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Theorem istmd 23965
Description: The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
istmd.1 𝐹 = (+π‘“β€˜πΊ)
istmd.2 𝐽 = (TopOpenβ€˜πΊ)
Assertion
Ref Expression
istmd (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))

Proof of Theorem istmd
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3960 . . 3 (𝐺 ∈ (Mnd ∩ TopSp) ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp))
21anbi1i 623 . 2 ((𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
3 fvexd 6906 . . . 4 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) ∈ V)
4 simpl 482 . . . . . . 7 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ 𝑓 = 𝐺)
54fveq2d 6895 . . . . . 6 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (+π‘“β€˜π‘“) = (+π‘“β€˜πΊ))
6 istmd.1 . . . . . 6 𝐹 = (+π‘“β€˜πΊ)
75, 6eqtr4di 2785 . . . . 5 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (+π‘“β€˜π‘“) = 𝐹)
8 id 22 . . . . . . . 8 (𝑗 = (TopOpenβ€˜π‘“) β†’ 𝑗 = (TopOpenβ€˜π‘“))
9 fveq2 6891 . . . . . . . . 9 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) = (TopOpenβ€˜πΊ))
10 istmd.2 . . . . . . . . 9 𝐽 = (TopOpenβ€˜πΊ)
119, 10eqtr4di 2785 . . . . . . . 8 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) = 𝐽)
128, 11sylan9eqr 2789 . . . . . . 7 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ 𝑗 = 𝐽)
1312, 12oveq12d 7432 . . . . . 6 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (𝑗 Γ—t 𝑗) = (𝐽 Γ—t 𝐽))
1413, 12oveq12d 7432 . . . . 5 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ ((𝑗 Γ—t 𝑗) Cn 𝑗) = ((𝐽 Γ—t 𝐽) Cn 𝐽))
157, 14eleq12d 2822 . . . 4 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ ((+π‘“β€˜π‘“) ∈ ((𝑗 Γ—t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
163, 15sbcied 3819 . . 3 (𝑓 = 𝐺 β†’ ([(TopOpenβ€˜π‘“) / 𝑗](+π‘“β€˜π‘“) ∈ ((𝑗 Γ—t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
17 df-tmd 23963 . . 3 TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpenβ€˜π‘“) / 𝑗](+π‘“β€˜π‘“) ∈ ((𝑗 Γ—t 𝑗) Cn 𝑗)}
1816, 17elrab2 3683 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
19 df-3an 1087 . 2 ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
202, 18, 193bitr4i 303 1 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  Vcvv 3469  [wsbc 3774   ∩ cin 3943  β€˜cfv 6542  (class class class)co 7414  TopOpenctopn 17394  +𝑓cplusf 18588  Mndcmnd 18685  TopSpctps 22821   Cn ccn 23115   Γ—t ctx 23451  TopMndctmd 23961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-tmd 23963
This theorem is referenced by:  tmdmnd  23966  tmdtps  23967  tmdcn  23974  istgp2  23982  oppgtmd  23988  efmndtmd  23992  submtmd  23995  prdstmdd  24015  nrgtrg  24594  mhmhmeotmd  33464  xrge0tmdALT  33483
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