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Theorem frlmssuvc1 18217
Description: A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
Hypotheses
Ref Expression
frlmssuvc1.f  |-  F  =  ( R freeLMod  I )
frlmssuvc1.u  |-  U  =  ( R unitVec  I )
frlmssuvc1.b  |-  B  =  ( Base `  F
)
frlmssuvc1.k  |-  K  =  ( Base `  R
)
frlmssuvc1.t  |-  .x.  =  ( .s `  F )
frlmssuvc1.z  |-  .0.  =  ( 0g `  R )
frlmssuvc1.c  |-  C  =  { x  e.  B  |  ( x supp  .0.  )  C_  J }
frlmssuvc1.r  |-  ( ph  ->  R  e.  Ring )
frlmssuvc1.i  |-  ( ph  ->  I  e.  V )
frlmssuvc1.j  |-  ( ph  ->  J  C_  I )
frlmssuvc1.l  |-  ( ph  ->  L  e.  J )
frlmssuvc1.x  |-  ( ph  ->  X  e.  K )
Assertion
Ref Expression
frlmssuvc1  |-  ( ph  ->  ( X  .x.  ( U `  L )
)  e.  C )
Distinct variable groups:    x, B    x, F    x, I    x, J    x, K    x, L    x, R    x,  .0.    ph, x    x, U    x, V    x,  .x.    x, X
Allowed substitution hint:    C( x)

Proof of Theorem frlmssuvc1
StepHypRef Expression
1 frlmssuvc1.r . . 3  |-  ( ph  ->  R  e.  Ring )
2 frlmssuvc1.i . . 3  |-  ( ph  ->  I  e.  V )
3 frlmssuvc1.f . . . 4  |-  F  =  ( R freeLMod  I )
43frlmlmod 18172 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  F  e.  LMod )
51, 2, 4syl2anc 661 . 2  |-  ( ph  ->  F  e.  LMod )
6 frlmssuvc1.j . . 3  |-  ( ph  ->  J  C_  I )
7 eqid 2441 . . . 4  |-  ( LSubSp `  F )  =  (
LSubSp `  F )
8 frlmssuvc1.b . . . 4  |-  B  =  ( Base `  F
)
9 frlmssuvc1.z . . . 4  |-  .0.  =  ( 0g `  R )
10 frlmssuvc1.c . . . 4  |-  C  =  { x  e.  B  |  ( x supp  .0.  )  C_  J }
113, 7, 8, 9, 10frlmsslss2 18197 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  C  e.  ( LSubSp `  F )
)
121, 2, 6, 11syl3anc 1218 . 2  |-  ( ph  ->  C  e.  ( LSubSp `  F ) )
13 frlmssuvc1.x . . 3  |-  ( ph  ->  X  e.  K )
14 frlmssuvc1.k . . . 4  |-  K  =  ( Base `  R
)
153frlmsca 18176 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
161, 2, 15syl2anc 661 . . . . 5  |-  ( ph  ->  R  =  (Scalar `  F ) )
1716fveq2d 5693 . . . 4  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  F )
) )
1814, 17syl5eq 2485 . . 3  |-  ( ph  ->  K  =  ( Base `  (Scalar `  F )
) )
1913, 18eleqtrd 2517 . 2  |-  ( ph  ->  X  e.  ( Base `  (Scalar `  F )
) )
20 frlmssuvc1.u . . . . . 6  |-  U  =  ( R unitVec  I )
2120, 3, 8uvcff 18214 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> B )
221, 2, 21syl2anc 661 . . . 4  |-  ( ph  ->  U : I --> B )
23 frlmssuvc1.l . . . . 5  |-  ( ph  ->  L  e.  J )
246, 23sseldd 3355 . . . 4  |-  ( ph  ->  L  e.  I )
2522, 24ffvelrnd 5842 . . 3  |-  ( ph  ->  ( U `  L
)  e.  B )
263, 14, 8frlmbasf 18186 . . . . 5  |-  ( ( I  e.  V  /\  ( U `  L )  e.  B )  -> 
( U `  L
) : I --> K )
272, 25, 26syl2anc 661 . . . 4  |-  ( ph  ->  ( U `  L
) : I --> K )
281adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  R  e.  Ring )
292adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  I  e.  V )
3024adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  L  e.  I )
31 eldifi 3476 . . . . . 6  |-  ( x  e.  ( I  \  J )  ->  x  e.  I )
3231adantl 466 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  x  e.  I )
33 disjdif 3749 . . . . . . 7  |-  ( J  i^i  ( I  \  J ) )  =  (/)
3433a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  ( J  i^i  ( I  \  J
) )  =  (/) )
3523adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  L  e.  J )
36 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  x  e.  ( I  \  J ) )
37 disjne 3722 . . . . . 6  |-  ( ( ( J  i^i  (
I  \  J )
)  =  (/)  /\  L  e.  J  /\  x  e.  ( I  \  J
) )  ->  L  =/=  x )
3834, 35, 36, 37syl3anc 1218 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  L  =/=  x )
3920, 28, 29, 30, 32, 38, 9uvcvv0 18213 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  ( ( U `  L ) `  x )  =  .0.  )
4027, 39suppss 6717 . . 3  |-  ( ph  ->  ( ( U `  L ) supp  .0.  )  C_  J )
41 oveq1 6096 . . . . 5  |-  ( x  =  ( U `  L )  ->  (
x supp  .0.  )  =  ( ( U `  L ) supp  .0.  )
)
4241sseq1d 3381 . . . 4  |-  ( x  =  ( U `  L )  ->  (
( x supp  .0.  )  C_  J  <->  ( ( U `
 L ) supp  .0.  )  C_  J ) )
4342, 10elrab2 3117 . . 3  |-  ( ( U `  L )  e.  C  <->  ( ( U `  L )  e.  B  /\  (
( U `  L
) supp  .0.  )  C_  J ) )
4425, 40, 43sylanbrc 664 . 2  |-  ( ph  ->  ( U `  L
)  e.  C )
45 eqid 2441 . . 3  |-  (Scalar `  F )  =  (Scalar `  F )
46 frlmssuvc1.t . . 3  |-  .x.  =  ( .s `  F )
47 eqid 2441 . . 3  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
4845, 46, 47, 7lssvscl 17034 . 2  |-  ( ( ( F  e.  LMod  /\  C  e.  ( LSubSp `  F ) )  /\  ( X  e.  ( Base `  (Scalar `  F
) )  /\  ( U `  L )  e.  C ) )  -> 
( X  .x.  ( U `  L )
)  e.  C )
495, 12, 19, 44, 48syl22anc 1219 1  |-  ( ph  ->  ( X  .x.  ( U `  L )
)  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   {crab 2717    \ cdif 3323    i^i cin 3325    C_ wss 3326   (/)c0 3635   -->wf 5412   ` cfv 5416  (class class class)co 6089   supp csupp 6688   Basecbs 14172  Scalarcsca 14239   .scvsca 14240   0gc0g 14376   Ringcrg 16643   LModclmod 16946   LSubSpclss 17011   freeLMod cfrlm 18169   unitVec cuvc 18205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-map 7214  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-fz 11436  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-hom 14260  df-cco 14261  df-0g 14378  df-prds 14384  df-pws 14386  df-mnd 15413  df-mhm 15462  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-subg 15676  df-ghm 15743  df-mgp 16590  df-ur 16602  df-rng 16645  df-subrg 16861  df-lmod 16948  df-lss 17012  df-lmhm 17101  df-sra 17251  df-rgmod 17252  df-dsmm 18155  df-frlm 18170  df-uvc 18206
This theorem is referenced by: (None)
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