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Theorem istmd 23578
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 3965 . . 3 (𝐺 ∈ (Mnd ∩ TopSp) ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp))
21anbi1i 625 . 2 ((𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
3 fvexd 6907 . . . 4 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) ∈ V)
4 simpl 484 . . . . . . 7 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ 𝑓 = 𝐺)
54fveq2d 6896 . . . . . 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 2791 . . . . 5 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (+π‘“β€˜π‘“) = 𝐹)
8 id 22 . . . . . . . 8 (𝑗 = (TopOpenβ€˜π‘“) β†’ 𝑗 = (TopOpenβ€˜π‘“))
9 fveq2 6892 . . . . . . . . 9 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) = (TopOpenβ€˜πΊ))
10 istmd.2 . . . . . . . . 9 𝐽 = (TopOpenβ€˜πΊ)
119, 10eqtr4di 2791 . . . . . . . 8 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) = 𝐽)
128, 11sylan9eqr 2795 . . . . . . 7 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ 𝑗 = 𝐽)
1312, 12oveq12d 7427 . . . . . 6 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (𝑗 Γ—t 𝑗) = (𝐽 Γ—t 𝐽))
1413, 12oveq12d 7427 . . . . 5 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ ((𝑗 Γ—t 𝑗) Cn 𝑗) = ((𝐽 Γ—t 𝐽) Cn 𝐽))
157, 14eleq12d 2828 . . . 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 3823 . . 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 23576 . . 3 TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpenβ€˜π‘“) / 𝑗](+π‘“β€˜π‘“) ∈ ((𝑗 Γ—t 𝑗) Cn 𝑗)}
1816, 17elrab2 3687 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
19 df-3an 1090 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475  [wsbc 3778   ∩ cin 3948  β€˜cfv 6544  (class class class)co 7409  TopOpenctopn 17367  +𝑓cplusf 18558  Mndcmnd 18625  TopSpctps 22434   Cn ccn 22728   Γ—t ctx 23064  TopMndctmd 23574
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 5307
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 2942  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-tmd 23576
This theorem is referenced by:  tmdmnd  23579  tmdtps  23580  tmdcn  23587  istgp2  23595  oppgtmd  23601  efmndtmd  23605  submtmd  23608  prdstmdd  23628  nrgtrg  24207  mhmhmeotmd  32907  xrge0tmdALT  32926
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