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Theorem reslmhm 16133
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 16114 . . . 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 16041 . . . 4  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  R  e.  LMod )
51, 4sylan 459 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  R  e.  LMod )
6 lmhmlmod2 16113 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
76adantr 453 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  T  e.  LMod )
85, 7jca 520 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( R  e.  LMod  /\  T  e.  LMod ) )
9 lmghm 16112 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
109adantr 453 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  F  e.  ( S  GrpHom  T ) )
113lsssubg 16038 . . . . 5  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
121, 11sylan 459 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
132resghm 15027 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
1410, 12, 13syl2anc 644 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
15 eqid 2438 . . . . 5  |-  (Scalar `  S )  =  (Scalar `  S )
16 eqid 2438 . . . . 5  |-  (Scalar `  T )  =  (Scalar `  T )
1715, 16lmhmsca 16111 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
182, 15resssca 13609 . . . 4  |-  ( X  e.  U  ->  (Scalar `  S )  =  (Scalar `  R ) )
1917, 18sylan9eq 2490 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  T )  =  (Scalar `  R ) )
20 simpll 732 . . . . . . 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 734 . . . . . . 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 2438 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
2322, 3lssss 16018 . . . . . . . . . 10  |-  ( X  e.  U  ->  X  C_  ( Base `  S
) )
2423adantl 454 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  C_  ( Base `  S
) )
2524adantr 453 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  X  C_  ( Base `  S
) )
262, 22ressbas2 13525 . . . . . . . . . . . 12  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  R )
)
2724, 26syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  =  ( Base `  R
) )
2827eleq2d 2505 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
b  e.  X  <->  b  e.  ( Base `  R )
) )
2928biimpar 473 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  b  e.  ( Base `  R ) )  ->  b  e.  X
)
3029adantrl 698 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  X )
3125, 30sseldd 3351 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  ( Base `  S ) )
32 eqid 2438 . . . . . . . 8  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
33 eqid 2438 . . . . . . . 8  |-  ( .s
`  S )  =  ( .s `  S
)
34 eqid 2438 . . . . . . . 8  |-  ( .s
`  T )  =  ( .s `  T
)
3515, 32, 22, 33, 34lmhmlin 16116 . . . . . . 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 1185 . . . . . 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 453 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  S  e.  LMod )
3837adantr 453 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  S  e.  LMod )
39 simplr 733 . . . . . . . 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 16036 . . . . . . . 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 1186 . . . . . . 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 5748 . . . . . . 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 5748 . . . . . . . 8  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4544oveq2d 6100 . . . . . . 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 2480 . . . . 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 2800 . . . 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 454 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  S )  =  (Scalar `  R ) )
5049fveq2d 5735 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  R
) ) )
512, 33ressvsca 13610 . . . . . . . . . 10  |-  ( X  e.  U  ->  ( .s `  S )  =  ( .s `  R
) )
5251adantl 454 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( .s `  S )  =  ( .s `  R
) )
5352oveqd 6101 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
a ( .s `  S ) b )  =  ( a ( .s `  R ) b ) )
5453fveq2d 5735 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
( F  |`  X ) `
 ( a ( .s `  S ) b ) )  =  ( ( F  |`  X ) `  (
a ( .s `  R ) b ) ) )
5554eqeq1d 2446 . . . . . 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 2727 . . . . 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 2918 . . . 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 203 . . 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 1135 . 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 2438 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
61 eqid 2438 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
62 eqid 2438 . . 3  |-  ( Base `  R )  =  (
Base `  R )
63 eqid 2438 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
6460, 16, 61, 62, 63, 34islmhm 16108 . 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 647 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322    |` cres 4883   ` cfv 5457  (class class class)co 6084   Basecbs 13474   ↾s cress 13475  Scalarcsca 13537   .scvsca 13538  SubGrpcsubg 14943    GrpHom cghm 15008   LModclmod 15955   LSubSpclss 16013   LMHom clmhm 16100
This theorem is referenced by:  lmhmlnmsplit  27176  pwssplit4  27182  frlmsplit2  27234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-sca 13550  df-vsca 13551  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-sbg 14819  df-subg 14946  df-ghm 15009  df-mgp 15654  df-rng 15668  df-ur 15670  df-lmod 15957  df-lss 16014  df-lmhm 16103
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