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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 3796 | . . 3 ⊢ (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd)) | |
2 | 1 | anbi1i 731 | . 2 ⊢ ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
3 | fvexd 6203 | . . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V) | |
4 | simpl 473 | . . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺) | |
5 | 4 | fveq2d 6195 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = (invg‘𝐺)) |
6 | istgp.2 | . . . . . 6 ⊢ 𝐼 = (invg‘𝐺) | |
7 | 5, 6 | syl6eqr 2674 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (invg‘𝑓) = 𝐼) |
8 | id 22 | . . . . . . 7 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓)) | |
9 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺)) | |
10 | istgp.1 | . . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺) | |
11 | 9, 10 | syl6eqr 2674 | . . . . . . 7 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽) |
12 | 8, 11 | sylan9eqr 2678 | . . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽) |
13 | 12, 12 | oveq12d 6668 | . . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽)) |
14 | 7, 13 | eleq12d 2695 | . . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽))) |
15 | 3, 14 | sbcied 3472 | . . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽))) |
16 | df-tgp 21877 | . . 3 ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)} | |
17 | 15, 16 | elrab2 3366 | . 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
18 | df-3an 1039 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) | |
19 | 2, 17, 18 | 3bitr4i 292 | 1 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 ∩ cin 3573 ‘cfv 5888 (class class class)co 6650 TopOpenctopn 16082 Grpcgrp 17422 invgcminusg 17423 Cn ccn 21028 TopMndctmd 21874 TopGrpctgp 21875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-tgp 21877 |
This theorem is referenced by: tgpgrp 21882 tgptmd 21883 tgpinv 21889 istgp2 21895 oppgtgp 21902 symgtgp 21905 subgtgp 21909 prdstgpd 21928 tlmtgp 21999 nrgtdrg 22497 |
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