Theorem istrg 21967

Description: Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypothesis

Ref	Expression
istrg.1	|- M = (mulGrp ` R )

Assertion

Ref	Expression
istrg	|- ( R e. TopRing <-> ( R e. TopGrp /\ R e. Ring /\ M e. TopMnd ) )

Proof of Theorem istrg

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( R e. ( TopGrp i^i Ring ) <-> ( R e. TopGrp /\ R e. Ring ) )
2	1	anbi1i 731	. 2 |- ( ( R e. ( TopGrp i^i Ring ) /\ M e. TopMnd ) <-> ( ( R e. TopGrp /\ R e. Ring ) /\ M e. TopMnd ) )
3		fveq2 6191	. . . . 5 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
4		istrg.1	. . . . 5 |- M = (mulGrp ` R )
5	3, 4	syl6eqr 2674	. . . 4 |- ( r = R -> (mulGrp ` r ) = M )
6	5	eleq1d 2686	. . 3 |- ( r = R -> ( (mulGrp ` r ) e. TopMnd <-> M e. TopMnd ) )
7		df-trg 21963	. . 3 |- TopRing = { r e. ( TopGrp i^i Ring ) | (mulGrp ` r ) e. TopMnd }
8	6, 7	elrab2 3366	. 2 |- ( R e. TopRing <-> ( R e. ( TopGrp i^i Ring ) /\ M e. TopMnd ) )
9		df-3an 1039	. 2 |- ( ( R e. TopGrp /\ R e. Ring /\ M e. TopMnd ) <-> ( ( R e. TopGrp /\ R e. Ring ) /\ M e. TopMnd ) )
10	2, 8, 9	3bitr4i 292	1 |- ( R e. TopRing <-> ( R e. TopGrp /\ R e. Ring /\ M e. TopMnd ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   ` cfv 5888  mulGrpcmgp 18489   Ringcrg 18547  TopMndctmd 21874   TopGrpctgp 21875   TopRingctrg 21959

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-trg 21963

This theorem is referenced by:  trgtmd  21968  trgtgp  21971  trgring  21974  nrgtrg  22494