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Theorem frlmssuvc2OLD 18701
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.) Obsolete version of frlmssuvc2 18699 as of 24-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frlmssuvc1OLD.f  |-  F  =  ( R freeLMod  I )
frlmssuvc1OLD.u  |-  U  =  ( R unitVec  I )
frlmssuvc1OLD.b  |-  B  =  ( Base `  F
)
frlmssuvc1OLD.k  |-  K  =  ( Base `  R
)
frlmssuvc1OLD.t  |-  .x.  =  ( .s `  F )
frlmssuvc1OLD.z  |-  .0.  =  ( 0g `  R )
frlmssuvc1OLD.c  |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } ) )  C_  J }
frlmssuvc1OLD.r  |-  ( ph  ->  R  e.  Ring )
frlmssuvc1OLD.i  |-  ( ph  ->  I  e.  V )
frlmssuvc1OLD.j  |-  ( ph  ->  J  C_  I )
frlmssuvc2OLD.l  |-  ( ph  ->  L  e.  ( I 
\  J ) )
frlmssuvc2OLD.x  |-  ( ph  ->  X  e.  ( K 
\  {  .0.  }
) )
Assertion
Ref Expression
frlmssuvc2OLD  |-  ( 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 frlmssuvc2OLD
StepHypRef Expression
1 frlmssuvc2OLD.l . . . . . . 7  |-  ( ph  ->  L  e.  ( I 
\  J ) )
21eldifad 3473 . . . . . 6  |-  ( ph  ->  L  e.  I )
3 frlmssuvc1OLD.f . . . . . . . . 9  |-  F  =  ( R freeLMod  I )
4 frlmssuvc1OLD.b . . . . . . . . 9  |-  B  =  ( Base `  F
)
5 frlmssuvc1OLD.k . . . . . . . . 9  |-  K  =  ( Base `  R
)
6 frlmssuvc1OLD.i . . . . . . . . 9  |-  ( ph  ->  I  e.  V )
7 frlmssuvc2OLD.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( K 
\  {  .0.  }
) )
87eldifad 3473 . . . . . . . . 9  |-  ( ph  ->  X  e.  K )
9 frlmssuvc1OLD.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Ring )
10 frlmssuvc1OLD.u . . . . . . . . . . . 12  |-  U  =  ( R unitVec  I )
1110, 3, 4uvcff 18695 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> B )
129, 6, 11syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  U : I --> B )
1312, 2ffvelrnd 6017 . . . . . . . . 9  |-  ( ph  ->  ( U `  L
)  e.  B )
14 frlmssuvc1OLD.t . . . . . . . . 9  |-  .x.  =  ( .s `  F )
15 eqid 2443 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
163, 4, 5, 6, 8, 13, 2, 14, 15frlmvscaval 18673 . . . . . . . 8  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =  ( X ( .r `  R ) ( ( U `  L ) `  L
) ) )
17 eqid 2443 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
1810, 9, 6, 2, 17uvcvv1 18693 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  L ) `  L
)  =  ( 1r
`  R ) )
1918oveq2d 6297 . . . . . . . 8  |-  ( ph  ->  ( X ( .r
`  R ) ( ( U `  L
) `  L )
)  =  ( X ( .r `  R
) ( 1r `  R ) ) )
205, 15, 17ringridm 17097 . . . . . . . . 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 2488 . . . . . . 7  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =  X )
23 eldifsni 4141 . . . . . . . 8  |-  ( X  e.  ( K  \  {  .0.  } )  ->  X  =/=  .0.  )
247, 23syl 16 . . . . . . 7  |-  ( ph  ->  X  =/=  .0.  )
2522, 24eqnetrd 2736 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =/=  .0.  )
26 fveq2 5856 . . . . . . . 8  |-  ( x  =  L  ->  (
( X  .x.  ( U `  L )
) `  x )  =  ( ( X 
.x.  ( U `  L ) ) `  L ) )
2726neeq1d 2720 . . . . . . 7  |-  ( x  =  L  ->  (
( ( X  .x.  ( U `  L ) ) `  x )  =/=  .0.  <->  ( ( X  .x.  ( U `  L ) ) `  L )  =/=  .0.  ) )
2827elrab 3243 . . . . . 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 3474 . . . . 5  |-  ( ph  ->  -.  L  e.  J
)
31 nelss 3548 . . . . 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 18653 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  F  e.  LMod )
349, 6, 33syl2anc 661 . . . . . . . 8  |-  ( ph  ->  F  e.  LMod )
