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Theorem ghomgrpilem2 25097
Description: Lemma for ghomgrpi 25098. (Contributed by Paul Chapman, 25-Feb-2008.)
Hypotheses
Ref Expression
ghomgrpilem1.1  |-  G  e. 
GrpOp
ghomgrpilem1.2  |-  H  e. 
GrpOp
ghomgrpilem1.3  |-  F  e.  ( G GrpOpHom  H )
ghomgrpilem1.4  |-  X  =  ran  G
ghomgrpilem1.5  |-  U  =  (GId `  G )
ghomgrpilem1.6  |-  N  =  ( inv `  G
)
ghomgrpilem1.7  |-  W  =  ran  H
ghomgrpilem1.8  |-  T  =  (GId `  H )
ghomgrpilem1.9  |-  M  =  ( inv `  H
)
ghomgrpilem1.10  |-  Z  =  ran  F
ghomgrpilem1.11  |-  S  =  ( H  |`  ( Z  X.  Z ) )
Assertion
Ref Expression
ghomgrpilem2  |-  S  e.  ( SubGrpOp `  H )

Proof of Theorem ghomgrpilem2
Dummy variables  x  y  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghomgrpilem1.2 . 2  |-  H  e. 
GrpOp
2 ghomgrpilem1.7 . 2  |-  W  =  ran  H
3 ghomgrpilem1.8 . 2  |-  T  =  (GId `  H )
4 ghomgrpilem1.9 . 2  |-  M  =  ( inv `  H
)
5 ghomgrpilem1.10 . . 3  |-  Z  =  ran  F
6 ghomgrpilem1.3 . . . . . 6  |-  F  e.  ( G GrpOpHom  H )
7 ghomgrpilem1.1 . . . . . . 7  |-  G  e. 
GrpOp
8 ghomgrpilem1.4 . . . . . . . 8  |-  X  =  ran  G
98, 2elghom 21951 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : X
--> W  /\  A. x  e.  X  A. y  e.  X  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) ) )
107, 1, 9mp2an 654 . . . . . 6  |-  ( F  e.  ( G GrpOpHom  H )  <-> 
( F : X --> W  /\  A. x  e.  X  A. y  e.  X  ( ( F `
 x ) H ( F `  y
) )  =  ( F `  ( x G y ) ) ) )
116, 10mpbi 200 . . . . 5  |-  ( F : X --> W  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) )
1211simpli 445 . . . 4  |-  F : X
--> W
13 frn 5597 . . . 4  |-  ( F : X --> W  ->  ran  F  C_  W )
1412, 13ax-mp 8 . . 3  |-  ran  F  C_  W
155, 14eqsstri 3378 . 2  |-  Z  C_  W
16 ghomgrpilem1.11 . 2  |-  S  =  ( H  |`  ( Z  X.  Z ) )
175eleq2i 2500 . . . . . . 7  |-  ( x  e.  Z  <->  x  e.  ran  F )
18 ffn 5591 . . . . . . . . 9  |-  ( F : X --> W  ->  F  Fn  X )
1912, 18ax-mp 8 . . . . . . . 8  |-  F  Fn  X
20 fvelrnb 5774 . . . . . . . 8  |-  ( F  Fn  X  ->  (
x  e.  ran  F  <->  E. z  e.  X  ( F `  z )  =  x ) )
2119, 20ax-mp 8 . . . . . . 7  |-  ( x  e.  ran  F  <->  E. z  e.  X  ( F `  z )  =  x )
2217, 21bitri 241 . . . . . 6  |-  ( x  e.  Z  <->  E. z  e.  X  ( F `  z )  =  x )
2322biimpi 187 . . . . 5  |-  ( x  e.  Z  ->  E. z  e.  X  ( F `  z )  =  x )
245eleq2i 2500 . . . . . . 7  |-  ( y  e.  Z  <->  y  e.  ran  F )
25 fvelrnb 5774 . . . . . . . 8  |-  ( F  Fn  X  ->  (
y  e.  ran  F  <->  E. w  e.  X  ( F `  w )  =  y ) )
2619, 25ax-mp 8 . . . . . . 7  |-  ( y  e.  ran  F  <->  E. w  e.  X  ( F `  w )  =  y )
2724, 26bitri 241 . . . . . 6  |-  ( y  e.  Z  <->  E. w  e.  X  ( F `  w )  =  y )
2827biimpi 187 . . . . 5  |-  ( y  e.  Z  ->  E. w  e.  X  ( F `  w )  =  y )
2923, 28anim12i 550 . . . 4  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( E. z  e.  X  ( F `  z )  =  x  /\  E. w  e.  X  ( F `  w )  =  y ) )
30 reeanv 2875 . . . 4  |-  ( E. z  e.  X  E. w  e.  X  (
( F `  z
)  =  x  /\  ( F `  w )  =  y )  <->  ( E. z  e.  X  ( F `  z )  =  x  /\  E. w  e.  X  ( F `  w )  =  y ) )
3129, 30sylibr 204 . . 3  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  E. z  e.  X  E. w  e.  X  ( ( F `  z )  =  x  /\  ( F `  w )  =  y ) )
