MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resghm Structured version   Unicode version

Theorem resghm 15784
Description: Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypothesis
Ref Expression
resghm.u  |-  U  =  ( Ss  X )
Assertion
Ref Expression
resghm  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )

Proof of Theorem resghm
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . 2  |-  ( Base `  U )  =  (
Base `  U )
2 eqid 2443 . 2  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2443 . 2  |-  ( +g  `  U )  =  ( +g  `  U )
4 eqid 2443 . 2  |-  ( +g  `  T )  =  ( +g  `  T )
5 resghm.u . . . 4  |-  U  =  ( Ss  X )
65subggrp 15705 . . 3  |-  ( X  e.  (SubGrp `  S
)  ->  U  e.  Grp )
76adantl 466 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  U  e.  Grp )
8 ghmgrp2 15771 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
98adantr 465 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  T  e.  Grp )
10 eqid 2443 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
1110, 2ghmf 15772 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
1210subgss 15703 . . . 4  |-  ( X  e.  (SubGrp `  S
)  ->  X  C_  ( Base `  S ) )
13 fssres 5599 . . . 4  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  X  C_  ( Base `  S
) )  ->  ( F  |`  X ) : X --> ( Base `  T
) )
1411, 12, 13syl2an 477 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X ) : X --> ( Base `  T )
)
1512adantl 466 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  X  C_  ( Base `  S ) )
165, 10ressbas2 14250 . . . . 5  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  U )
)
1715, 16syl 16 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  X  =  ( Base `  U )
)
1817feq2d 5568 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( ( F  |`  X ) : X --> ( Base `  T
)  <->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
) )
1914, 18mpbid 210 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
)
20 eleq2 2504 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( a  e.  X  <->  a  e.  (
Base `  U )
) )
21 eleq2 2504 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( b  e.  X  <->  b  e.  (
Base `  U )
) )
2220, 21anbi12d 710 . . . . 5  |-  ( X  =  ( Base `  U
)  ->  ( (
a  e.  X  /\  b  e.  X )  <->  ( a  e.  ( Base `  U )  /\  b  e.  ( Base `  U
) ) ) )
2317, 22syl 16 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( (
a  e.  X  /\  b  e.  X )  <->  ( a  e.  ( Base `  U )  /\  b  e.  ( Base `  U
) ) ) )
2423biimpar 485 . . 3  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  ( Base `  U )  /\  b  e.  ( Base `  U
) ) )  -> 
( a  e.  X  /\  b  e.  X
) )
25 simpll 753 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  F  e.  ( S  GrpHom  T ) )
2615sselda 3377 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  a  e.  X )  ->  a  e.  ( Base `  S
) )
2726adantrr 716 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  a  e.  ( Base `  S )
)
2815sselda 3377 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  b  e.  X )  ->  b  e.  ( Base `  S
) )
2928adantrl 715 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  b  e.  ( Base `  S )
)
30 eqid 2443 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
3110, 30, 4ghmlin 15773 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  a  e.  ( Base `  S
)  /\  b  e.  ( Base `  S )
)  ->  ( F `  ( a ( +g  `  S ) b ) )  =  ( ( F `  a ) ( +g  `  T
) ( F `  b ) ) )
3225, 27, 29, 31syl3anc 1218 . . . 4  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( F `  ( a ( +g  `  S ) b ) )  =  ( ( F `  a ) ( +g  `  T
) ( F `  b ) ) )
335, 30ressplusg 14301 . . . . . . . 8  |-  ( X  e.  (SubGrp `  S
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3433ad2antlr 726 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3534oveqd 6129 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
( +g  `  S ) b )  =  ( a ( +g  `  U
) b ) )
3635fveq2d 5716 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( ( F  |`  X ) `  ( a ( +g  `  S ) b ) )  =  ( ( F  |`  X ) `  ( a ( +g  `  U ) b ) ) )
3730subgcl 15712 . . . . . . . 8  |-  ( ( X  e.  (SubGrp `  S )  /\  a  e.  X  /\  b  e.  X )  ->  (
a ( +g  `  S
) b )  e.  X )
38373expb 1188 . . . . . . 7  |-  ( ( X  e.  (SubGrp `  S )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
( +g  `  S ) b )  e.  X
)
3938adantll 713 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
( +g  `  S ) b )  e.  X
)
40 fvres 5725 . . . . . 6  |-  ( ( a ( +g  `  S
) b )  e.  X  ->  ( ( F  |`  X ) `  ( a ( +g  `  S ) b ) )  =  ( F `
 ( a ( +g  `  S ) b ) ) )
4139, 40syl 16 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( ( F  |`  X ) `  ( a ( +g  `  S ) b ) )  =  ( F `
 ( a ( +g  `  S ) b ) ) )
4236, 41eqtr3d 2477 . . . 4  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( ( F  |`  X ) `  ( a ( +g  `  U ) b ) )  =  ( F `
 ( a ( +g  `  S ) b ) ) )
43 fvres 5725 . . . . . 6  |-  ( a  e.  X  ->  (
( F  |`  X ) `
 a )  =  ( F `  a
) )
44 fvres 5725 . . . . . 6  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4543, 44oveqan12d 6131 . . . . 5  |-  ( ( a  e.  X  /\  b  e.  X )  ->  ( ( ( F  |`  X ) `  a
) ( +g  `  T
) ( ( F  |`  X ) `  b
) )  =  ( ( F `  a
) ( +g  `  T
) ( F `  b ) ) )
4645adantl 466 . . . 4  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( (
( F  |`  X ) `
 a ) ( +g  `  T ) ( ( F  |`  X ) `  b
) )  =  ( ( F `  a
) ( +g  `  T
) ( F `  b ) ) )
4732, 42, 463eqtr4d 2485 . . 3  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( ( F  |`  X ) `  ( a ( +g  `  U ) b ) )  =  ( ( ( F  |`  X ) `
 a ) ( +g  `  T ) ( ( F  |`  X ) `  b
) ) )
4824, 47syldan 470 . 2  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  ( Base `  U )  /\  b  e.  ( Base `  U
) ) )  -> 
( ( F  |`  X ) `  (
a ( +g  `  U
) b ) )  =  ( ( ( F  |`  X ) `  a ) ( +g  `  T ) ( ( F  |`  X ) `  b ) ) )
491, 2, 3, 4, 7, 9, 19, 48isghmd 15777 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3349    |` cres 4863   -->wf 5435   ` cfv 5439  (class class class)co 6112   Basecbs 14195   ↾s cress 14196   +g cplusg 14259   Grpcgrp 15431  SubGrpcsubg 15696    GrpHom cghm 15765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mnd 15436  df-grp 15566  df-subg 15699  df-ghm 15766
This theorem is referenced by:  ghmima  15788  resrhm  16916  reslmhm  17155
  Copyright terms: Public domain W3C validator