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Theorem reslmhm 16083
Description: Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
reslmhm.u  |-  U  =  ( LSubSp `  S )
reslmhm.r  |-  R  =  ( Ss  X )
Assertion
Ref Expression
reslmhm  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R LMHom  T ) )

Proof of Theorem reslmhm
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmlmod1 16064 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
2 reslmhm.r . . . . 5  |-  R  =  ( Ss  X )
3 reslmhm.u . . . . 5  |-  U  =  ( LSubSp `  S )
42, 3lsslmod 15991 . . . 4  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  R  e.  LMod )
51, 4sylan 458 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  R  e.  LMod )
6 lmhmlmod2 16063 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
76adantr 452 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  T  e.  LMod )
85, 7jca 519 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( R  e.  LMod  /\  T  e.  LMod ) )
9 lmghm 16062 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
109adantr 452 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  F  e.  ( S  GrpHom  T ) )
113lsssubg 15988 . . . . 5  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
121, 11sylan 458 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
132resghm 14977 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
1410, 12, 13syl2anc 643 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
15 eqid 2404 . . . . 5  |-  (Scalar `  S )  =  (Scalar `  S )
16 eqid 2404 . . . . 5  |-  (Scalar `  T )  =  (Scalar `  T )
1715, 16lmhmsca 16061 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
182, 15resssca 13559 . . . 4  |-  ( X  e.  U  ->  (Scalar `  S )  =  (Scalar `  R ) )
1917, 18sylan9eq 2456 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  T )  =  (Scalar `  R ) )
20 simpll 731 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  F  e.  ( S LMHom  T ) )
21 simprl 733 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
a  e.  ( Base `  (Scalar `  S )
) )
22 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
2322, 3lssss 15968 . . . . . . . . . 10  |-  ( X  e.  U  ->  X  C_  ( Base `  S
) )
2423adantl 453 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  C_  ( Base `  S
) )
2524adantr 452 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  X  C_  ( Base `  S
) )
262, 22ressbas2 13475 . . . . . . . . . . . 12  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  R )
)
2724, 26syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  =  ( Base `  R
) )
2827eleq2d 2471 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
b  e.  X  <->  b  e.  ( Base `  R )
) )
2928biimpar 472 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  b  e.  ( Base `  R ) )  ->  b  e.  X
)
3029adantrl 697 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  X )
3125, 30sseldd 3309 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  ( Base `  S ) )
32 eqid 2404 . . . . . . . 8  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
33 eqid 2404 . . . . . . . 8  |-  ( .s
`  S )  =  ( .s `  S
)
34 eqid 2404 . . . . . . . 8  |-  ( .s
`  T )  =  ( .s `  T
)
3515, 32, 22, 33, 34lmhmlin 16066 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( Base `  (Scalar `  S ) )  /\  b  e.  ( Base `  S ) )  -> 
( F `  (
a ( .s `  S ) b ) )  =  ( a ( .s `  T
) ( F `  b ) ) )
3620, 21, 31, 35syl3anc 1184 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
( F `  (
a ( .s `  S ) b ) )  =  ( a ( .s `  T
) ( F `  b ) ) )
371adantr 452 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  S  e.  LMod )
3837adantr 452 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  S  e.  LMod )
39 simplr 732 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  X  e.  U )
4015, 33, 32, 3lssvscl 15986 . . . . . . . 8  |-  ( ( ( S  e.  LMod  /\  X  e.  U )  /\  ( a  e.  ( Base `  (Scalar `  S ) )  /\  b  e.  X )
)  ->  ( a
( .s `  S
) b )  e.  X )
4138, 39, 21, 30, 40syl22anc 1185 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
( a ( .s
`  S ) b )  e.  X )
42 fvres 5704 . . . . . . 7  |-  ( ( a ( .s `  S ) b )  e.  X  ->  (
( F  |`  X ) `
 ( a ( .s `  S ) b ) )  =  ( F `  (
a ( .s `  S ) b ) ) )
4341, 42syl 16 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
