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Theorem gimco 14748
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 14745 . . 3  |-  ( F  e.  ( T GrpIso  U
)  <->  ( F  e.  ( T  GrpHom  U )  /\  `' F  e.  ( U  GrpHom  T ) ) )
2 isgim2 14745 . . 3  |-  ( G  e.  ( S GrpIso  T
)  <->  ( G  e.  ( S  GrpHom  T )  /\  `' G  e.  ( T  GrpHom  S ) ) )
3 ghmco 14718 . . . . 5  |-  ( ( F  e.  ( T 
GrpHom  U )  /\  G  e.  ( S  GrpHom  T ) )  ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
4 cnvco 4881 . . . . . 6  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
5 ghmco 14718 . . . . . . 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 2380 . . . . 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 14745 . 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 1696   `'ccnv 4704    o. ccom 4709  (class class class)co 5874    GrpHom cghm 14696   GrpIso cgim 14737
This theorem is referenced by:  gictr  14755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697  df-gim 14739
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