MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gicsym Structured version   Unicode version

Theorem gicsym 15053
Description: Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
gicsym  |-  ( R 
~=ph𝑔  S  ->  S  ~=ph𝑔 
R )

Proof of Theorem gicsym
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 brgic 15048 . 2  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 n0 3629 . . 3  |-  ( ( R GrpIso  S )  =/=  (/) 
<->  E. f  f  e.  ( R GrpIso  S ) )
3 gimcnv 15046 . . . . 5  |-  ( f  e.  ( R GrpIso  S
)  ->  `' f  e.  ( S GrpIso  R ) )
4 brgici 15049 . . . . 5  |-  ( `' f  e.  ( S GrpIso  R )  ->  S  ~=ph𝑔  R )
53, 4syl 16 . . . 4  |-  ( f  e.  ( R GrpIso  S
)  ->  S  ~=ph𝑔  R )
65exlimiv 1644 . . 3  |-  ( E. f  f  e.  ( R GrpIso  S )  ->  S  ~=ph𝑔 
R )
72, 6sylbi 188 . 2  |-  ( ( R GrpIso  S )  =/=  (/)  ->  S  ~=ph𝑔 
R )
81, 7sylbi 188 1  |-  ( R 
~=ph𝑔  S  ->  S  ~=ph𝑔 
R )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    e. wcel 1725    =/= wne 2598   (/)c0 3620   class class class wbr 4204   `'ccnv 4869  (class class class)co 6073   GrpIso cgim 15036    ~=ph𝑔 cgic 15037
This theorem is referenced by:  gicer  15055  cygznlem3  16842  cygth  16844  cyggic  16845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-mnd 14682  df-grp 14804  df-ghm 14996  df-gim 15038  df-gic 15039
  Copyright terms: Public domain W3C validator