Theorem istmd 18065

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 e. TopMnd <-> ( G e. Mnd /\ G e. TopSp /\ F e. ( ( J tX 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 3498	. . 3 |- ( G e. ( Mnd i^i TopSp ) <-> ( G e. Mnd /\ G e. TopSp ) )
2	1	anbi1i 677	. 2 |- ( ( G e. ( Mnd i^i TopSp ) /\ F e. ( ( J tX J ) Cn J ) ) <-> ( ( G e. Mnd /\ G e. TopSp ) /\ F e. ( ( J tX J ) Cn J ) ) )
3		fvex 5709	. . . . 5 |- ( TopOpen ` f ) e. _V
4	3	a1i 11	. . . 4 |- ( f = G -> ( TopOpen ` f ) e. _V )
5		simpl 444	. . . . . . 7 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> f = G )
6	5	fveq2d 5699	. . . . . 6 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( + f ` f ) = ( + f ` G ) )
7		istmd.1	. . . . . 6 |- F = ( + f ` G )
8	6, 7	syl6eqr 2462	. . . . 5 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( + f ` f ) = F )
9		id 20	. . . . . . . 8 |- ( j = ( TopOpen ` f ) -> j = ( TopOpen ` f ) )
10		fveq2 5695	. . . . . . . . 9 |- ( f = G -> ( TopOpen ` f ) = ( TopOpen ` G ) )
11		istmd.2	. . . . . . . . 9 |- J = ( TopOpen ` G )
12	10, 11	syl6eqr 2462	. . . . . . . 8 |- ( f = G -> ( TopOpen ` f ) = J )
13	9, 12	sylan9eqr 2466	. . . . . . 7 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> j = J )
14	13, 13	oveq12d 6066	. . . . . 6 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( j tX j ) = ( J tX J ) )
15	14, 13	oveq12d 6066	. . . . 5 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( ( j tX j ) Cn j ) = ( ( J tX J ) Cn J ) )
16	8, 15	eleq12d 2480	. . . 4 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( ( + f ` f ) e. ( ( j tX j ) Cn j ) <-> F e. ( ( J tX J ) Cn J ) ) )
17	4, 16	sbcied 3165	. . 3 |- ( f = G -> ( [. ( TopOpen ` f ) / j ]. ( + f ` f ) e. ( ( j tX j ) Cn j ) <-> F e. ( ( J tX J ) Cn J ) ) )
18		df-tmd 18063	. . 3 |- TopMnd = { f e. ( Mnd i^i TopSp ) | [. ( TopOpen ` f ) / j ]. ( + f ` f ) e. ( ( j tX j ) Cn j ) }
19	17, 18	elrab2 3062	. 2 |- ( G e. TopMnd <-> ( G e. ( Mnd i^i TopSp ) /\ F e. ( ( J tX J ) Cn J ) ) )
20		df-3an 938	. 2 |- ( ( G e. Mnd /\ G e. TopSp /\ F e. ( ( J tX J ) Cn J ) ) <-> ( ( G e. Mnd /\ G e. TopSp ) /\ F e. ( ( J tX J ) Cn J ) ) )
21	2, 19, 20	3bitr4i 269	1 |- ( G e. TopMnd <-> ( G e. Mnd /\ G e. TopSp /\ F e. ( ( J tX J ) Cn J ) ) )

Syntax hints:   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   _Vcvv 2924   [.wsbc 3129   i^i cin 3287   ` cfv 5421  (class class class)co 6048   TopOpenctopn 13612   Mndcmnd 14647   + fcplusf 14650   TopSpctps 16924   Cn ccn 17250   tX ctx 17553  TopMndctmd 18061

This theorem is referenced by:  tmdmnd  18066  tmdtps  18067  tmdcn  18074  istgp2  18082  oppgtmd  18088  symgtgp  18092  submtmd  18095  prdstmdd  18114  nrgtrg  18686  mhmhmeotmd  24274  xrge0tmdOLD  24292

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-nul 4306

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051  df-tmd 18063