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Theorem gimco 14732
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 14729 . . 3  |-  ( F  e.  ( T GrpIso  U
)  <->  ( F  e.  ( T  GrpHom  U )  /\  `' F  e.  ( U  GrpHom  T ) ) )
2 isgim2 14729 . . 3  |-  ( G  e.  ( S GrpIso  T
)  <->  ( G  e.  ( S  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) ) )
3 ghmco 14702 . . . . 5  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
4 cnvco 4865 . . . . . 6  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
5 ghmco 14702 . . . . . . 7  |-  ( ( `' G  e.  ( T  GrpHom  S )  /\  `' F  e.  ( U  GrpHom  T ) )  ->  ( `' G  o.  `' F )  e.  ( U  GrpHom  S ) )
65ancoms 439 . . . . . 6  |-  ( ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) )  ->  ( `' G  o.  `' F )  e.  ( U  GrpHom  S ) )
74, 6syl5eqel 2367 . . . . 5  |-  ( ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) )  ->  `' ( F  o.  G )  e.  ( U  GrpHom  S ) )
83, 7anim12i 549 . . . 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 799 . . 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 465 . 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 14729 . 2  |-  ( ( F  o.  G )  e.  ( S GrpIso  U
)  <->  ( ( F  o.  G )  e.  ( S  GrpHom  U )  /\  `' ( F  o.  G )  e.  ( U  GrpHom  S ) ) )
1210, 11sylibr 203 1  |-  ( ( F  e.  ( T GrpIso  U )  /\  G  e.  ( S GrpIso  T ) )  ->  ( F  o.  G )  e.  ( S GrpIso  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   `'ccnv 4688    o. ccom 4693  (class class class)co 5858    GrpHom cghm 14680   GrpIso cgim 14721
This theorem is referenced by:  gictr  14739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-map 6774  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-ghm 14681  df-gim 14723
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