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

Theorem resghm 15012
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 2435 . 2  |-  ( Base `  U )  =  (
Base `  U )
2 eqid 2435 . 2  |-  ( Base `  T )  =  (
Base `  T )
3 eqid 2435 . 2  |-  ( +g  `  U )  =  ( +g  `  U )
4 eqid 2435 . 2  |-  ( +g  `  T )  =  ( +g  `  T )
5 resghm.u . . . 4  |-  U  =  ( Ss  X )
65subggrp 14937 . . 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 14999 . . 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 2435 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
1110, 2ghmf 15000 . . . 4  |-  ( F  e.  ( S  GrpHom  T )  ->  F :
( Base `  S ) --> ( Base `  T )
)
1210subgss 14935 . . . 4  |-  ( X  e.  (SubGrp `  S
)  ->  X  C_  ( Base `  S ) )
13 fssres 5602 . . . 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 13510 . . . . 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 5573 . . 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 2496 . . . . . 6  |-  ( X  =  ( Base `  U
)  ->  ( a  e.  X  <->  a  e.  (
Base `  U )
) )
21 eleq2 2496 . . . . . 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 3340 . . . . . 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 3340 . . . . . 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 2435 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
3110, 30, 4ghmlin 15001 . . . . 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 13561 . . . . . . . 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 6090 . . . . . 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 5724 . . . . 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 14944 . . . . . . . 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 5737 . . . . . 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 2469 . . . 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 5737 . . . . . 6  |-  ( a  e.  X  ->  (
( F  |`  X ) `
 a )  =  ( F `  a
) )
44 fvres 5737 . . . . . 6  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4543, 44oveqan12d 6092 . . . . 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 2477 . . 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 15005 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 1652    e. wcel 1725    C_ wss 3312    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13459   ↾s cress 13460   +g cplusg 13519   Grpcgrp 14675  SubGrpcsubg 14928    GrpHom cghm 14993
This theorem is referenced by:  ghmima  15016  resrhm  15887  reslmhm  16118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mnd 14680  df-grp 14802  df-subg 14931  df-ghm 14994
  Copyright terms: Public domain W3C validator