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Theorem istmd 24082
Description: The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
istmd.1 𝐹 = (+𝑓𝐺)
istmd.2 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
istmd (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×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 3967 . . 3 (𝐺 ∈ (Mnd ∩ TopSp) ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp))
21anbi1i 624 . 2 ((𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
3 fvexd 6921 . . . 4 (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4 simpl 482 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
54fveq2d 6910 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (+𝑓𝑓) = (+𝑓𝐺))
6 istmd.1 . . . . . 6 𝐹 = (+𝑓𝐺)
75, 6eqtr4di 2795 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (+𝑓𝑓) = 𝐹)
8 id 22 . . . . . . . 8 (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9 fveq2 6906 . . . . . . . . 9 (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10 istmd.2 . . . . . . . . 9 𝐽 = (TopOpen‘𝐺)
119, 10eqtr4di 2795 . . . . . . . 8 (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
128, 11sylan9eqr 2799 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
1312, 12oveq12d 7449 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (𝑗 ×t 𝑗) = (𝐽 ×t 𝐽))
1413, 12oveq12d 7449 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((𝑗 ×t 𝑗) Cn 𝑗) = ((𝐽 ×t 𝐽) Cn 𝐽))
157, 14eleq12d 2835 . . . 4 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
163, 15sbcied 3832 . . 3 (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
17 df-tmd 24080 . . 3 TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
1816, 17elrab2 3695 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
19 df-3an 1089 . 2 ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
202, 18, 193bitr4i 303 1 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 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  +𝑓cplusf 18650  Mndcmnd 18747  TopSpctps 22938   Cn ccn 23232   ×t ctx 23568  TopMndctmd 24078
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-tmd 24080
This theorem is referenced by:  tmdmnd  24083  tmdtps  24084  tmdcn  24091  istgp2  24099  oppgtmd  24105  efmndtmd  24109  submtmd  24112  prdstmdd  24132  nrgtrg  24711  mhmhmeotmd  33926  xrge0tmdALT  33945
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