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Theorem ghomgrpilem2 29201
Description: Lemma for ghomgrpi 29202. (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, 2elghomOLD 25491 . . . . . . 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 672 . . . . . 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 208 . . . . 5  |-  ( F : X --> W  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) )
1211simpli 458 . . . 4  |-  F : X
--> W
13 frn 5743 . . . 4  |-  ( F : X --> W  ->  ran  F  C_  W )
1412, 13ax-mp 5 . . 3  |-  ran  F  C_  W
155, 14eqsstri 3529 . 2  |-  Z  C_  W
16 ghomgrpilem1.11 . 2  |-  S  =  ( H  |`  ( Z  X.  Z ) )
175eleq2i 2535 . . . . . . 7  |-  ( x  e.  Z  <->  x  e.  ran  F )
18 ffn 5737 . . . . . . . . 9  |-  ( F : X --> W  ->  F  Fn  X )
1912, 18ax-mp 5 . . . . . . . 8  |-  F  Fn  X
20 fvelrnb 5920 . . . . . . . 8  |-  ( F  Fn  X  ->  (
x  e.  ran  F  <->  E. z  e.  X  ( F `  z )  =  x ) )
2119, 20ax-mp 5 . . . . . . 7  |-  ( x  e.  ran  F  <->  E. z  e.  X  ( F `  z )  =  x )
2217, 21bitri 249 . . . . . 6  |-  ( x  e.  Z  <->  E. z  e.  X  ( F `  z )  =  x )
2322biimpi 194 . . . . 5  |-  ( x  e.  Z  ->  E. z  e.  X  ( F `  z )  =  x )
245eleq2i 2535 . . . . . . 7  |-  ( y  e.  Z  <->  y  e.  ran  F )
25 fvelrnb 5920 . . . . . . . 8  |-  ( F  Fn  X  ->  (
y  e.  ran  F  <->  E. w  e.  X  ( F `  w )  =  y ) )
2619, 25ax-mp 5 . . . . . . 7  |-  ( y  e.  ran  F  <->  E. w  e.  X  ( F `  w )  =  y )
2724, 26bitri 249 . . . . . 6  |-  ( y  e.  Z  <->  E. w  e.  X  ( F `  w )  =  y )
2827biimpi 194 . . . . 5  |-  ( y  e.  Z  ->  E. w  e.  X  ( F `  w )  =  y )
2923, 28anim12i 566 . . . 4  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( E. z  e.  X  ( F `  z )  =  x  /\  E. w  e.  X  ( F `  w )  =  y ) )
30 reeanv 3025 . . . 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 212 . . 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 29200 . . . . . 6  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( F `  z ) H ( F `  w ) )  =  ( F `
 ( z G w ) ) )
358grpocl 25328 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  z  e.  X  /\  w  e.  X )  ->  (
z G w )  e.  X )
367, 35mp3an1 1311 . . . . . . 7  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( z G w )  e.  X )
37 dffn3 5744 . . . . . . . . . 10  |-  ( F  Fn  X  <->  F : X
--> ran  F )
3819, 37mpbi 208 . . . . . . . . 9  |-  F : X
--> ran  F
39 feq3 5721 . . . . . . . . . 10  |-  ( Z  =  ran  F  -> 
( F : X --> Z 
<->  F : X --> ran  F
) )
405, 39ax-mp 5 . . . . . . . . 9  |-  ( F : X --> Z  <->  F : X
--> ran  F )
4138, 40mpbir 209 . . . . . . . 8  |-  F : X
--> Z
4241ffvelrni 6031 . . . . . . 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 2545 . . . . 5  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( F `  z ) H ( F `  w ) )  e.  Z )
45 oveq12 6305 . . . . . 6  |-  ( ( ( F `  z
)  =  x  /\  ( F `  w )  =  y )  -> 
( ( F `  z ) H ( F `  w ) )  =  ( x H y ) )
4645eleq1d 2526 . . . . 5  |-  ( ( ( F `  z
)  =  x  /\  ( F `  w )  =  y )  -> 
( ( ( F `
 z ) H ( F `  w
) )  e.  Z  <->  ( x H y )  e.  Z ) )
4744, 46syl5ibcom 220 . . . 4  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( ( F `
 z )  =  x  /\  ( F `
 w )  =  y )  ->  (
x H y )  e.  Z ) )
4847rexlimivv 2954 . . 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 25345 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  U  e.  X )
517, 50ax-mp 5 . . . . . . 7  |-  U  e.  X
527, 1, 6, 8, 32, 33, 2, 3, 4, 5, 16ghomgrpilem1 29200 . . . . . . 7  |-  ( ( U  e.  X  /\  U  e.  X )  ->  ( ( F `  U ) H ( F `  U ) )  =  ( F `
 ( U G U ) ) )
5351, 51, 52mp2an 672 . . . . . 6  |-  ( ( F `  U ) H ( F `  U ) )  =  ( F `  ( U G U ) )
548, 32grpolid 25347 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  U  e.  X )  ->  ( U G U )  =  U )
