp 17245 invgcminusg 17246 Cn ccn 20838 TopMndctmd 21684 TopGrpctgp 21685 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-tgp 21687 |
This theorem is referenced by: tgpgrp 21692 tgptmd 21693 tgpinv 21699 istgp2 21705 oppgtgp 21712 symgtgp 21715 subgtgp 21719 prdstgpd 21738 tlmtgp 21809 nrgtdrg 22307 |
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