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Theorem frlmssuvc2 18346
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.) (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 )
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 3449 . . . . . 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 3449 . . . . . . . . 9  |-  ( ph  ->  X  e.  K )
9 frlmssuvc1.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Ring )
10 frlmssuvc1.u . . . . . . . . . . . 12  |-  U  =  ( R unitVec  I )
1110, 3, 4uvcff 18342 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> B )
129, 6, 11syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  U : I --> B )
1312, 2ffvelrnd 5954 . . . . . . . . 9  |-  ( ph  ->  ( U `  L
)  e.  B )
14 frlmssuvc1.t . . . . . . . . 9  |-  .x.  =  ( .s `  F )
15 eqid 2454 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
163, 4, 5, 6, 8, 13, 2, 14, 15frlmvscaval 18320 . . . . . . . 8  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =  ( X ( .r `  R ) ( ( U `  L ) `  L
) ) )
17 eqid 2454 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
1810, 9, 6, 2, 17uvcvv1 18340 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  L ) `  L
)  =  ( 1r
`  R ) )
1918oveq2d 6217 . . . . . . . 8  |-  ( ph  ->  ( X ( .r
`  R ) ( ( U `  L
) `  L )
)  =  ( X ( .r `  R
) ( 1r `  R ) ) )
205, 15, 17rngridm 16793 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  X  e.  K )  ->  ( X ( .r `  R ) ( 1r
`  R ) )  =  X )
219, 8, 20syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( X ( .r
`  R ) ( 1r `  R ) )  =  X )
2216, 19, 213eqtrd 2499 . . . . . . 7  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =  X )
23 eldifsni 4110 . . . . . . . 8  |-  ( X  e.  ( K  \  {  .0.  } )  ->  X  =/=  .0.  )
247, 23syl 16 . . . . . . 7  |-  ( ph  ->  X  =/=  .0.  )
2522, 24eqnetrd 2745 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =/=  .0.  )
26 fveq2 5800 . . . . . . . 8  |-  ( x  =  L  ->  (
( X  .x.  ( U `  L )
) `  x )  =  ( ( X 
.x.  ( U `  L ) ) `  L ) )
2726neeq1d 2729 . . . . . . 7  |-  ( x  =  L  ->  (
( ( X  .x.  ( U `  L ) ) `  x )  =/=  .0.  <->  ( ( X  .x.  ( U `  L ) ) `  L )  =/=  .0.  ) )
2827elrab 3224 . . . . . 6  |-  ( L  e.  { x  e.  I  |  ( ( X  .x.  ( U `
 L ) ) `
 x )  =/= 
.0.  }  <->  ( L  e.  I  /\  ( ( X  .x.  ( U `
 L ) ) `
 L )  =/= 
.0.  ) )
292, 25, 28sylanbrc 664 . . . . 5  |-  ( ph  ->  L  e.  { x  e.  I  |  (
( X  .x.  ( U `  L )
) `  x )  =/=  .0.  } )
301eldifbd 3450 . . . . 5  |-  ( ph  ->  -.  L  e.  J
)
31 nelss 3524 . . . . 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 661 . . . 4  |-  ( ph  ->  -.  { x  e.  I  |  ( ( X  .x.  ( U `
 L ) ) `
 x )  =/= 
.0.  }  C_  J )
333frlmlmod 18300 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  F  e.  LMod )
349, 6, 33syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  F  e.  LMod )
353frlmsca 18304 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
369, 6, 35syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  R  =  (Scalar `  F ) )
3736fveq2d 5804 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  F )
) )
385, 37syl5eq 2507 . . . . . . . . . 10  |-  ( ph  ->  K  =  ( Base `  (Scalar `  F )
) )
398, 38eleqtrd 2544 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( Base `  (Scalar `  F )
) )
40 eqid 2454 . . . . . . . . . 10  |-  (Scalar `  F )  =  (Scalar `  F )
41 eqid 2454 . . . . . . . . . 10  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
424, 40, 14, 41lmodvscl 17089 . . . . . . . . 9  |-  ( ( F  e.  LMod  /\  X  e.  ( Base `  (Scalar `  F ) )  /\  ( U `  L )  e.  B )  -> 
( X  .x.  ( U `  L )
)  e.  B )
4334, 39, 13, 42syl3anc 1219 . . . . . . . 8  |-  ( ph  ->  ( X  .x.  ( U `  L )
)  e.  B )
443, 5, 4frlmbasf 18314 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( X  .x.  ( U `
 L ) )  e.  B )  -> 
( X  .x.  ( U `  L )
) : I --> K )
456, 43, 44syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( X  .x.  ( U `  L )
) : I --> K )
46 ffn 5668 . . . . . . 7  |-  ( ( X  .x.  ( U `
 L ) ) : I --> K  -> 
( X  .x.  ( U `  L )
)  Fn  I )
4745, 46syl 16 . . . . . 6  |-  ( ph  ->  ( X  .x.  ( U `  L )
)  Fn  I )
48 frlmssuvc1.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
49 fvex 5810 . . . . . . . 8  |-  ( 0g
`  R )  e. 
