Users' Mathboxes Mathbox for Paul Chapman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ghomgrpilem2 Structured version   Unicode version

Theorem ghomgrpilem2 25102
Description: Lemma for ghomgrpi 25103. (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 21956 . . . . . . 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 655 . . . . . 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 201 . . . . 5  |-  ( F : X --> W  /\  A. x  e.  X  A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) )
1211simpli 446 . . . 4  |-  F : X
--> W
13 frn 5600 . . . 4  |-  ( F : X --> W  ->  ran  F  C_  W )
1412, 13ax-mp 5 . . 3  |-  ran  F  C_  W
155, 14eqsstri 3380 . 2  |-  Z  C_  W
16 ghomgrpilem1.11 . 2  |-  S  =  ( H  |`  ( Z  X.  Z ) )
175eleq2i 2502 . . . . . . 7  |-  ( x  e.  Z  <->  x  e.  ran  F )
18 ffn 5594 . . . . . . . . 9  |-  ( F : X --> W  ->  F  Fn  X )
1912, 18ax-mp 5 . . . . . . . 8  |-  F  Fn  X
20 fvelrnb 5777 . . . . . . . 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 242 . . . . . 6  |-  ( x  e.  Z  <->  E. z  e.  X  ( F `  z )  =  x )
2322biimpi 188 . . . . 5  |-  ( x  e.  Z  ->  E. z  e.  X  ( F `  z )  =  x )
245eleq2i 2502 . . . . . . 7  |-  ( y  e.  Z  <->  y  e.  ran  F )
25 fvelrnb 5777 . . . . . . . 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 242 . . . . . 6  |-  ( y  e.  Z  <->  E. w  e.  X  ( F `  w )  =  y )
2827biimpi 188 . . . . 5  |-  ( y  e.  Z  ->  E. w  e.  X  ( F `  w )  =  y )
2923, 28anim12i 551 . . . 4  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( E. z  e.  X  ( F `  z )  =  x  /\  E. w  e.  X  ( F `  w )  =  y ) )
30 reeanv 2877 . . . 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 205 . . 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 25101 . . . . . 6  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( F `  z ) H ( F `  w ) )  =  ( F `
 ( z G w ) ) )
358grpocl 21793 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  z  e.  X  /\  w  e.  X )  ->  (
z G w )  e.  X )
367, 35mp3an1 1267 . . . . . . 7  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( z G w )  e.  X )
37 dffn3 5601 . . . . . . . . . 10  |-  ( F  Fn  X  <->  F : X
--> ran  F )
3819, 37mpbi 201 . . . . . . . . 9  |-  F : X
--> ran  F
39 feq3 5581 . . . . . . . . . 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 202 . . . . . . . 8  |-  F : X
--> Z
4241ffvelrni 5872 . . . . . . 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 2512 . . . . 5  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( F `  z ) H ( F `  w ) )  e.  Z )
45 oveq12 6093 . . . . . 6  |-  ( ( ( F `  z
)  =  x  /\  ( F `  w )  =  y )  -> 
( ( F `  z ) H ( F `  w ) )  =  ( x H y ) )
4645eleq1d 2504 . . . . 5  |-  ( ( ( F `  z
)  =  x  /\  ( F `  w )  =  y )  -> 
( ( ( F `
 z ) H ( F `  w
) )  e.  Z  <->  ( x H y )  e.  Z ) )
4744, 46syl5ibcom 213 . . . 4  |-  ( ( z  e.  X  /\  w  e.  X )  ->  ( ( ( F `
 z )  =  x  /\  ( F `
 w )  =  y )  ->  (
x H y )  e.  Z ) )
4847rexlimivv 2837 . . 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 21810 . . . . . . . 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 25101 . . . . . . 7  |-  ( ( U  e.  X  /\  U  e.  X )  ->  ( ( F `  U ) H ( F `  U ) )  =  ( F `
 ( U G U ) ) )
5351, 51, 52mp2an 655 . . . . . 6  |-  ( ( F `  U ) H ( F `  U ) )  =  ( F `  ( U G U ) )
548, 32grpolid 21812 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  U  e.  X )  ->  ( U G U )  =  U )
557, 51, 54mp2an 655 . . . . . . 7  |-  ( U G U )  =  U
5655fveq2i 5734 . . . . . 6  |-  ( F `
 ( U G U ) )  =  ( F `  U
)
5753, 56eqtri 2458 . . . . 5  |-  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
