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Theorem istmd 22097
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 3945 . . 3 (𝐺 ∈ (Mnd ∩ TopSp) ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp))
21anbi1i 602 . 2 ((𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
3 fvexd 6344 . . . 4 (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4 simpl 468 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
54fveq2d 6336 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (+𝑓𝑓) = (+𝑓𝐺))
6 istmd.1 . . . . . 6 𝐹 = (+𝑓𝐺)
75, 6syl6eqr 2822 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (+𝑓𝑓) = 𝐹)
8 id 22 . . . . . . . 8 (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9 fveq2 6332 . . . . . . . . 9 (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10 istmd.2 . . . . . . . . 9 𝐽 = (TopOpen‘𝐺)
119, 10syl6eqr 2822 . . . . . . . 8 (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
128, 11sylan9eqr 2826 . . . . . . 7 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
1312, 12oveq12d 6810 . . . . . 6 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → (𝑗 ×t 𝑗) = (𝐽 ×t 𝐽))
1413, 12oveq12d 6810 . . . . 5 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((𝑗 ×t 𝑗) Cn 𝑗) = ((𝐽 ×t 𝐽) Cn 𝐽))
157, 14eleq12d 2843 . . . 4 ((𝑓 = 𝐺𝑗 = (TopOpen‘𝑓)) → ((+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
163, 15sbcied 3622 . . 3 (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
17 df-tmd 22095 . . 3 TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
1816, 17elrab2 3516 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
19 df-3an 1072 . 2 ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
202, 18, 193bitr4i 292 1 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  Vcvv 3349  [wsbc 3585  cin 3720  cfv 6031  (class class class)co 6792  TopOpenctopn 16289  +𝑓cplusf 17446  Mndcmnd 17501  TopSpctps 20956   Cn ccn 21248   ×t ctx 21583  TopMndctmd 22093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795  df-tmd 22095
This theorem is referenced by:  tmdmnd  22098  tmdtps  22099  tmdcn  22106  istgp2  22114  oppgtmd  22120  symgtgp  22124  submtmd  22127  prdstmdd  22146  nrgtrg  22713  mhmhmeotmd  30307  xrge0tmdOLD  30325
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