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Theorem reslmhm 17811
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 17792 . . . 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 17719 . . . 4  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  R  e.  LMod )
51, 4sylan 469 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  R  e.  LMod )
6 lmhmlmod2 17791 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
76adantr 463 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  T  e.  LMod )
85, 7jca 530 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( R  e.  LMod  /\  T  e.  LMod ) )
9 lmghm 17790 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
109adantr 463 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  F  e.  ( S  GrpHom  T ) )
113lsssubg 17716 . . . . 5  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
121, 11sylan 469 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
132resghm 16400 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
1410, 12, 13syl2anc 659 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
15 eqid 2382 . . . . 5  |-  (Scalar `  S )  =  (Scalar `  S )
16 eqid 2382 . . . . 5  |-  (Scalar `  T )  =  (Scalar `  T )
1715, 16lmhmsca 17789 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
182, 15resssca 14784 . . . 4  |-  ( X  e.  U  ->  (Scalar `  S )  =  (Scalar `  R ) )
1917, 18sylan9eq 2443 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  T )  =  (Scalar `  R ) )
20 simpll 751 . . . . . . 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 754 . . . . . . 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 2382 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
2322, 3lssss 17696 . . . . . . . . . 10  |-  ( X  e.  U  ->  X  C_  ( Base `  S
) )
2423adantl 464 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  C_  ( Base `  S
) )
2524adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  X  C_  ( Base `  S
) )
262, 22ressbas2 14692 . . . . . . . . . . . 12  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  R )
)
2724, 26syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  =  ( Base `  R
) )
2827eleq2d 2452 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
b  e.  X  <->  b  e.  ( Base `  R )
) )
2928biimpar 483 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  b  e.  ( Base `  R ) )  ->  b  e.  X
)
3029adantrl 713 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  X )
3125, 30sseldd 3418 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  ( Base `  S ) )
32 eqid 2382 . . . . . . . 8  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
33 eqid 2382 . . . . . . . 8  |-  ( .s
`  S )  =  ( .s `  S
)
34 eqid 2382 . . . . . . . 8  |-  ( .s
`  T )  =  ( .s `  T
)
3515, 32, 22, 33, 34lmhmlin 17794 . . . . . . 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 1226 . . . . . 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 463 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  S  e.  LMod )
3837adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  S  e.  LMod )
39 simplr 753 . . . . . . . 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 17714 . . . . . . . 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 1227 . . . . . . 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 5788 . . . . . . 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 5788 . . . . . . . 8  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4544oveq2d 6212 . . . . . . 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 2433 . . . . 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 2803 . . . 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 464 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  S )  =  (Scalar `  R ) )
5049fveq2d 5778 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  R
) ) )
512, 33ressvsca 14785 . . . . . . . . . 10  |-  ( X  e.  U  ->  ( .s `  S )  =  ( .s `  R
) )
5251adantl 464 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( .s `  S )  =  ( .s `  R
) )
5352oveqd 6213 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
a ( .s `  S ) b )  =  ( a ( .s `  R ) b ) )
5453fveq2d 5778 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
( F  |`  X ) `
 ( a ( .s `  S ) b ) )  =  ( ( F  |`  X ) `  (
a ( .s `  R ) b ) ) )
5554eqeq1d 2384 . . . . . 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 2821 . . . . 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 2993 . . . 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 210 . . 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 1174 . 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 2382 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
61 eqid 2382 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
62 eqid 2382 . . 3  |-  ( Base `  R )  =  (
Base `  R )
63 eqid 2382 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
6460, 16, 61, 62, 63, 34islmhm 17786 . 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 662 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R LMHom  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732    C_ wss 3389    |` cres 4915   ` cfv 5496  (class class class)co 6196   Basecbs 14634   ↾s cress 14635  Scalarcsca 14705   .scvsca 14706  SubGrpcsubg 16312    GrpHom cghm 16381   LModclmod 17625   LSubSpclss 17691   LMHom clmhm 17778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-sca 14718  df-vsca 14719  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-sbg 16176  df-subg 16315  df-ghm 16382  df-mgp 17255  df-ur 17267  df-ring 17313  df-lmod 17627  df-lss 17692  df-lmhm 17781
This theorem is referenced by:  frlmsplit2  18892  lmhmlnmsplit  31199  pwssplit4  31201
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