Users' Mathboxes Mathbox for Paul Chapman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ghomgsg Structured version   Unicode version

Theorem ghomgsg 25104
Description: A group homomorphism from  G to  H is also a group homomorphism from  G to its image in  H. (Contributed by Paul Chapman, 3-Mar-2008.)
Hypotheses
Ref Expression
ghomgsg.1  |-  Y  =  ran  F
ghomgsg.2  |-  S  =  ( H  |`  ( Y  X.  Y ) )
Assertion
Ref Expression
ghomgsg  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  e.  ( G GrpOpHom  S ) )

Proof of Theorem ghomgsg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . 4  |-  ran  G  =  ran  G
2 ghomgsg.1 . . . 4  |-  Y  =  ran  F
3 ghomgsg.2 . . . 4  |-  S  =  ( H  |`  ( Y  X.  Y ) )
4 eqid 2436 . . . 4  |-  ran  S  =  ran  S
51, 2, 3, 4ghomfo 25102 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G
-onto->
ran  S )
6 fof 5653 . . 3  |-  ( F : ran  G -onto-> ran  S  ->  F : ran  G --> ran  S )
75, 6syl 16 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G --> ran  S )
8 eqid 2436 . . . . . 6  |-  ran  H  =  ran  H
91, 8elghom 21951 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
109biimp3a 1283 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) )
1110simprd 450 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) )
12 ffvelrn 5868 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  x  e. 
ran  G )  -> 
( F `  x
)  e.  ran  S
)
13 ffvelrn 5868 . . . . . . . 8  |-  ( ( F : ran  G --> ran  S  /\  y  e. 
ran  G )  -> 
( F `  y
)  e.  ran  S
)
1412, 13anim12dan 811 . . . . . . 7  |-  ( ( F : ran  G --> ran  S  /\  ( x  e.  ran  G  /\  y  e.  ran  G ) )  ->  ( ( F `  x )  e.  ran  S  /\  ( F `  y )  e.  ran  S ) )
157, 14sylan 458 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( x  e.  ran  G  /\  y  e.  ran  G ) )  ->  ( ( F `
 x )  e. 
ran  S  /\  ( F `  y )  e.  ran  S ) )
162, 3ghomgrp 25101 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
174subgoov 21893 . . . . . . 7  |-  ( ( S  e.  ( SubGrpOp `  H )  /\  (
( F `  x
)  e.  ran  S  /\  ( F `  y
)  e.  ran  S
) )  ->  (
( F `  x
) S ( F `
 y ) )  =  ( ( F `
 x ) H ( F `  y
) ) )
1816, 17sylan 458 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( ( F `  x )  e.  ran  S  /\  ( F `  y )  e.  ran  S ) )  ->  ( ( F `
 x ) S ( F `  y
) )  =  ( ( F `  x
) H ( F `
 y ) ) )
1915, 18syldan 457 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( x  e.  ran  G  /\  y  e.  ran  G ) )  ->  ( ( F `
 x ) S ( F `  y
) )  =  ( ( F `  x
) H ( F `
 y ) ) )
2019eqeq1d 2444 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( x  e.  ran  G  /\  y  e.  ran  G ) )  ->  ( ( ( F `  x ) S ( F `  y ) )  =  ( F `  (
x G y ) )  <->  ( ( F `
 x ) H ( F `  y
) )  =  ( F `  ( x G y ) ) ) )
21202ralbidva 2745 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `  (
x G y ) )  <->  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) )
2211, 21mpbird 224 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `
 ( x G y ) ) )
23 issubgo 21891 . . . . 5  |-  ( S  e.  ( SubGrpOp `  H
)  <->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
2416, 23sylib 189 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
2524simp2d 970 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
261, 4elghom 21951 . . . . 5  |-  ( ( G  e.  GrpOp  /\  S  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  S )  <->  ( F : ran  G --> ran  S  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
2726biimprd 215 . . . 4  |-  ( ( G  e.  GrpOp  /\  S  e.  GrpOp )  ->  (
( F : ran  G --> ran  S  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x
) S ( F `
 y ) )  =  ( F `  ( x G y ) ) )  ->  F  e.  ( G GrpOpHom  S ) ) )
28273adant3 977 . . 3  |-  ( ( G  e.  GrpOp  /\  S  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F : ran  G --> ran  S  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `
 ( x G y ) ) )  ->  F  e.  ( G GrpOpHom  S ) ) )
2925, 28syld3an2 1231 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F : ran  G --> ran  S  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) S ( F `  y ) )  =  ( F `
 ( x G y ) ) )  ->  F  e.  ( G GrpOpHom  S ) ) )
307, 22, 29mp2and 661 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  e.  ( G GrpOpHom  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320    X. cxp 4876   ran crn 4879    |` cres 4880   -->wf 5450   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081   GrpOpcgr 21774   SubGrpOpcsubgo 21889   GrpOpHom cghom 21945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781  df-subgo 21890  df-ghom 21946
  Copyright terms: Public domain W3C validator