)
58 ffvelrn 5871 . . . . . . 7  |-  ( ( F : X --> W  /\  U  e.  X )  ->  ( F `  U
)  e.  W )
5912, 51, 58mp2an 655 . . . . . 6  |-  ( F `
 U )  e.  W
60 eqid 2438 . . . . . . 7  |-  (GId `  H )  =  (GId
`  H )
612, 60grpoid 21816 . . . . . 6  |-  ( ( H  e.  GrpOp  /\  ( F `  U )  e.  W )  ->  (
( F `  U
)  =  (GId `  H )  <->  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
) ) )
621, 59, 61mp2an 655 . . . . 5  |-  ( ( F `  U )  =  (GId `  H
)  <->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) )
6357, 62mpbir 202 . . . 4  |-  ( F `
 U )  =  (GId `  H )
643, 63eqtr4i 2461 . . 3  |-  T  =  ( F `  U
)
65 ffvelrn 5871 . . . 4  |-  ( ( F : X --> Z  /\  U  e.  X )  ->  ( F `  U
)  e.  Z )
6641, 51, 65mp2an 655 . . 3  |-  ( F `
 U )  e.  Z
6764, 66eqeltri 2508 . 2  |-  T  e.  Z
688, 33grpoinvcl 21819 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  z  e.  X )  ->  ( N `  z )  e.  X )
697, 68mpan 653 . . . . . . . . . 10  |-  ( z  e.  X  ->  ( N `  z )  e.  X )
707, 1, 6, 8, 32, 33, 2, 3, 4, 5, 16ghomgrpilem1 25101 . . . . . . . . . 10  |-  ( ( z  e.  X  /\  ( N `  z )  e.  X )  -> 
( ( F `  z ) H ( F `  ( N `
 z ) ) )  =  ( F `
 ( z G ( N `  z
) ) ) )
7169, 70mpdan 651 . . . . . . . . 9  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  ( F `  ( z G ( N `  z ) ) ) )
728, 32, 33grporinv 21822 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  z  e.  X )  ->  (
z G ( N `
 z ) )  =  U )
737, 72mpan 653 . . . . . . . . . 10  |-  ( z  e.  X  ->  (
z G ( N `
 z ) )  =  U )
7473fveq2d 5735 . . . . . . . . 9  |-  ( z  e.  X  ->  ( F `  ( z G ( N `  z ) ) )  =  ( F `  U ) )
7571, 74eqtrd 2470 . . . . . . . 8  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  ( F `  U ) )
7675, 64syl6eqr 2488 . . . . . . 7  |-  ( z  e.  X  ->  (
( F `  z
) H ( F `
 ( N `  z ) ) )  =  T )
7712ffvelrni 5872 . . . . . . . 8  |-  ( z  e.  X  ->  ( F `  z )  e.  W )
7812ffvelrni 5872 . . . . . . . . 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 21823 . . . . . . . . 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 1267 . . . . . . . 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 644 . . . . . . 7  |-  ( z  e.  X  ->  (
( M `  ( F `  z )
)  =  ( F `
 ( N `  z ) )  <->  ( ( F `  z ) H ( F `  ( N `  z ) ) )  =  T ) )
8376, 82mpbird 225 . . . . . 6  |-  ( z  e.  X  ->  ( M `  ( F `  z ) )  =  ( F `  ( N `  z )
) )
8441ffvelrni 5872 . . . . . . 7  |-  ( ( N `  z )  e.  X  ->  ( F `  ( N `  z ) )  e.  Z )
8569, 84syl 16 . . . . . 6  |-  ( z  e.  X  ->  ( F `  ( N `  z ) )  e.  Z )
8683, 85eqeltrd 2512 . . . . 5  |-  ( z  e.  X  ->  ( M `  ( F `  z ) )  e.  Z )
87 fveq2 5731 . . . . . 6  |-  ( ( F `  z )  =  x  ->  ( M `  ( F `  z ) )  =  ( M `  x
) )
8887eleq1d 2504 . . . . 5  |-  ( ( F `  z )  =  x  ->  (
( M `  ( F `  z )
)  e.  Z  <->  ( M `  x )  e.  Z
) )
8986, 88syl5ibcom 213 . . . 4  |-  ( z  e.  X  ->  (
( F `  z
)  =  x  -> 
( M `  x
)  e.  Z ) )
9089rexlimiv 2826 . . 3  |-  ( E. z  e.  X  ( F `  z )  =  x  ->  ( M `  x )  e.  Z )
9122, 90sylbi 189 . 2  |-  ( x  e.  Z  ->  ( M `  x )  e.  Z )
921, 2, 3, 4, 15, 16, 49, 67, 91issubgoi 21903 1  |-  S  e.  ( SubGrpOp `  H )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322    X. cxp 4879   ran crn 4882    |` cres 4883    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   GrpOpcgr 21779  GIdcgi 21780   invcgn 21781   SubGrpOpcsubgo 21894   GrpOpHom cghom 21950
This theorem is referenced by:  ghomgrpi  25103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786  df-subgo 21895  df-ghom 21951
  Copyright terms: Public domain W3C validator