32 ghomgrpilem1.5 . . . . . . 7  |-  U  =  (GId `  G )
33 ghomgrpilem1.6 . . . . . . 7  |-  N  =  ( inv `  G
)
347, 1, 6, 8, 32, 33, 2, 3, 4, 5, 16ghomgrpilem1 25096 . . . . . 6  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( F `  z ) H ( F `  w ) )  =  ( F `
 ( z G w ) ) )
358grpocl 21788 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  z  e.  X  /\  w  e.  X )  ->  (
z G w )  e.  X )
367, 35mp3an1 1266 . . . . . . 7  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( z G w )  e.  X )
37 dffn3 5598 . . . . . . . . . 10  |-  ( F  Fn  X  <->  F : X
--> ran  F )
3819, 37mpbi 200 . . . . . . . . 9  |-  F : X
--> ran  F
39 feq3 5578 . . . . . . . . . 10  |-  ( Z  =  ran  F  -> 
( F : X --> Z 
<->  F : X --> ran  F
) )
405, 39ax-mp 8 . . . . . . . . 9  |-  ( F : X --> Z  <->  F : X
--> ran  F )
4138, 40mpbir 201 . . . . . . . 8  |-  F : X
--> Z
4241ffvelrni 5869 . . . . . . 7  |-  ( ( z G w )  e.  X  ->  ( F `  ( z G w ) )  e.  Z )
4336, 42syl 16 . . . . . 6  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( F `  (
z G w ) )  e.  Z )
4434, 43eqeltrd 2510 . . . . 5  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( F `  z ) H ( F `  w ) )  e.  Z )
45 oveq12 6090 . . . . . 6  |-  ( ( ( F `  z
)  =  x  /\  ( F `  w )  =  y )  -> 
( ( F `  z ) H ( F `  w ) )  =  ( x H y ) )
4645eleq1d 2502 . . . . 5  |-  ( ( ( F `  z
)  =  x  /\  ( F `  w )  =  y )  -> 
( ( ( F `
 z ) H ( F `  w
) )  e.  Z  <->  ( x H y )  e.  Z ) )
4744, 46syl5ibcom 212 . . . 4  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( ( F `
 z )  =  x  /\  ( F `
 w )  =  y )  ->  (
x H y )  e.  Z ) )
4847rexlimivv 2835 . . 3  |-  ( E. z  e.  X  E. w  e.  X  (
( F `  z
)  =  x  /\  ( F `  w )  =  y )  -> 
( x H y )  e.  Z )
4931, 48syl 16 . 2  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( x H y )  e.  Z )
508, 32grpoidcl 21805 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  U  e.  X )
517, 50ax-mp 8 . . . . . . 7  |-  U  e.  X
527, 1, 6, 8, 32, 33, 2, 3, 4, 5, 16ghomgrpilem1 25096 . . . . . . 7  |-  ( ( U  e.  X  /\  U  e.  X )  ->  ( ( F `  U ) H ( F `  U ) )  =  ( F `
 ( U G U ) ) )
5351, 51, 52mp2an 654 . . . . . 6  |-  ( ( F `  U ) H ( F `  U ) )  =  ( F `  ( U G U ) )
548, 32grpolid 21807 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  U  e.  X )  ->  ( U G U )  =  U )
557, 51, 54mp2an 654 . . . . . . 7  |-  ( U G U )  =  U
5655fveq2i 5731 . . . . . 6  |-  ( F `
 ( U G U ) )  =  ( F `  U
)
5753, 56eqtri 2456 . . . . 5  |-  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
)
58 ffvelrn 5868 . . . . . . 7  |-  ( ( F : X --> W  /\  U  e.  X )  ->  ( F `  U
)  e.  W )
5912, 51, 58mp2an 654 . . . . . 6  |-  ( F `
 U )  e.  W
60 eqid 2436 . . . . . . 7  |-  (GId `  H )  =  (GId
`  H )
612, 60grpoid 21811 . . . . . 6  |-  ( ( H  e.  GrpOp  /\  ( F `  U )  e.  W )  ->  (
( F `  U
)  =  (GId `  H )  <->  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
) ) )
621, 59, 61mp2an 654 . . . . 5  |-  ( ( F `  U )  =  (GId `  H
)  <->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) )
6357, 62mpbir 201 . . . 4  |-  ( F `
 U )  =  (GId `  H )
643, 63eqtr4i 2459 . . 3  |-  T  =  ( F `  U
)
65 ffvelrn 5868 . . . 4  |-  ( ( F : X --> Z  /\  U  e.  X )  ->  ( F `  U
