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Theorem gimco 16876
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 16873 . . 3  |-  ( F  e.  ( T GrpIso  U
)  <->  ( F  e.  ( T  GrpHom  U )  /\  `' F  e.  ( U  GrpHom  T ) ) )
2 isgim2 16873 . . 3  |-  ( G  e.  ( S GrpIso  T
)  <->  ( G  e.  ( S  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) ) )
3 ghmco 16846 . . . . 5  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
4 cnvco 5031 . . . . . 6  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
5 ghmco 16846 . . . . . . 7  |-  ( ( `' G  e.  ( T  GrpHom  S )  /\  `' F  e.  ( U  GrpHom  T ) )  ->  ( `' G  o.  `' F )  e.  ( U  GrpHom  S ) )
65ancoms 454 . . . . . 6  |-  ( ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) )  ->  ( `' G  o.  `' F )  e.  ( U  GrpHom  S ) )
74, 6syl5eqel 2512 . . . . 5  |-  ( ( `' F  e.  ( U  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) )  ->  `' ( F  o.  G )  e.  ( U  GrpHom  S ) )
83, 7anim12i 568 . . . 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 833 . . 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 481 . 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 16873 . 2  |-  ( ( F  o.  G )  e.  ( S GrpIso  U
)  <->  ( ( F  o.  G )  e.  ( S  GrpHom  U )  /\  `' ( F  o.  G )  e.  ( U  GrpHom  S ) ) )
1210, 11sylibr 215 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 370    e. wcel 1867   `'ccnv 4844    o. ccom 4849  (class class class)co 6296    GrpHom cghm 16824   GrpIso cgim 16865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7473  df-0g 15292  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-mhm 16526  df-grp 16617  df-ghm 16825  df-gim 16867
This theorem is referenced by:  gictr  16883
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