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Theorem frlmssuvc1 27214
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.)
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 " ( _V  \  {  .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 27185 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  F  e.  LMod )
51, 2, 4syl2anc 643 . 2  |-  ( ph  ->  F  e.  LMod )
6 frlmssuvc1.j . . 3  |-  ( ph  ->  J  C_  I )
7 eqid 2435 . . . 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 " ( _V  \  {  .0.  } ) )  C_  J }
113, 7, 8, 9, 10frlmsslss2 27213 . . 3  |-  ( ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  C  e.  ( LSubSp `  F )
)
121, 2, 6, 11syl3anc 1184 . 2  |-  ( ph  ->  C  e.  ( LSubSp `  F ) )
13 frlmssuvc1.x . . 3  |-  ( ph  ->  X  e.  K )
14 frlmssuvc1.k . . . 4  |-  K  =  ( Base `  R
)
153frlmsca 27189 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
161, 2, 15syl2anc 643 . . . . 5  |-  ( ph  ->  R  =  (Scalar `  F ) )
1716fveq2d 5724 . . . 4  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  F )
) )
1814, 17syl5eq 2479 . . 3  |-  ( ph  ->  K  =  ( Base `  (Scalar `  F )
) )
1913, 18eleqtrd 2511 . 2  |-  ( ph  ->  X  e.  ( Base `  (Scalar `  F )
) )
20 frlmssuvc1.u . . . . . 6  |-  U  =  ( R unitVec  I )
2120, 3, 8uvcff 27208 . . . . 5  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> B )
221, 2, 21syl2anc 643 . . . 4  |-  ( ph  ->  U : I --> B )
23 frlmssuvc1.l . . . . 5  |-  ( ph  ->  L  e.  J )
246, 23sseldd 3341 . . . 4  |-  ( ph  ->  L  e.  I )
2522, 24ffvelrnd 5863 . . 3  |-  ( ph  ->  ( U `  L
)  e.  B )
263, 14, 8frlmbasf 27196 . . . . 5  |-  ( ( I  e.  V  /\  ( U `  L )  e.  B )  -> 
( U `  L
) : I --> K )
272, 25, 26syl2anc 643 . . . 4  |-  ( ph  ->  ( U `  L
) : I --> K )
281adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  R  e.  Ring )
292adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  I  e.  V )
3024adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  L  e.  I )
31 eldifi 3461 . . . . . 6  |-  ( x  e.  ( I  \  J )  ->  x  e.  I )
3231adantl 453 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  x  e.  I )
33 disjdif 3692 . . . . . . 7  |-  ( J  i^i  ( I  \  J ) )  =  (/)
3433a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  ( J  i^i  ( I  \  J
) )  =  (/) )
3523adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  L  e.  J )
36 simpr 448 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  x  e.  ( I  \  J ) )
37 disjne 3665 . . . . . 6  |-  ( ( ( J  i^i  (
I  \  J )
)  =  (/)  /\  L  e.  J  /\  x  e.  ( I  \  J
) )  ->  L  =/=  x )
3834, 35, 36, 37syl3anc 1184 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  L  =/=  x )
3920, 28, 29, 30, 32, 38, 9uvcvv0 27207 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  ( ( U `  L ) `  x )  =  .0.  )
4027, 39suppss 5855 . . 3  |-  ( ph  ->  ( `' ( U `
 L ) "
( _V  \  {  .0.  } ) )  C_  J )
41 cnveq 5038 . . . . . 6  |-  ( x  =  ( U `  L )  ->  `' x  =  `' ( U `  L )
)
4241imaeq1d 5194 . . . . 5  |-  ( x  =  ( U `  L )  ->  ( `' x " ( _V 
\  {  .0.  }
) )  =  ( `' ( U `  L ) " ( _V  \  {  .0.  }
) ) )
4342sseq1d 3367 . . . 4  |-  ( x  =  ( U `  L )  ->  (
( `' x "
( _V  \  {  .0.  } ) )  C_  J 
<->  ( `' ( U `
 L ) "
( _V  \  {  .0.  } ) )  C_  J ) )
4443, 10elrab2 3086 . . 3  |-  ( ( U `  L )  e.  C  <->  ( ( U `  L )  e.  B  /\  ( `' ( U `  L ) " ( _V  \  {  .0.  }
) )  C_  J
) )
4525, 40, 44sylanbrc 646 . 2  |-  ( ph  ->  ( U `  L
)  e.  C )
46 eqid 2435 . . 3  |-  (Scalar `  F )  =  (Scalar `  F )
47 frlmssuvc1.t . . 3  |-  .x.  =  ( .s `  F )
48 eqid 2435 . . 3  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
4946, 47, 48, 7lssvscl 16023 . 2  |-  ( ( ( F  e.  LMod  /\  C  e.  ( LSubSp `  F ) )  /\  ( X  e.  ( Base `  (Scalar `  F
) )  /\  ( U `  L )  e.  C ) )  -> 
( X  .x.  ( U `  L )
)  e.  C )
505, 12, 19, 45, 49syl22anc 1185 1  |-  ( ph  ->  ( X  .x.  ( U `  L )
)  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   `'ccnv 4869   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   Ringcrg 15652   LModclmod 15942   LSubSpclss 16000   freeLMod cfrlm 27180   unitVec cuvc 27181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-hom 13545  df-cco 13546  df-prds 13663  df-pws 13665  df-0g 13719  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-ghm 14996  df-mgp 15641  df-rng 15655  df-ur 15657  df-subrg 15858  df-lmod 15944  df-lss 16001  df-lmhm 16090  df-sra 16236  df-rgmod 16237  df-dsmm 27166  df-frlm 27182  df-uvc 27183
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