Theorem istmd 24103

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 3992	. . 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 6935	. . . 4 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) ∈ V)
4		simpl 482	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑓 = 𝐺)
5	4	fveq2d 6924	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (+𝑓‘𝑓) = (+𝑓‘𝐺))
6		istmd.1	. . . . . 6 ⊢ 𝐹 = (+𝑓‘𝐺)
7	5, 6	eqtr4di 2798	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (+𝑓‘𝑓) = 𝐹)
8		id 22	. . . . . . . 8 ⊢ (𝑗 = (TopOpen‘𝑓) → 𝑗 = (TopOpen‘𝑓))
9		fveq2 6920	. . . . . . . . 9 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = (TopOpen‘𝐺))
10		istmd.2	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝐺)
11	9, 10	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑓 = 𝐺 → (TopOpen‘𝑓) = 𝐽)
12	8, 11	sylan9eqr 2802	. . . . . . 7 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → 𝑗 = 𝐽)
13	12, 12	oveq12d 7466	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → (𝑗 ×t 𝑗) = (𝐽 ×t 𝐽))
14	13, 12	oveq12d 7466	. . . . 5 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((𝑗 ×t 𝑗) Cn 𝑗) = ((𝐽 ×t 𝐽) Cn 𝐽))
15	7, 14	eleq12d 2838	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpen‘𝑓)) → ((+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
16	3, 15	sbcied 3850	. . 3 ⊢ (𝑓 = 𝐺 → ([(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
17		df-tmd 24101	. . 3 ⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
18	16, 17	elrab2 3711	. 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)))
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 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488  [wsbc 3804   ∩ cin 3975  ‘cfv 6573  (class class class)co 7448  TopOpenctopn 17481  +_𝑓cplusf 18675  Mndcmnd 18772  TopSpctps 22959   Cn ccn 23253   ×_t ctx 23589  TopMndctmd 24099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-tmd 24101

This theorem is referenced by:  tmdmnd  24104  tmdtps  24105  tmdcn  24112  istgp2  24120  oppgtmd  24126  efmndtmd  24130  submtmd  24133  prdstmdd  24153  nrgtrg  24732  mhmhmeotmd  33873  xrge0tmdALT  33892