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Theorem frlmssuvc1 26916
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 26887 . . 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 2388 . . . 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 26915 . . 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 26891 . . . . . 6  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
161, 2, 15syl2anc 643 . . . . 5  |-  ( ph  ->  R  =  (Scalar `  F ) )
1716fveq2d 5673 . . . 4  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  F )
) )
1814, 17syl5eq 2432 . . 3  |-  ( ph  ->  K  =  ( Base `  (Scalar `  F )
) )
1913, 18eleqtrd 2464 . 2  |-  ( ph  ->  X  e.  ( Base `  (Scalar `  F )
) )
20 frlmssuvc1.u . . . . . 6  |-  U  =  ( R unitVec  I )
2120, 3, 8uvcff 26910 . . . . 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 3293 . . . 4  |-  ( ph  ->  L  e.  I )
2522, 24ffvelrnd 5811 . . 3  |-  ( ph  ->  ( U `  L
)  e.  B )
263, 14, 8frlmbasf 26898 . . . . 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 3413 . . . . . 6  |-  ( x  e.  ( I  \  J )  ->  x  e.  I )
3231adantl 453 . . . . 5  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  x  e.  I )
33 disjdif 3644 . . . . . . 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 3617 . . . . . 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 26909 . . . 4  |-  ( (
ph  /\  x  e.  ( I  \  J ) )  ->  ( ( U `  L ) `  x )  =  .0.  )
4027, 39suppss 5803 . . 3  |-  ( ph  ->  ( `' ( U `
 L ) "
( _V  \  {  .0.  } ) )  C_  J )
41 cnveq 4987 . . . . . 6  |-  ( x  =  ( U `  L )  ->  `' x  =  `' ( U `  L )
)
4241imaeq1d 5143 . . . . 5  |-  ( x  =  ( U `  L )  ->  ( `' x " ( _V 
\  {  .0.  }
) )  =  ( `' ( U `  L ) " ( _V  \  {  .0.  }
) ) )
4342sseq1d 3319 . . . 4  |-  ( x  =  ( U `  L )  ->  (
( `' x "
( _V  \  {  .0.  } ) )  C_  J 
<->  ( `' ( U `
 L ) "
( _V  \  {  .0.  } ) )  C_  J ) )
4443, 10elrab2 3038 . . 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 2388 . . 3  |-  (Scalar `  F )  =  (Scalar `  F )
47 frlmssuvc1.t . . 3  |-  .x.  =  ( .s `  F )
48 eqid 2388 . . 3  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
4946, 47, 48, 7lssvscl 15959 . 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 1649    e. wcel 1717    =/= wne 2551   {crab 2654   _Vcvv 2900    \ cdif 3261    i^i cin 3263    C_ wss 3264   (/)c0 3572   {csn 3758   `'ccnv 4818   "cima 4822   -->wf 5391   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Scalarcsca 13460   .scvsca 13461   0gc0g 13651   Ringcrg 15588   LModclmod 15878   LSubSpclss 15936   freeLMod cfrlm 26882   unitVec cuvc 26883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-fz 10977  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-hom 13481  df-cco 13482  df-prds 13599  df-pws 13601  df-0g 13655  df-mnd 14618  df-mhm 14666  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-subg 14869  df-ghm 14932  df-mgp 15577  df-rng 15591  df-ur 15593  df-subrg 15794  df-lmod 15880  df-lss 15937  df-lmhm 16026  df-sra 16172  df-rgmod 16173  df-dsmm 26868  df-frlm 26884  df-uvc 26885
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