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Theorem reslmhm 15825
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 15806 . . . 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 15733 . . . 4  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  R  e.  LMod )
51, 4sylan 457 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  R  e.  LMod )
6 lmhmlmod2 15805 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
76adantr 451 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  T  e.  LMod )
85, 7jca 518 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( R  e.  LMod  /\  T  e.  LMod ) )
9 lmghm 15804 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
109adantr 451 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  F  e.  ( S  GrpHom  T ) )
113lsssubg 15730 . . . . 5  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
121, 11sylan 457 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
132resghm 14715 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
1410, 12, 13syl2anc 642 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
15 eqid 2296 . . . . 5  |-  (Scalar `  S )  =  (Scalar `  S )
16 eqid 2296 . . . . 5  |-  (Scalar `  T )  =  (Scalar `  T )
1715, 16lmhmsca 15803 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
182, 15resssca 13299 . . . 4  |-  ( X  e.  U  ->  (Scalar `  S )  =  (Scalar `  R ) )
1917, 18sylan9eq 2348 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  T )  =  (Scalar `  R ) )
20 simpll 730 . . . . . . 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 732 . . . . . . 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 2296 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
2322, 3lssss 15710 . . . . . . . . . 10  |-  ( X  e.  U  ->  X  C_  ( Base `  S
) )
2423adantl 452 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  C_  ( Base `  S
) )
2524adantr 451 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  X  C_  ( Base `  S
) )
262, 22ressbas2 13215 . . . . . . . . . . . 12  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  R )
)
2724, 26syl 15 . . . . . . . . . . 11  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  =  ( Base `  R
) )
2827eleq2d 2363 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
b  e.  X  <->  b  e.  ( Base `  R )
) )
2928biimpar 471 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  b  e.  ( Base `  R ) )  ->  b  e.  X
)
3029adantrl 696 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  X )
3125, 30sseldd 3194 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  ( Base `  S ) )
32 eqid 2296 . . . . . . . 8  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
33 eqid 2296 . . . . . . . 8  |-  ( .s
`  S )  =  ( .s `  S
)
34 eqid 2296 . . . . . . . 8  |-  ( .s
`  T )  =  ( .s `  T
)
3515, 32, 22, 33, 34lmhmlin 15808 . . . . . . 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 1182 . . . . . 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 451 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  S  e.  LMod )
3837adantr 451 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  S  e.  LMod )
39 simplr 731 . . . . . . . 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 15728 . . . . . . . 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 1183 . . . . . . 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 5558 . . . . . . 7  |-  ( ( a ( .s `  S ) b )  e.  X  ->  (
( F  |`  X ) `
 ( a ( .s `  S ) b ) )  =  ( F `  (
a ( .s `  S ) b ) ) )
4341, 42syl 15 . . . . . 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 5558 . . . . . . . 8  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4544oveq2d 5890 . . . . . . 7  |-  ( b  e.  X  ->  (
a ( .s `  T ) ( ( F  |`  X ) `  b ) )  =  ( a ( .s
`  T ) ( F `  b ) ) )
4630, 45syl 15 . . . . . 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 2338 . . . . 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 2648 . . . 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 452 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  S )  =  (Scalar `  R ) )
5049fveq2d 5545 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  R
) ) )
512, 33ressvsca 13300 . . . . . . . . . 10  |-  ( X  e.  U  ->  ( .s `  S )  =  ( .s `  R
) )
5251adantl 452 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( .s `  S )  =  ( .s `  R
) )
5352oveqd 5891 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
a ( .s `  S ) b )  =  ( a ( .s `  R ) b ) )
5453fveq2d 5545 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
( F  |`  X ) `
 ( a ( .s `  S ) b ) )  =  ( ( F  |`  X ) `  (
a ( .s `  R ) b ) ) )
5554eqeq1d 2304 . . . . . 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 2576 . . . . 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 2761 . . . 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 201 . . 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 1132 . 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 2296 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
61 eqid 2296 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
62 eqid 2296 . . 3  |-  ( Base `  R )  =  (
Base `  R )
63 eqid 2296 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
6460, 16, 61, 62, 63, 34islmhm 15800 . 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 645 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165    |` cres 4707   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165  Scalarcsca 13227   .scvsca 13228  SubGrpcsubg 14631    GrpHom cghm 14696   LModclmod 15643   LSubSpclss 15705   LMHom clmhm 15792
This theorem is referenced by:  lmhmlnmsplit  27288  pwssplit4  27294  frlmsplit2  27346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lmhm 15795
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