( ( F  |`  X ) `  (
a ( .s `  S ) b ) )  =  ( F `
 ( a ( .s `  S ) b ) ) )
44 fvres 5704 . . . . . . . 8  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4544oveq2d 6056 . . . . . . 7  |-  ( b  e.  X  ->  (
a ( .s `  T ) ( ( F  |`  X ) `  b ) )  =  ( a ( .s
`  T ) ( F `  b ) ) )
4630, 45syl 16 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
( a ( .s
`  T ) ( ( F  |`  X ) `
 b ) )  =  ( a ( .s `  T ) ( F `  b
) ) )
4736, 43, 463eqtr4d 2446 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
( ( F  |`  X ) `  (
a ( .s `  S ) b ) )  =  ( a ( .s `  T
) ( ( F  |`  X ) `  b
) ) )
4847ralrimivva 2758 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  A. a  e.  ( Base `  (Scalar `  S ) ) A. b  e.  ( Base `  R ) ( ( F  |`  X ) `  ( a ( .s
`  S ) b ) )  =  ( a ( .s `  T ) ( ( F  |`  X ) `  b ) ) )
4918adantl 453 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  S )  =  (Scalar `  R ) )
5049fveq2d 5691 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  R
) ) )
512, 33ressvsca 13560 . . . . . . . . . 10  |-  ( X  e.  U  ->  ( .s `  S )  =  ( .s `  R
) )
5251adantl 453 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( .s `  S )  =  ( .s `  R
) )
5352oveqd 6057 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
a ( .s `  S ) b )  =  ( a ( .s `  R ) b ) )
5453fveq2d 5691 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
( F  |`  X ) `
 ( a ( .s `  S ) b ) )  =  ( ( F  |`  X ) `  (
a ( .s `  R ) b ) ) )
5554eqeq1d 2412 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
( ( F  |`  X ) `  (
a ( .s `  S ) b ) )  =  ( a ( .s `  T
) ( ( F  |`  X ) `  b
) )  <->  ( ( F  |`  X ) `  ( a ( .s
`  R ) b ) )  =  ( a ( .s `  T ) ( ( F  |`  X ) `  b ) ) ) )
5655ralbidv 2686 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( A. b  e.  ( Base `  R ) ( ( F  |`  X ) `
 ( a ( .s `  S ) b ) )  =  ( a ( .s
`  T ) ( ( F  |`  X ) `
 b ) )  <->  A. b  e.  ( Base `  R ) ( ( F  |`  X ) `
 ( a ( .s `  R ) b ) )  =  ( a ( .s
`  T ) ( ( F  |`  X ) `
 b ) ) ) )
5750, 56raleqbidv 2876 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( A. a  e.  ( Base `  (Scalar `  S
) ) A. b  e.  ( Base `  R
) ( ( F  |`  X ) `  (
a ( .s `  S ) b ) )  =  ( a ( .s `  T
) ( ( F  |`  X ) `  b
) )  <->  A. a  e.  ( Base `  (Scalar `  R ) ) A. b  e.  ( Base `  R ) ( ( F  |`  X ) `  ( a ( .s
`  R ) b ) )  =  ( a ( .s `  T ) ( ( F  |`  X ) `  b ) ) ) )
5848, 57mpbid 202 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  A. a  e.  ( Base `  (Scalar `  R ) ) A. b  e.  ( Base `  R ) ( ( F  |`  X ) `  ( a ( .s
`  R ) b ) )  =  ( a ( .s `  T ) ( ( F  |`  X ) `  b ) ) )
5914, 19, 583jca 1134 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
( F  |`  X )  e.  ( R  GrpHom  T )  /\  (Scalar `  T )  =  (Scalar `  R )  /\  A. a  e.  ( Base `  (Scalar `  R )
) A. b  e.  ( Base `  R
) ( ( F  |`  X ) `  (
a ( .s `  R ) b ) )  =  ( a ( .s `  T
) ( ( F  |`  X ) `  b
) ) ) )
60 eqid 2404 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
61 eqid 2404 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
62 eqid 2404 . . 3  |-  ( Base `  R )  =  (
Base `  R )
63 eqid 2404 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
6460, 16, 61, 62, 63, 34islmhm 16058 . 2  |-  ( ( F  |`  X )  e.  ( R LMHom  T )  <-> 
( ( R  e. 
LMod  /\  T  e.  LMod )  /\  ( ( F  |`  X )  e.  ( R  GrpHom  T )  /\  (Scalar `  T )  =  (Scalar `  R )  /\  A. a  e.  (
Base `  (Scalar `  R
) ) A. b  e.  ( Base `  R
) ( ( F  |`  X ) `  (
a ( .s `  R ) b ) )  =  ( a ( .s `  T
) ( ( F  |`  X ) `  b
) ) ) ) )
658, 59, 64sylanbrc 646 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R LMHom  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280    |` cres 4839   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487   .scvsca 13488  SubGrpcsubg 14893    GrpHom cghm 14958   LModclmod 15905   LSubSpclss 15963   LMHom clmhm 16050
This theorem is referenced by:  lmhmlnmsplit  27053  pwssplit4  27059  frlmsplit2  27111
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-sca 13500  df-vsca 13501  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-ghm 14959  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lmhm 16053
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