Theorem istmd 23133

Description: The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)

Hypotheses

Ref	Expression
istmd.1	⊢ 𝐹 = (+𝑓‘𝐺)
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.

Step	Hyp	Ref	Expression
1		elin 3899	. . 3 ⊢ (𝐺 ∈ (Mnd ∩ TopSp) ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp))
2	1	anbi1i 623	. 2 ⊢ ((𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
3		fvexd 6771	. . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4		simpl 482	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
5	4	fveq2d 6760	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (+𝑓‘𝑓) = (+𝑓‘𝐺))
6		istmd.1	. . . . . 6 ⊢ 𝐹 = (+𝑓‘𝐺)
7	5, 6	eqtr4di 2797	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (+𝑓‘𝑓) = 𝐹)
8		id 22	. . . . . . . 8 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9		fveq2 6756	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10		istmd.2	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝐺)
11	9, 10	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
12	8, 11	sylan9eqr 2801	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
13	12, 12	oveq12d 7273	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 ×t 𝑗) = (𝐽 ×t 𝐽))
14	13, 12	oveq12d 7273	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((𝑗 ×t 𝑗) Cn 𝑗) = ((𝐽 ×t 𝐽) Cn 𝐽))
15	7, 14	eleq12d 2833	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
16	3, 15	sbcied 3756	. . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
17		df-tmd 23131	. . 3 ⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
18	16, 17	elrab2 3620	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
19		df-3an 1087	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
20	2, 18, 19	3bitr4i 302	1 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422  [wsbc 3711   ∩ cin 3882  ‘cfv 6418  (class class class)co 7255  TopOpenctopn 17049  +_𝑓cplusf 18238  Mndcmnd 18300  TopSpctps 21989   Cn ccn 22283   ×_t ctx 22619  TopMndctmd 23129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-tmd 23131

This theorem is referenced by:  tmdmnd  23134  tmdtps  23135  tmdcn  23142  istgp2  23150  oppgtmd  23156  efmndtmd  23160  submtmd  23163  prdstmdd  23183  nrgtrg  23760  mhmhmeotmd  31779  xrge0tmdALT  31798