353frlmsca 18657 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
369, 6, 35syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  R  =  (Scalar `  F ) )
3736fveq2d 5860 . . . . . . . . . 10  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  F )
) )
385, 37syl5eq 2496 . . . . . . . . 9  |-  ( ph  ->  K  =  ( Base `  (Scalar `  F )
) )
398, 38eleqtrd 2533 . . . . . . . 8  |-  ( ph  ->  X  e.  ( Base `  (Scalar `  F )
) )
40 eqid 2443 . . . . . . . . 9  |-  (Scalar `  F )  =  (Scalar `  F )
41 eqid 2443 . . . . . . . . 9  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
424, 40, 14, 41lmodvscl 17403 . . . . . . . 8  |-  ( ( F  e.  LMod  /\  X  e.  ( Base `  (Scalar `  F ) )  /\  ( U `  L )  e.  B )  -> 
( X  .x.  ( U `  L )
)  e.  B )
4334, 39, 13, 42syl3anc 1229 . . . . . . 7  |-  ( ph  ->  ( X  .x.  ( U `  L )
)  e.  B )
443, 5, 4frlmbasf 18667 . . . . . . 7  |-  ( ( I  e.  V  /\  ( X  .x.  ( U `
 L ) )  e.  B )  -> 
( X  .x.  ( U `  L )
) : I --> K )
456, 43, 44syl2anc 661 . . . . . 6  |-  ( ph  ->  ( X  .x.  ( U `  L )
) : I --> K )
46 ffn 5721 . . . . . 6  |-  ( ( X  .x.  ( U `
 L ) ) : I --> K  -> 
( X  .x.  ( U `  L )
)  Fn  I )
47 fnniniseg2OLD 5996 . . . . . 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 20 . . . . 5  |-  ( ph  ->  ( `' ( X 
.x.  ( U `  L ) ) "
( _V  \  {  .0.  } ) )  =  { x  e.  I  |  ( ( X 
.x.  ( U `  L ) ) `  x )  =/=  .0.  } )
4948sseq1d 3516 . . . 4  |-  ( ph  ->  ( ( `' ( X  .x.  ( U `
 L ) )
" ( _V  \  {  .0.  } ) ) 
C_  J  <->  { x  e.  I  |  (
( X  .x.  ( U `  L )
) `  x )  =/=  .0.  }  C_  J
) )
5032, 49mtbird 301 . . 3  |-  ( ph  ->  -.  ( `' ( X  .x.  ( U `
 L ) )
" ( _V  \  {  .0.  } ) ) 
C_  J )
5150intnand 916 . 2  |-  ( ph  ->  -.  ( ( X 
.x.  ( U `  L ) )  e.  B  /\  ( `' ( X  .x.  ( U `  L )
) " ( _V 
\  {  .0.  }
) )  C_  J
) )
52 cnveq 5166 . . . . 5  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  `' x  =  `' ( X  .x.  ( U `  L ) ) )
5352imaeq1d 5326 . . . 4  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  ( `' x " ( _V  \  {  .0.  } ) )  =  ( `' ( X  .x.  ( U `
 L ) )
" ( _V  \  {  .0.  } ) ) )
5453sseq1d 3516 . . 3  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  ( ( `' x " ( _V 
\  {  .0.  }
) )  C_  J  <->  ( `' ( X  .x.  ( U `  L ) ) " ( _V 
\  {  .0.  }
) )  C_  J
) )
55 frlmssuvc1OLD.c . . 3  |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } ) )  C_  J }
5654, 55elrab2 3245 . 2  |-  ( ( X  .x.  ( U `
 L ) )  e.  C  <->  ( ( X  .x.  ( U `  L ) )  e.  B  /\  ( `' ( X  .x.  ( U `  L )
) " ( _V 
\  {  .0.  }
) )  C_  J
) )
5751, 56sylnibr 305 1  |-  ( ph  ->  -.  ( X  .x.  ( U `  L ) )  e.  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   {crab 2797   _Vcvv 3095    \ cdif 3458    C_ wss 3461   {csn 4014   `'ccnv 4988   "cima 4992    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   Basecbs 14509   .rcmulr 14575  Scalarcsca 14577   .scvsca 14578   0gc0g 14714   1rcur 17027   Ringcrg 17072   LModclmod 17386   freeLMod cfrlm 18650   unitVec cuvc 18686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-hom 14598  df-cco 14599  df-0g 14716  df-prds 14722  df-pws 14724  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-sbg 15933  df-subg 16072  df-mgp 17016  df-ur 17028  df-ring 17074  df-subrg 17301  df-lmod 17388  df-lss 17453  df-sra 17692  df-rgmod 17693  df-dsmm 18636  df-frlm 18651  df-uvc 18687
This theorem is referenced by: (None)
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