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Theorem resghm 16154
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 2441 . 2  |-  ( Base `  U )  =  (
Base `  U )
2 eqid 2441 . 2  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2441 . 2  |-  ( +g  `  U )  =  ( +g  `  U )
4 eqid 2441 . 2  |-  ( +g  `  T )  =  ( +g  `  T )
5 resghm.u . . . 4  |-  U  =  ( Ss  X )
65subggrp 16075 . . 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 16141 . . 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 2441 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
1110, 2ghmf 16142 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
1210subgss 16073 . . . 4  |-  ( X  e.  (SubGrp `  S
)  ->  X  C_  ( Base `  S ) )
13 fssres 5738 . . . 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 14562 . . . . 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 5705 . . 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 2514 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( a  e.  X  <->  a  e.  (
Base `  U )
) )
21 eleq2 2514 . . . . . 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 3487 . . . . . 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 3487 . . . . . 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 2441 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
3110, 30, 4ghmlin 16143 . . . . 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 1227 . . . 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 14613 . . . . . . . 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 6295 . . . . . 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 5857 . . . . 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 16082 . . . . . . . 8  |-  ( ( X  e.  (SubGrp `  S )  /\  a  e.  X  /\  b  e.  X )  ->  (
a ( +g  `  S
) b )  e.  X )
38373expb 1196 . . . . . . 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 5867 . . . . . 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 2484 . . . 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 5867 . . . . . 6  |-  ( a  e.  X  ->  (
( F  |`  X ) `
 a )  =  ( F `  a
) )
44 fvres 5867 . . . . . 6  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4543, 44oveqan12d 6297 . . . . 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 2492 . . 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 16147 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 1381    e. wcel 1802    C_ wss 3459    |` cres 4988   -->wf 5571   ` cfv 5575  (class class class)co 6278   Basecbs 14506   ↾s cress 14507   +g cplusg 14571   Grpcgrp 15924  SubGrpcsubg 16066    GrpHom cghm 16135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-recs 7041  df-rdg 7075  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-grp 15928  df-subg 16069  df-ghm 16136
This theorem is referenced by:  ghmima  16158  resrhm  17329  reslmhm  17569
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