_V
5048, 49eqeltri 2538 . . . . . . 7  |-  .0.  e.  _V
5150a1i 11 . . . . . 6  |-  ( ph  ->  .0.  e.  _V )
52 suppvalfn 6808 . . . . . 6  |-  ( ( ( X  .x.  ( U `  L )
)  Fn  I  /\  I  e.  V  /\  .0.  e.  _V )  -> 
( ( X  .x.  ( U `  L ) ) supp  .0.  )  =  { x  e.  I  |  ( ( X 
.x.  ( U `  L ) ) `  x )  =/=  .0.  } )
5347, 6, 51, 52syl3anc 1219 . . . . 5  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) supp  .0.  )  =  { x  e.  I  |  ( ( X 
.x.  ( U `  L ) ) `  x )  =/=  .0.  } )
5453sseq1d 3492 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  ( U `  L ) ) supp  .0.  )  C_  J  <->  { x  e.  I  |  (
( X  .x.  ( U `  L )
) `  x )  =/=  .0.  }  C_  J
) )
5532, 54mtbird 301 . . 3  |-  ( ph  ->  -.  ( ( X 
.x.  ( U `  L ) ) supp  .0.  )  C_  J )
5655intnand 907 . 2  |-  ( ph  ->  -.  ( ( X 
.x.  ( U `  L ) )  e.  B  /\  ( ( X  .x.  ( U `
 L ) ) supp 
.0.  )  C_  J
) )
57 oveq1 6208 . . . 4  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  ( x supp  .0.  )  =  ( ( X  .x.  ( U `
 L ) ) supp 
.0.  ) )
5857sseq1d 3492 . . 3  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  ( (
x supp  .0.  )  C_  J 
<->  ( ( X  .x.  ( U `  L ) ) supp  .0.  )  C_  J ) )
59 frlmssuvc1.c . . 3  |-  C  =  { x  e.  B  |  ( x supp  .0.  )  C_  J }
6058, 59elrab2 3226 . 2  |-  ( ( X  .x.  ( U `
 L ) )  e.  C  <->  ( ( X  .x.  ( U `  L ) )  e.  B  /\  ( ( X  .x.  ( U `
 L ) ) supp 
.0.  )  C_  J
) )
6156, 60sylnibr 305 1  |-  ( ph  ->  -.  ( X  .x.  ( U `  L ) )  e.  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078    \ cdif 3434    C_ wss 3437   {csn 3986    Fn wfn 5522   -->wf 5523   ` cfv 5527  (class class class)co 6201   supp csupp 6801   Basecbs 14293   .rcmulr 14359  Scalarcsca 14361   .scvsca 14362   0gc0g 14498   1rcur 16726   Ringcrg 16769   LModclmod 17072   freeLMod cfrlm 18297   unitVec cuvc 18333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-sca 14374  df-vsca 14375  df-ip 14376  df-tset 14377  df-ple 14378  df-ds 14380  df-hom 14382  df-cco 14383  df-0g 14500  df-prds 14506  df-pws 14508  df-mnd 15535  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-mgp 16715  df-ur 16727  df-rng 16771  df-subrg 16987  df-lmod 17074  df-lss 17138  df-sra 17377  df-rgmod 17378  df-dsmm 18283  df-frlm 18298  df-uvc 18334
This theorem is referenced by:  frlmlbs  18351
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