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Theorem frlmssuvc2 27115
Description: A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not 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 )
frlmssuvc2.l  |-  ( ph  ->  L  e.  ( I 
\  J ) )
frlmssuvc2.x  |-  ( ph  ->  X  e.  ( K 
\  {  .0.  }
) )
Assertion
Ref Expression
frlmssuvc2  |-  ( 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 frlmssuvc2
StepHypRef Expression
1 frlmssuvc2.l . . . . . . 7  |-  ( ph  ->  L  e.  ( I 
\  J ) )
21eldifad 3292 . . . . . 6  |-  ( ph  ->  L  e.  I )
3 frlmssuvc1.f . . . . . . . . 9  |-  F  =  ( R freeLMod  I )
4 frlmssuvc1.b . . . . . . . . 9  |-  B  =  ( Base `  F
)
5 frlmssuvc1.k . . . . . . . . 9  |-  K  =  ( Base `  R
)
6 frlmssuvc1.i . . . . . . . . 9  |-  ( ph  ->  I  e.  V )
7 frlmssuvc2.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( K 
\  {  .0.  }
) )
87eldifad 3292 . . . . . . . . 9  |-  ( ph  ->  X  e.  K )
9 frlmssuvc1.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Ring )
10 frlmssuvc1.u . . . . . . . . . . . 12  |-  U  =  ( R unitVec  I )
1110, 3, 4uvcff 27108 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> B )
129, 6, 11syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  U : I --> B )
1312, 2ffvelrnd 5830 . . . . . . . . 9  |-  ( ph  ->  ( U `  L
)  e.  B )
14 frlmssuvc1.t . . . . . . . . 9  |-  .x.  =  ( .s `  F )
15 eqid 2404 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
163, 4, 5, 6, 8, 13, 2, 14, 15frlmvscaval 27099 . . . . . . . 8  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =  ( X ( .r `  R ) ( ( U `  L ) `  L
) ) )
17 eqid 2404 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
1810, 9, 6, 2, 17uvcvv1 27106 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  L ) `  L
)  =  ( 1r
`  R ) )
1918oveq2d 6056 . . . . . . . 8  |-  ( ph  ->  ( X ( .r
`  R ) ( ( U `  L
) `  L )
)  =  ( X ( .r `  R
) ( 1r `  R ) ) )
205, 15, 17rngridm 15643 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  X  e.  K )  ->  ( X ( .r `  R ) ( 1r
`  R ) )  =  X )
219, 8, 20syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( X ( .r
`  R ) ( 1r `  R ) )  =  X )
2216, 19, 213eqtrd 2440 . . . . . . 7  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =  X )
23 eldifsni 3888 . . . . . . . 8  |-  ( X  e.  ( K  \  {  .0.  } )  ->  X  =/=  .0.  )
247, 23syl 16 . . . . . . 7  |-  ( ph  ->  X  =/=  .0.  )
2522, 24eqnetrd 2585 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =/=  .0.  )
26 fveq2 5687 . . . . . . . 8  |-  ( x  =  L  ->  (
( X  .x.  ( U `  L )
) `  x )  =  ( ( X 
.x.  ( U `  L ) ) `  L ) )
2726neeq1d 2580 . . . . . . 7  |-  ( x  =  L  ->  (
( ( X  .x.  ( U `  L ) ) `  x )  =/=  .0.  <->  ( ( X  .x.  ( U `  L ) ) `  L )  =/=  .0.  ) )
2827elrab 3052 . . . . . 6  |-  ( L  e.  { x  e.  I  |  ( ( X  .x.  ( U `
 L ) ) `
 x )  =/= 
.0.  }  <->  ( L  e.  I  /\  ( ( X  .x.  ( U `
 L ) ) `
 L )  =/= 
.0.  ) )
292, 25, 28sylanbrc 646 . . . . 5  |-  ( ph  ->  L  e.  { x  e.  I  |  (
( X  .x.  ( U `  L )
) `  x )  =/=  .0.  } )
301eldifbd 3293 . . . . 5  |-  ( ph  ->  -.  L  e.  J
)
31 nelss 26622 . . . . 5  |-  ( ( L  e.  { x  e.  I  |  (
( X  .x.  ( U `  L )
) `  x )  =/=  .0.  }  /\  -.  L  e.  J )  ->  -.  { x  e.  I  |  ( ( X  .x.  ( U `
 L ) ) `
 x )  =/= 
.0.  }  C_  J )
3229, 30, 31syl2anc 643 . . . 4  |-  ( ph  ->  -.  { x  e.  I  |  ( ( X  .x.  ( U `
 L ) ) `
 x )  =/= 
.0.  }  C_  J )