557, 51, 54mp2an 672 . . . . . . 7  |-  ( U G U )  =  U
5655fveq2i 5875 . . . . . 6  |-  ( F `
 ( U G U ) )  =  ( F `  U
)
5753, 56eqtri 2486 . . . . 5  |-  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
)
58 ffvelrn 6030 . . . . . . 7  |-  ( ( F : X --> W  /\  U  e.  X )  ->  ( F `  U
)  e.  W )
5912, 51, 58mp2an 672 . . . . . 6  |-  ( F `
 U )  e.  W
60 eqid 2457 . . . . . . 7  |-  (GId `  H )  =  (GId
`  H )
612, 60grpoid 25351 . . . . . 6  |-  ( ( H  e.  GrpOp  /\  ( F `  U )  e.  W )  ->  (
( F `  U
)  =  (GId `  H )  <->  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
) ) )
621, 59, 61mp2an 672 . . . . 5  |-  ( ( F `  U )  =  (GId `  H
)  <->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) )
6357, 62mpbir 209 . . . 4  |-  ( F `
 U )  =  (GId `  H )
643, 63eqtr4i 2489 . . 3  |-  T  =  ( F `  U
)
65 ffvelrn 6030 . . . 4  |-  ( ( F : X --> Z  /\  U  e.  X )  ->  ( F `  U
)  e.  Z )
6641, 51, 65mp2an 672 . . 3  |-  ( F `
 U )  e.  Z
6764, 66eqeltri 2541 . 2  |-  T  e.  Z
688, 33grpoinvcl 25354 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  z  e.  X )  ->  ( N `  z )  e.  X )
697, 68mpan 670 . . . . . . . . . 10  |-  ( z  e.  X  ->  ( N `  z )  e.  X )
707, 1, 6, 8, 32, 33, 2, 3, 4, 5, 16ghomgrpilem1 29200 . . . . . . . . . 10  |-  ( ( z  e.  X  /\  ( N `  z )  e.  X )  -> 
( ( F `  z ) H ( F `  ( N `
 z ) ) )  =  ( F `
 ( z G ( N `  z
) ) ) )
7169, 70mpdan 668 . . . . . . . . 9  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  ( F `  ( z G ( N `  z ) ) ) )
728, 32, 33grporinv 25357 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  z  e.  X )  ->  (
z G ( N `
 z ) )  =  U )
737, 72mpan 670 . . . . . . . . . 10  |-  ( z  e.  X  ->  (
z G ( N `
 z ) )  =  U )
7473fveq2d 5876 . . . . . . . . 9  |-  ( z  e.  X  ->  ( F `  ( z G ( N `  z ) ) )  =  ( F `  U ) )
7571, 74eqtrd 2498 . . . . . . . 8  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  ( F `  U ) )
7675, 64syl6eqr 2516 . . . . . . 7  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  T )
7712ffvelrni 6031 . . . . . . . 8  |-  ( z  e.  X  ->  ( F `  z )  e.  W )
7812ffvelrni 6031 . . . . . . . . 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 25358 . . . . . . . . 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 1311 . . . . . . . 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 661 . . . . . . 7  |-  ( z  e.  X  ->  (
( M `  ( F `  z )
)  =  ( F `
 ( N `  z ) )  <->  ( ( F `  z ) H ( F `  ( N `  z ) ) )  =  T ) )
8376, 82mpbird 232 . . . . . 6  |-  ( z  e.  X  ->  ( M `  ( F `  z ) )  =  ( F `  ( N `  z )
) )
8441ffvelrni 6031 . . . . . . 7  |-  ( ( N `  z )  e.  X  ->  ( F `  ( N `  z ) )  e.  Z )
8569, 84syl 16 . . . . . 6  |-  ( z  e.  X  ->  ( F `  ( N `  z ) )  e.  Z )
8683, 85eqeltrd 2545 . . . . 5  |-  ( z  e.  X  ->  ( M `  ( F `  z ) )  e.  Z )
87 fveq2 5872 . . . . . 6  |-  ( ( F `  z )  =  x  ->  ( M `  ( F `  z ) )  =  ( M `  x
) )
8887eleq1d 2526 . . . . 5  |-  ( ( F `  z )  =  x  ->  (
( M `  ( F `  z )
)  e.  Z  <->  ( M `  x )  e.  Z
) )
8986, 88syl5ibcom 220 . . . 4  |-  ( z  e.  X  ->  (
( F `  z
)  =  x  -> 
( M `  x
)  e.  Z ) )
9089rexlimiv 2943 . . 3  |-  ( E. z  e.  X  ( F `  z )  =  x  ->  ( M `  x )  e.  Z )
9122, 90sylbi 195 . 2  |-  ( x  e.  Z  ->  ( M `  x )  e.  Z )
921, 2, 3, 4, 15, 16, 49, 67, 91issubgoi 25438 1  |-  S  e.  ( SubGrpOp `  H )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471    X. cxp 5006   ran crn 5009    |` cres 5010    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   GrpOpcgr 25314  GIdcgi 25315   invcgn 25316   SubGrpOpcsubgo 25429   GrpOpHom cghomOLD 25485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-grpo 25319  df-gid 25320  df-ginv 25321  df-subgo 25430  df-ghomOLD 25486
This theorem is referenced by:  ghomgrpi  29202
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