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Theorem resghm 15756
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 15677 . . 3  |-  ( X  e.  (SubGrp `  S
)  ->  U  e.  Grp )
76adantl 463 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  U  e.  Grp )
8 ghmgrp2 15743 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
98adantr 462 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  T  e.  Grp )
10 eqid 2441 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
1110, 2ghmf 15744 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
1210subgss 15675 . . . 4  |-  ( X  e.  (SubGrp `  S
)  ->  X  C_  ( Base `  S ) )
13 fssres 5575 . . . 4  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  X  C_  ( Base `  S
) )  ->  ( F  |`  X ) : X --> ( Base `  T
) )
1411, 12, 13syl2an 474 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X ) : X --> ( Base `  T )
)
1512adantl 463 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  X  C_  ( Base `  S ) )
165, 10ressbas2 14225 . . . . 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 5544 . . 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 2502 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( a  e.  X  <->  a  e.  (
Base `  U )
) )
21 eleq2 2502 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( b  e.  X  <->  b  e.  (
Base `  U )
) )
2220, 21anbi12d 705 . . . . 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 482 . . 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 748 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  F  e.  ( S  GrpHom  T ) )
2615sselda 3353 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  a  e.  X )  ->  a  e.  ( Base `  S
) )
2726adantrr 711 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  a  e.  ( Base `  S )
)
2815sselda 3353 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  b  e.  X )  ->  b  e.  ( Base `  S
) )
2928adantrl 710 . . . . 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 15745 . . . . 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 1213 . . . 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 14276 . . . . . . . 8  |-  ( X  e.  (SubGrp `  S
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3433ad2antlr 721 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3534oveqd 6107 . . . . . 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 5692 . . . . 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 15684 . . . . . . . 8  |-  ( ( X  e.  (SubGrp `  S )  /\  a  e.  X  /\  b  e.  X )  ->  (
a ( +g  `  S
) b )  e.  X )
38373expb 1183 . . . . . . 7  |-  ( ( X  e.  (SubGrp `  S )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
( +g  `  S ) b )  e.  X
)
3938adantll 708 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
( +g  `  S ) b )  e.  X
)
40 fvres 5701 . . . . . 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 2475 . . . 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 5701 . . . . . 6  |-  ( a  e.  X  ->  (
( F  |`  X ) `
 a )  =  ( F `  a
) )
44 fvres 5701 . . . . . 6  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4543, 44oveqan12d 6109 . . . . 5  |-  ( ( a  e.  X  /\  b  e.  X )  ->  ( ( ( F  |`  X ) `  a
) ( +g  `  T
) ( ( F  |`  X ) `  b
) )  =  ( ( F `  a
) ( +g  `  T
) ( F `  b ) ) )
4645adantl 463 . . . 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 2483 . . 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 467 . 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 15749 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 1364    e. wcel 1761    C_ wss 3325    |` cres 4838   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   ↾s cress 14171   +g cplusg 14234   Grpcgrp 15406  SubGrpcsubg 15668    GrpHom cghm 15737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mnd 15411  df-grp 15538  df-subg 15671  df-ghm 15738
This theorem is referenced by:  ghmima  15760  resrhm  16874  reslmhm  17111
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