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Theorem gictr 15064
Description: Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
gictr  |-  ( ( R  ~=ph𝑔 
S  /\  S  ~=ph𝑔  T )  ->  R  ~=ph𝑔 
T )

Proof of Theorem gictr
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brgic 15058 . 2  |-  ( R 
~=ph𝑔  S 
<->  ( R GrpIso  S )  =/=  (/) )
2 brgic 15058 . 2  |-  ( S 
~=ph𝑔  T 
<->  ( S GrpIso  T )  =/=  (/) )
3 n0 3639 . . 3  |-  ( ( R GrpIso  S )  =/=  (/) 
<->  E. f  f  e.  ( R GrpIso  S ) )
4 n0 3639 . . 3  |-  ( ( S GrpIso  T )  =/=  (/) 
<->  E. g  g  e.  ( S GrpIso  T ) )
5 eeanv 1938 . . . 4  |-  ( E. f E. g ( f  e.  ( R GrpIso  S )  /\  g  e.  ( S GrpIso  T ) )  <->  ( E. f 
f  e.  ( R GrpIso  S )  /\  E. g  g  e.  ( S GrpIso  T ) ) )
6 gimco 15057 . . . . . . 7  |-  ( ( g  e.  ( S GrpIso  T )  /\  f  e.  ( R GrpIso  S ) )  ->  ( g  o.  f )  e.  ( R GrpIso  T ) )
7 brgici 15059 . . . . . . 7  |-  ( ( g  o.  f )  e.  ( R GrpIso  T
)  ->  R  ~=ph𝑔  T )
86, 7syl 16 . . . . . 6  |-  ( ( g  e.  ( S GrpIso  T )  /\  f  e.  ( R GrpIso  S ) )  ->  R  ~=ph𝑔  T )
98ancoms 441 . . . . 5  |-  ( ( f  e.  ( R GrpIso  S )  /\  g  e.  ( S GrpIso  T ) )  ->  R  ~=ph𝑔  T )
109exlimivv 1646 . . . 4  |-  ( E. f E. g ( f  e.  ( R GrpIso  S )  /\  g  e.  ( S GrpIso  T ) )  ->  R  ~=ph𝑔  T )
115, 10sylbir 206 . . 3  |-  ( ( E. f  f  e.  ( R GrpIso  S )  /\  E. g  g  e.  ( S GrpIso  T
) )  ->  R  ~=ph𝑔  T )
123, 4, 11syl2anb 467 . 2  |-  ( ( ( R GrpIso  S )  =/=  (/)  /\  ( S GrpIso  T )  =/=  (/) )  ->  R  ~=ph𝑔 
T )
131, 2, 12syl2anb 467 1  |-  ( ( R  ~=ph𝑔 
S  /\  S  ~=ph𝑔  T )  ->  R  ~=ph𝑔 
T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    e. wcel 1726    =/= wne 2601   (/)c0 3630   class class class wbr 4214    o. ccom 4884  (class class class)co 6083   GrpIso cgim 15046    ~=ph𝑔 cgic 15047
This theorem is referenced by:  gicer  15065  cyggic  16855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-1o 6726  df-map 7022  df-0g 13729  df-mnd 14692  df-mhm 14740  df-grp 14814  df-ghm 15006  df-gim 15048  df-gic 15049
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