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Theorem istgp 23451
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.)
Hypotheses
Ref Expression
istgp.1 𝐽 = (TopOpenβ€˜πΊ)
istgp.2 𝐼 = (invgβ€˜πΊ)
Assertion
Ref Expression
istgp (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))

Proof of Theorem istgp
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3930 . . 3 (𝐺 ∈ (Grp ∩ TopMnd) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd))
21anbi1i 625 . 2 ((𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
3 fvexd 6861 . . . 4 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) ∈ V)
4 simpl 484 . . . . . . 7 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ 𝑓 = 𝐺)
54fveq2d 6850 . . . . . 6 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (invgβ€˜π‘“) = (invgβ€˜πΊ))
6 istgp.2 . . . . . 6 𝐼 = (invgβ€˜πΊ)
75, 6eqtr4di 2791 . . . . 5 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (invgβ€˜π‘“) = 𝐼)
8 id 22 . . . . . . 7 (𝑗 = (TopOpenβ€˜π‘“) β†’ 𝑗 = (TopOpenβ€˜π‘“))
9 fveq2 6846 . . . . . . . 8 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) = (TopOpenβ€˜πΊ))
10 istgp.1 . . . . . . . 8 𝐽 = (TopOpenβ€˜πΊ)
119, 10eqtr4di 2791 . . . . . . 7 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) = 𝐽)
128, 11sylan9eqr 2795 . . . . . 6 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ 𝑗 = 𝐽)
1312, 12oveq12d 7379 . . . . 5 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (𝑗 Cn 𝑗) = (𝐽 Cn 𝐽))
147, 13eleq12d 2828 . . . 4 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ ((invgβ€˜π‘“) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
153, 14sbcied 3788 . . 3 (𝑓 = 𝐺 β†’ ([(TopOpenβ€˜π‘“) / 𝑗](invgβ€˜π‘“) ∈ (𝑗 Cn 𝑗) ↔ 𝐼 ∈ (𝐽 Cn 𝐽)))
16 df-tgp 23447 . . 3 TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpenβ€˜π‘“) / 𝑗](invgβ€˜π‘“) ∈ (𝑗 Cn 𝑗)}
1715, 16elrab2 3652 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ (Grp ∩ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
18 df-3an 1090 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd) ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
192, 17, 183bitr4i 303 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3447  [wsbc 3743   ∩ cin 3913  β€˜cfv 6500  (class class class)co 7361  TopOpenctopn 17311  Grpcgrp 18756  invgcminusg 18757   Cn ccn 22598  TopMndctmd 23444  TopGrpctgp 23445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-tgp 23447
This theorem is referenced by:  tgpgrp  23452  tgptmd  23453  tgpinv  23459  istgp2  23465  oppgtgp  23472  subgtgp  23479  symgtgp  23480  prdstgpd  23499  tlmtgp  23570  nrgtdrg  24080
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