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Theorem gimco 16442
Description: The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
gimco  |-  ( ( F  e.  ( T GrpIso  U )  /\  G  e.  ( S GrpIso  T ) )  ->  ( F  o.  G )  e.  ( S GrpIso  U ) )

Proof of Theorem gimco
StepHypRef Expression
1 isgim2 16439 . . 3  |-  ( F  e.  ( T GrpIso  U
)  <->  ( F  e.  ( T  GrpHom  U )  /\  `' F  e.  ( U  GrpHom  T ) ) )
2 isgim2 16439 . . 3  |-  ( G  e.  ( S GrpIso  T
)  <->  ( G  e.  ( S  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) ) )
3 ghmco 16412 . . . . 5  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
4 cnvco 5198 . . . . . 6  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
5 ghmco 16412 . . . . . . 7  |-  ( ( `' G  e.  ( T  GrpHom  S )  /\  `' F  e.  ( U  GrpHom  T ) )  ->  ( `' G  o.  `' F )  e.  ( U  GrpHom  S ) )
65ancoms 453 . . . . . 6  |-  ( ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) )  ->  ( `' G  o.  `' F )  e.  ( U  GrpHom  S ) )
74, 6syl5eqel 2549 . . . . 5  |-  ( ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) )  ->  `' ( F  o.  G )  e.  ( U  GrpHom  S ) )
83, 7anim12i 566 . . . 4  |-  ( ( ( F  e.  ( T  GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  /\  ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) ) )  ->  ( ( F  o.  G )  e.  ( S  GrpHom  U )  /\  `' ( F  o.  G )  e.  ( U  GrpHom  S ) ) )
98an4s 826 . . 3  |-  ( ( ( F  e.  ( T  GrpHom  U )  /\  `' F  e.  ( U  GrpHom  T ) )  /\  ( G  e.  ( S  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) ) )  ->  (
( F  o.  G
)  e.  ( S 
GrpHom  U )  /\  `' ( F  o.  G
)  e.  ( U 
GrpHom  S ) ) )
101, 2, 9syl2anb 479 . 2  |-  ( ( F  e.  ( T GrpIso  U )  /\  G  e.  ( S GrpIso  T ) )  ->  ( ( F  o.  G )  e.  ( S  GrpHom  U )  /\  `' ( F  o.  G )  e.  ( U  GrpHom  S ) ) )
11 isgim2 16439 . 2  |-  ( ( F  o.  G )  e.  ( S GrpIso  U
)  <->  ( ( F  o.  G )  e.  ( S  GrpHom  U )  /\  `' ( F  o.  G )  e.  ( U  GrpHom  S ) ) )
1210, 11sylibr 212 1  |-  ( ( F  e.  ( T GrpIso  U )  /\  G  e.  ( S GrpIso  T ) )  ->  ( F  o.  G )  e.  ( S GrpIso  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1819   `'ccnv 5007    o. ccom 5012  (class class class)co 6296    GrpHom cghm 16390   GrpIso cgim 16431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-grp 16183  df-ghm 16391  df-gim 16433
This theorem is referenced by:  gictr  16449
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