333frlmlmod 27085 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  F  e.  LMod )
349, 6, 33syl2anc 643 . . . . . . . 8  |-  ( ph  ->  F  e.  LMod )
353frlmsca 27089 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
369, 6, 35syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  R  =  (Scalar `  F ) )
3736fveq2d 5691 . . . . . . . . . 10  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  F )
) )
385, 37syl5eq 2448 . . . . . . . . 9  |-  ( ph  ->  K  =  ( Base `  (Scalar `  F )
) )
398, 38eleqtrd 2480 . . . . . . . 8  |-  ( ph  ->  X  e.  ( Base `  (Scalar `  F )
) )
40 eqid 2404 . . . . . . . . 9  |-  (Scalar `  F )  =  (Scalar `  F )
41 eqid 2404 . . . . . . . . 9  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
424, 40, 14, 41lmodvscl 15922 . . . . . . . 8  |-  ( ( F  e.  LMod  /\  X  e.  ( Base `  (Scalar `  F ) )  /\  ( U `  L )  e.  B )  -> 
( X  .x.  ( U `  L )
)  e.  B )
4334, 39, 13, 42syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( X  .x.  ( U `  L )
)  e.  B )
443, 5, 4frlmbasf 27096 . . . . . . 7  |-  ( ( I  e.  V  /\  ( X  .x.  ( U `
 L ) )  e.  B )  -> 
( X  .x.  ( U `  L )
) : I --> K )
456, 43, 44syl2anc 643 . . . . . 6  |-  ( ph  ->  ( X  .x.  ( U `  L )
) : I --> K )
46 ffn 5550 . . . . . 6  |-  ( ( X  .x.  ( U `
 L ) ) : I --> K  -> 
( X  .x.  ( U `  L )
)  Fn  I )
47 fnniniseg2 5813 . . . . . 6  |-  ( ( X  .x.  ( U `
 L ) )  Fn  I  ->  ( `' ( X  .x.  ( U `  L ) ) " ( _V 
\  {  .0.  }
) )  =  {
x  e.  I  |  ( ( X  .x.  ( U `  L ) ) `  x )  =/=  .0.  } )
4845, 46, 473syl 19 . . . . 5  |-  ( ph  ->  ( `' ( X 
.x.  ( U `  L ) ) "
( _V  \  {  .0.  } ) )  =  { x  e.  I  |  ( ( X 
.x.  ( U `  L ) ) `  x )  =/=  .0.  } )
4948sseq1d 3335 . . . 4  |-  ( ph  ->  ( ( `' ( X  .x.  ( U `
 L ) )
" ( _V  \  {  .0.  } ) ) 
C_  J  <->  { x  e.  I  |  (
( X  .x.  ( U `  L )
) `  x )  =/=  .0.  }  C_  J
) )
5032, 49mtbird 293 . . 3  |-  ( ph  ->  -.  ( `' ( X  .x.  ( U `
 L ) )
" ( _V  \  {  .0.  } ) ) 
C_  J )
5150intnand 883 . 2  |-  ( ph  ->  -.  ( ( X 
.x.  ( U `  L ) )  e.  B  /\  ( `' ( X  .x.  ( U `  L )
) " ( _V 
\  {  .0.  }
) )  C_  J
) )
52 cnveq 5005 . . . . 5  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  `' x  =  `' ( X  .x.  ( U `  L ) ) )
5352imaeq1d 5161 . . . 4  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  ( `' x " ( _V  \  {  .0.  } ) )  =  ( `' ( X  .x.  ( U `
 L ) )
" ( _V  \  {  .0.  } ) ) )
5453sseq1d 3335 . . 3  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  ( ( `' x " ( _V 
\  {  .0.  }
) )  C_  J  <->  ( `' ( X  .x.  ( U `  L ) ) " ( _V 
\  {  .0.  }
) )  C_  J
) )
55 frlmssuvc1.c . . 3  |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } ) )  C_  J }
5654, 55elrab2 3054 . 2  |-  ( ( X  .x.  ( U `
 L ) )  e.  C  <->  ( ( X  .x.  ( U `  L ) )  e.  B  /\  ( `' ( X  .x.  ( U `  L )
) " ( _V 
\  {  .0.  }
) )  C_  J
) )
5751, 56sylnibr 297 1  |-  ( ph  ->  -.  ( X  .x.  ( U `  L ) )  e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774   `'ccnv 4836   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   0gc0g 13678   Ringcrg 15615   1rcur 15617   LModclmod 15905   freeLMod cfrlm 27080   unitVec cuvc 27081
This theorem is referenced by:  frlmlbs  27117
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-mgp 15604  df-rng 15618  df-ur 15620  df-subrg 15821  df-lmod 15907  df-lss 15964  df-sra 16199  df-rgmod 16200  df-dsmm 27066  df-frlm 27082  df-uvc 27083
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