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

Theorem ghomfo 29883
Description: A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
Hypotheses
Ref Expression
ghomfo.1  |-  X  =  ran  G
ghomfo.2  |-  Y  =  ran  F
ghomfo.3  |-  S  =  ( H  |`  ( Y  X.  Y ) )
ghomfo.4  |-  Z  =  ran  S
Assertion
Ref Expression
ghomfo  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )

Proof of Theorem ghomfo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghomfo.1 . . . . . 6  |-  X  =  ran  G
2 eqid 2402 . . . . . 6  |-  ran  H  =  ran  H
31, 2elghomOLD 25779 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : X
--> ran  H  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) ) ) )
43biimp3a 1330 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F : X
--> ran  H  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) ) )
54simpld 457 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X --> ran  H )
6 ffn 5714 . . 3  |-  ( F : X --> ran  H  ->  F  Fn  X )
75, 6syl 17 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  Fn  X
)
8 ghomfo.3 . . . . . 6  |-  S  =  ( H  |`  ( Y  X.  Y ) )
98dmeqi 5025 . . . . 5  |-  dom  S  =  dom  ( H  |`  ( Y  X.  Y
) )
10 ghomfo.2 . . . . . . . . 9  |-  Y  =  ran  F
1110, 8ghomgrp 29882 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  (
SubGrpOp `  H ) )
12 issubgo 25719 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  H
)  <->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
1311, 12sylib 196 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( H  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  H )
)
1413simp2d 1010 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  GrpOp )
15 ghomfo.4 . . . . . . 7  |-  Z  =  ran  S
1615grpofo 25615 . . . . . 6  |-  ( S  e.  GrpOp  ->  S :
( Z  X.  Z
) -onto-> Z )
17 fof 5778 . . . . . 6  |-  ( S : ( Z  X.  Z ) -onto-> Z  ->  S : ( Z  X.  Z ) --> Z )
18 fdm 5718 . . . . . 6  |-  ( S : ( Z  X.  Z ) --> Z  ->  dom  S  =  ( Z  X.  Z ) )
1914, 16, 17, 184syl 19 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  dom  S  =  ( Z  X.  Z
) )
20 frn 5720 . . . . . . . . 9  |-  ( F : X --> ran  H  ->  ran  F  C_  ran  H )
215, 20syl 17 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  C_  ran  H )
2210, 21syl5eqss 3486 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Y  C_  ran  H )
23 xpss12 4929 . . . . . . 7  |-  ( ( Y  C_  ran  H  /\  Y  C_  ran  H )  ->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) )
2422, 22, 23syl2anc 659 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) )
25 ssdmres 5115 . . . . . . . 8  |-  ( ( Y  X.  Y ) 
C_  dom  H  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
262grpofo 25615 . . . . . . . . . 10  |-  ( H  e.  GrpOp  ->  H :
( ran  H  X.  ran  H ) -onto-> ran  H
)
27 fof 5778 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) -onto-> ran 
H  ->  H :
( ran  H  X.  ran  H ) --> ran  H
)
28 fdm 5718 . . . . . . . . . 10  |-  ( H : ( ran  H  X.  ran  H ) --> ran 
H  ->  dom  H  =  ( ran  H  X.  ran  H ) )
2926, 27, 283syl 18 . . . . . . . . 9  |-  ( H  e.  GrpOp  ->  dom  H  =  ( ran  H  X.  ran  H ) )
3029sseq2d 3470 . . . . . . . 8  |-  ( H  e.  GrpOp  ->  ( ( Y  X.  Y )  C_  dom  H  <->  ( Y  X.  Y )  C_  ( ran  H  X.  ran  H
) ) )
3125, 30syl5rbbr 260 . . . . . . 7  |-  ( H  e.  GrpOp  ->  ( ( Y  X.  Y )  C_  ( ran  H  X.  ran  H )  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) ) )
32313ad2ant2 1019 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( Y  X.  Y )  C_  ( ran  H  X.  ran  H )  <->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) ) )
3324, 32mpbid 210 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  dom  ( H  |`  ( Y  X.  Y
) )  =  ( Y  X.  Y ) )
349, 19, 333eqtr3a 2467 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( Z  X.  Z )  =  ( Y  X.  Y ) )
35 xpid11 5045 . . . 4  |-  ( ( Z  X.  Z )  =  ( Y  X.  Y )  <->  Z  =  Y )
3634, 35sylib 196 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  Z  =  Y )
3736, 10syl6req 2460 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ran  F  =  Z )
38 df-fo 5575 . 2  |-  ( F : X -onto-> Z  <->  ( F  Fn  X  /\  ran  F  =  Z ) )
397, 37, 38sylanbrc 662 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : X -onto-> Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754    C_ wss 3414    X. cxp 4821   dom cdm 4823   ran crn 4824    |` cres 4825    Fn wfn 5564   -->wf 5565   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278   GrpOpcgr 25602   SubGrpOpcsubgo 25717   GrpOpHom cghomOLD 25773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-grpo 25607  df-gid 25608  df-ginv 25609  df-subgo 25718  df-ghomOLD 25774
This theorem is referenced by:  ghomcl  29884  ghomgsg  29885  ghomf1olem  29886
  Copyright terms: Public domain W3C validator