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Theorem gimco 15057
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 15054 . . 3  |-  ( F  e.  ( T GrpIso  U
)  <->  ( F  e.  ( T  GrpHom  U )  /\  `' F  e.  ( U  GrpHom  T ) ) )
2 isgim2 15054 . . 3  |-  ( G  e.  ( S GrpIso  T
)  <->  ( G  e.  ( S  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) ) )
3 ghmco 15027 . . . . 5  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
4 cnvco 5058 . . . . . 6  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
5 ghmco 15027 . . . . . . 7  |-  ( ( `' G  e.  ( T  GrpHom  S )  /\  `' F  e.  ( U  GrpHom  T ) )  ->  ( `' G  o.  `' F )  e.  ( U  GrpHom  S ) )
65ancoms 441 . . . . . 6  |-  ( ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) )  ->  ( `' G  o.  `' F )  e.  ( U  GrpHom  S ) )
74, 6syl5eqel 2522 . . . . 5  |-  ( ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) )  ->  `' ( F  o.  G )  e.  ( U  GrpHom  S ) )
83, 7anim12i 551 . . . 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 801 . . 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 467 . 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 15054 . 2  |-  ( ( F  o.  G )  e.  ( S GrpIso  U
)  <->  ( ( F  o.  G )  e.  ( S  GrpHom  U )  /\  `' ( F  o.  G )  e.  ( U  GrpHom  S ) ) )
1210, 11sylibr 205 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 360    e. wcel 1726   `'ccnv 4879    o. ccom 4884  (class class class)co 6083    GrpHom cghm 15005   GrpIso cgim 15046
This theorem is referenced by:  gictr  15064
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-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-riota 6551  df-map 7022  df-0g 13729  df-mnd 14692  df-mhm 14740  df-grp 14814  df-ghm 15006  df-gim 15048
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