Theorem istmd 21878

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 3796	. . 3 |- ( G e. ( Mnd i^i TopSp ) <-> ( G e. Mnd /\ G e. TopSp ) )
2	1	anbi1i 731	. 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		fvexd 6203	. . . 4 |- ( f = G -> ( TopOpen ` f ) e. _V )
4		simpl 473	. . . . . . 7 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> f = G )
5	4	fveq2d 6195	. . . . . 6 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( +f ` f ) = ( +f ` G ) )
6		istmd.1	. . . . . 6 |- F = ( +f ` G )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( +f ` f ) = F )
8		id 22	. . . . . . . 8 |- ( j = ( TopOpen ` f ) -> j = ( TopOpen ` f ) )
9		fveq2 6191	. . . . . . . . 9 |- ( f = G -> ( TopOpen ` f ) = ( TopOpen ` G ) )
10		istmd.2	. . . . . . . . 9 |- J = ( TopOpen ` G )
11	9, 10	syl6eqr 2674	. . . . . . . 8 |- ( f = G -> ( TopOpen ` f ) = J )
12	8, 11	sylan9eqr 2678	. . . . . . 7 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> j = J )
13	12, 12	oveq12d 6668	. . . . . 6 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( j tX j ) = ( J tX J ) )
14	13, 12	oveq12d 6668	. . . . 5 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( ( j tX j ) Cn j ) = ( ( J tX J ) Cn J ) )
15	7, 14	eleq12d 2695	. . . 4 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( ( +f ` f ) e. ( ( j tX j ) Cn j ) <-> F e. ( ( J tX J ) Cn J ) ) )
16	3, 15	sbcied 3472	. . 3 |- ( f = G -> ( [. ( TopOpen ` f ) / j ]. ( +f ` f ) e. ( ( j tX j ) Cn j ) <-> F e. ( ( J tX J ) Cn J ) ) )
17		df-tmd 21876	. . 3 |- TopMnd = { f e. ( Mnd i^i TopSp ) | [. ( TopOpen ` f ) / j ]. ( +f ` f ) e. ( ( j tX j ) Cn j ) }
18	16, 17	elrab2 3366	. 2 |- ( G e. TopMnd <-> ( G e. ( Mnd i^i TopSp ) /\ F e. ( ( J tX J ) Cn J ) ) )
19		df-3an 1039	. 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 ) ) )
20	2, 18, 19	3bitr4i 292	1 |- ( G e. TopMnd <-> ( G e. Mnd /\ G e. TopSp /\ F e. ( ( J tX J ) Cn J ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   ` cfv 5888  (class class class)co 6650   TopOpenctopn 16082   +fcplusf 17239   Mndcmnd 17294   TopSpctps 20736   Cn ccn 21028   tX ctx 21363  TopMndctmd 21874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-tmd 21876

This theorem is referenced by:  tmdmnd  21879  tmdtps  21880  tmdcn  21887  istgp2  21895  oppgtmd  21901  symgtgp  21905  submtmd  21908  prdstmdd  21927  nrgtrg  22494  mhmhmeotmd  29973  xrge0tmdOLD  29991