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Theorem resghm 14950
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 2388 . 2  |-  ( Base `  U )  =  (
Base `  U )
2 eqid 2388 . 2  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2388 . 2  |-  ( +g  `  U )  =  ( +g  `  U )
4 eqid 2388 . 2  |-  ( +g  `  T )  =  ( +g  `  T )
5 resghm.u . . . 4  |-  U  =  ( Ss  X )
65subggrp 14875 . . 3  |-  ( X  e.  (SubGrp `  S
)  ->  U  e.  Grp )
76adantl 453 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  U  e.  Grp )
8 ghmgrp2 14937 . . 3  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
98adantr 452 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  T  e.  Grp )
10 eqid 2388 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
1110, 2ghmf 14938 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
1210subgss 14873 . . . 4  |-  ( X  e.  (SubGrp `  S
)  ->  X  C_  ( Base `  S ) )
13 fssres 5551 . . . 4  |-  ( ( F : ( Base `  S ) --> ( Base `  T )  /\  X  C_  ( Base `  S
) )  ->  ( F  |`  X ) : X --> ( Base `  T
) )
1411, 12, 13syl2an 464 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X ) : X --> ( Base `  T )
)
1512adantl 453 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  X  C_  ( Base `  S ) )
165, 10ressbas2 13448 . . . . 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 5522 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( ( F  |`  X ) : X --> ( Base `  T
)  <->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
) )
1914, 18mpbid 202 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X ) : (
Base `  U ) --> ( Base `  T )
)
20 eleq2 2449 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( a  e.  X  <->  a  e.  (
Base `  U )
) )
21 eleq2 2449 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( b  e.  X  <->  b  e.  (
Base `  U )
) )
2220, 21anbi12d 692 . . . . 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 472 . . 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 731 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  F  e.  ( S  GrpHom  T ) )
2615sselda 3292 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  a  e.  X )  ->  a  e.  ( Base `  S
) )
2726adantrr 698 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  a  e.  ( Base `  S )
)
2815sselda 3292 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  b  e.  X )  ->  b  e.  ( Base `  S
) )
2928adantrl 697 . . . . 5  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  b  e.  ( Base `  S )
)
30 eqid 2388 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
3110, 30, 4ghmlin 14939 . . . . 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 1184 . . . 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 13499 . . . . . . . 8  |-  ( X  e.  (SubGrp `  S
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3433ad2antlr 708 . . . . . . 7  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( +g  `  S )  =  ( +g  `  U ) )
3534oveqd 6038 . . . . . 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 5673 . . . . 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 14882 . . . . . . . 8  |-  ( ( X  e.  (SubGrp `  S )  /\  a  e.  X  /\  b  e.  X )  ->  (
a ( +g  `  S
) b )  e.  X )
38373expb 1154 . . . . . . 7  |-  ( ( X  e.  (SubGrp `  S )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
( +g  `  S ) b )  e.  X
)
3938adantll 695 . . . . . 6  |-  ( ( ( F  e.  ( S  GrpHom  T )  /\  X  e.  (SubGrp `  S
) )  /\  (
a  e.  X  /\  b  e.  X )
)  ->  ( a
( +g  `  S ) b )  e.  X
)
40 fvres 5686 . . . . . 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 2422 . . . 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 5686 . . . . . 6  |-  ( a  e.  X  ->  (
( F  |`  X ) `
 a )  =  ( F `  a
) )
44 fvres 5686 . . . . . 6  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4543, 44oveqan12d 6040 . . . . 5  |-  ( ( a  e.  X  /\  b  e.  X )  ->  ( ( ( F  |`  X ) `  a
) ( +g  `  T
) ( ( F  |`  X ) `  b
) )  =  ( ( F `  a
) ( +g  `  T
) ( F `  b ) ) )
4645adantl 453 . . . 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 2430 . . 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 457 . 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 14943 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3264    |` cres 4821   -->wf 5391   ` cfv 5395  (class class class)co 6021   Basecbs 13397   ↾s cress 13398   +g cplusg 13457   Grpcgrp 14613  SubGrpcsubg 14866    GrpHom cghm 14931
This theorem is referenced by:  ghmima  14954  resrhm  15825  reslmhm  16056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mnd 14618  df-grp 14740  df-subg 14869  df-ghm 14932
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