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

Theorem reslmhm 17055
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 17036 . . . 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 16963 . . . 4  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  R  e.  LMod )
51, 4sylan 468 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  R  e.  LMod )
6 lmhmlmod2 17035 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
76adantr 462 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  T  e.  LMod )
85, 7jca 529 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( R  e.  LMod  /\  T  e.  LMod ) )
9 lmghm 17034 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
109adantr 462 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  F  e.  ( S  GrpHom  T ) )
113lsssubg 16960 . . . . 5  |-  ( ( S  e.  LMod  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
121, 11sylan 468 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  e.  (SubGrp `  S )
)
132resghm 15743 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  (SubGrp `  S )
)  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
1410, 12, 13syl2anc 654 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R  GrpHom  T ) )
15 eqid 2433 . . . . 5  |-  (Scalar `  S )  =  (Scalar `  S )
16 eqid 2433 . . . . 5  |-  (Scalar `  T )  =  (Scalar `  T )
1715, 16lmhmsca 17033 . . . 4  |-  ( F  e.  ( S LMHom  T
)  ->  (Scalar `  T
)  =  (Scalar `  S ) )
182, 15resssca 14299 . . . 4  |-  ( X  e.  U  ->  (Scalar `  S )  =  (Scalar `  R ) )
1917, 18sylan9eq 2485 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  T )  =  (Scalar `  R ) )
20 simpll 746 . . . . . . 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 748 . . . . . . 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 2433 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
2322, 3lssss 16940 . . . . . . . . . 10  |-  ( X  e.  U  ->  X  C_  ( Base `  S
) )
2423adantl 463 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  C_  ( Base `  S
) )
2524adantr 462 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  X  C_  ( Base `  S
) )
262, 22ressbas2 14212 . . . . . . . . . . . 12  |-  ( X 
C_  ( Base `  S
)  ->  X  =  ( Base `  R )
)
2724, 26syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  X  =  ( Base `  R
) )
2827eleq2d 2500 . . . . . . . . . 10  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
b  e.  X  <->  b  e.  ( Base `  R )
) )
2928biimpar 482 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  b  e.  ( Base `  R ) )  ->  b  e.  X
)
3029adantrl 708 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  X )
3125, 30sseldd 3345 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  -> 
b  e.  ( Base `  S ) )
32 eqid 2433 . . . . . . . 8  |-  ( Base `  (Scalar `  S )
)  =  ( Base `  (Scalar `  S )
)
33 eqid 2433 . . . . . . . 8  |-  ( .s
`  S )  =  ( .s `  S
)
34 eqid 2433 . . . . . . . 8  |-  ( .s
`  T )  =  ( .s `  T
)
3515, 32, 22, 33, 34lmhmlin 17038 . . . . . . 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 1211 . . . . . 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 462 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  S  e.  LMod )
3837adantr 462 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  /\  ( a  e.  (
Base `  (Scalar `  S
) )  /\  b  e.  ( Base `  R
) ) )  ->  S  e.  LMod )
39 simplr 747 . . . . . . . 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 16958 . . . . . . . 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 1212 . . . . . . 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 5692 . . . . . . 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 5692 . . . . . . . 8  |-  ( b  e.  X  ->  (
( F  |`  X ) `
 b )  =  ( F `  b
) )
4544oveq2d 6096 . . . . . . 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 2475 . . . . 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 2798 . . . 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 463 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (Scalar `  S )  =  (Scalar `  R ) )
5049fveq2d 5683 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( Base `  (Scalar `  S
) )  =  (
Base `  (Scalar `  R
) ) )
512, 33ressvsca 14300 . . . . . . . . . 10  |-  ( X  e.  U  ->  ( .s `  S )  =  ( .s `  R
) )
5251adantl 463 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  ( .s `  S )  =  ( .s `  R
) )
5352oveqd 6097 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
a ( .s `  S ) b )  =  ( a ( .s `  R ) b ) )
5453fveq2d 5683 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  X  e.  U )  ->  (
( F  |`  X ) `
 ( a ( .s `  S ) b ) )  =  ( ( F  |`  X ) `  (
a ( .s `  R ) b ) ) )
5554eqeq1d 2441 . . . . . 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 2725 . . . . 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 2921 . . . 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 1161 . 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 2433 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
61 eqid 2433 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
62 eqid 2433 . . 3  |-  ( Base `  R )  =  (
Base `  R )
63 eqid 2433 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
6460, 16, 61, 62, 63, 34islmhm 17030 . 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 657 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 369    /\ w3a 958    = wceq 1362    e. wcel 1755   A.wral 2705    C_ wss 3316    |` cres 4829   ` cfv 5406  (class class class)co 6080   Basecbs 14157   ↾s cress 14158  Scalarcsca 14224   .scvsca 14225  SubGrpcsubg 15655    GrpHom cghm 15724   LModclmod 16872   LSubSpclss 16935   LMHom clmhm 17022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-sca 14237  df-vsca 14238  df-0g 14363  df-mnd 15398  df-grp 15525  df-minusg 15526  df-sbg 15527  df-subg 15658  df-ghm 15725  df-mgp 16566  df-rng 16580  df-ur 16582  df-lmod 16874  df-lss 16936  df-lmhm 17025
This theorem is referenced by:  frlmsplit2  18039  lmhmlnmsplit  29285  pwssplit4  29287
  Copyright terms: Public domain W3C validator