Theorem istmd 24018

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 3917	. . 3 ⊢ (𝐺 ∈ (Mnd ∩ TopSp) ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp))
2	1	anbi1i 624	. 2 ⊢ ((𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
3		fvexd 6849	. . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4		simpl 482	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
5	4	fveq2d 6838	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (+𝑓‘𝑓) = (+𝑓‘𝐺))
6		istmd.1	. . . . . 6 ⊢ 𝐹 = (+𝑓‘𝐺)
7	5, 6	eqtr4di 2789	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (+𝑓‘𝑓) = 𝐹)
8		id 22	. . . . . . . 8 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9		fveq2 6834	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10		istmd.2	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝐺)
11	9, 10	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
12	8, 11	sylan9eqr 2793	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
13	12, 12	oveq12d 7376	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 ×t 𝑗) = (𝐽 ×t 𝐽))
14	13, 12	oveq12d 7376	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((𝑗 ×t 𝑗) Cn 𝑗) = ((𝐽 ×t 𝐽) Cn 𝐽))
15	7, 14	eleq12d 2830	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
16	3, 15	sbcied 3784	. . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
17		df-tmd 24016	. . 3 ⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
18	16, 17	elrab2 3649	. 2 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
19		df-3an 1088	. 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 303	1 ⊢ (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3440  [wsbc 3740   ∩ cin 3900  ‘cfv 6492  (class class class)co 7358  TopOpenctopn 17341  +_𝑓cplusf 18562  Mndcmnd 18659  TopSpctps 22876   Cn ccn 23168   ×_t ctx 23504  TopMndctmd 24014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-tmd 24016

This theorem is referenced by:  tmdmnd  24019  tmdtps  24020  tmdcn  24027  istgp2  24035  oppgtmd  24041  efmndtmd  24045  submtmd  24048  prdstmdd  24068  nrgtrg  24634  mhmhmeotmd  34084  xrge0tmdALT  34103