)  e.  Z )
6641, 51, 65mp2an 654 . . 3  |-  ( F `
 U )  e.  Z
6764, 66eqeltri 2506 . 2  |-  T  e.  Z
688, 33grpoinvcl 21814 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  z  e.  X )  ->  ( N `  z )  e.  X )
697, 68mpan 652 . . . . . . . . . 10  |-  ( z  e.  X  ->  ( N `  z )  e.  X )
707, 1, 6, 8, 32, 33, 2, 3, 4, 5, 16ghomgrpilem1 25096 . . . . . . . . . 10  |-  ( ( z  e.  X  /\  ( N `  z )  e.  X )  -> 
( ( F `  z ) H ( F `  ( N `
 z ) ) )  =  ( F `
 ( z G ( N `  z
) ) ) )
7169, 70mpdan 650 . . . . . . . . 9  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  ( F `  ( z G ( N `  z ) ) ) )
728, 32, 33grporinv 21817 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  z  e.  X )  ->  (
z G ( N `
 z ) )  =  U )
737, 72mpan 652 . . . . . . . . . 10  |-  ( z  e.  X  ->  (
z G ( N `
 z ) )  =  U )
7473fveq2d 5732 . . . . . . . . 9  |-  ( z  e.  X  ->  ( F `  ( z G ( N `  z ) ) )  =  ( F `  U ) )
7571, 74eqtrd 2468 . . . . . . . 8  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  ( F `  U ) )
7675, 64syl6eqr 2486 . . . . . . 7  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  T )
7712ffvelrni 5869 . . . . . . . 8  |-  ( z  e.  X  ->  ( F `  z )  e.  W )
7812ffvelrni 5869 . . . . . . . . 9  |-  ( ( N `  z )  e.  X  ->  ( F `  ( N `  z ) )  e.  W )
7969, 78syl 16 . . . . . . . 8  |-  ( z  e.  X  ->  ( F `  ( N `  z ) )  e.  W )
802, 3, 4grpoinvid1 21818 . . . . . . . . 9  |-  ( ( H  e.  GrpOp  /\  ( F `  z )  e.  W  /\  ( F `  ( N `  z ) )  e.  W )  ->  (
( M `  ( F `  z )
)  =  ( F `
 ( N `  z ) )  <->  ( ( F `  z ) H ( F `  ( N `  z ) ) )  =  T ) )
811, 80mp3an1 1266 . . . . . . . 8  |-  ( ( ( F `  z
)  e.  W  /\  ( F `  ( N `
 z ) )  e.  W )  -> 
( ( M `  ( F `  z ) )  =  ( F `
 ( N `  z ) )  <->  ( ( F `  z ) H ( F `  ( N `  z ) ) )  =  T ) )
8277, 79, 81syl2anc 643 . . . . . . 7  |-  ( z  e.  X  ->  (
( M `  ( F `  z )
)  =  ( F `
 ( N `  z ) )  <->  ( ( F `  z ) H ( F `  ( N `  z ) ) )  =  T ) )
8376, 82mpbird 224 . . . . . 6  |-  ( z  e.  X  ->  ( M `  ( F `  z ) )  =  ( F `  ( N `  z )
) )
8441ffvelrni 5869 . . . . . . 7  |-  ( ( N `  z )  e.  X  ->  ( F `  ( N `  z ) )  e.  Z )
8569, 84syl 16 . . . . . 6  |-  ( z  e.  X  ->  ( F `  ( N `  z ) )  e.  Z )
8683, 85eqeltrd 2510 . . . . 5  |-  ( z  e.  X  ->  ( M `  ( F `  z ) )  e.  Z )
87 fveq2 5728 . . . . . 6  |-  ( ( F `  z )  =  x  ->  ( M `  ( F `  z ) )  =  ( M `  x
) )
8887eleq1d 2502 . . . . 5  |-  ( ( F `  z )  =  x  ->  (
( M `  ( F `  z )
)  e.  Z  <->  ( M `  x )  e.  Z
) )
8986, 88syl5ibcom 212 . . . 4  |-  ( z  e.  X  ->  (
( F `  z
)  =  x  -> 
( M `  x
)  e.  Z ) )
9089rexlimiv 2824 . . 3  |-  ( E. z  e.  X  ( F `  z )  =  x  ->  ( M `  x )  e.  Z )
9122, 90sylbi 188 . 2  |-  ( x  e.  Z  ->  ( M `  x )  e.  Z )
921, 2, 3, 4, 15, 16, 49, 67, 91issubgoi 21898 1  |-  S  e.  ( SubGrpOp `  H )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320    X. cxp 4876   ran crn 4879    |` cres 4880    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   GrpOpcgr 21774  GIdcgi 21775   invcgn 21776   SubGrpOpcsubgo 21889   GrpOpHom cghom 21945
This theorem is referenced by:  ghomgrpi  25098
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
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