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Theorem frlmssuvc2OLD 18331
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 18329 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 3438 . . . . . 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 3438 . . . . . . . . 9  |-  ( ph  ->  X  e.  K )
9 frlmssuvc1OLD.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Ring )
10 frlmssuvc1OLD.u . . . . . . . . . . . 12  |-  U  =  ( R unitVec  I )
1110, 3, 4uvcff 18325 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  U : I --> B )
129, 6, 11syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  U : I --> B )
1312, 2ffvelrnd 5943 . . . . . . . . 9  |-  ( ph  ->  ( U `  L
)  e.  B )
14 frlmssuvc1OLD.t . . . . . . . . 9  |-  .x.  =  ( .s `  F )
15 eqid 2451 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
163, 4, 5, 6, 8, 13, 2, 14, 15frlmvscaval 18303 . . . . . . . 8  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =  ( X ( .r `  R ) ( ( U `  L ) `  L
) ) )
17 eqid 2451 . . . . . . . . . 10  |-  ( 1r
`  R )  =  ( 1r `  R
)
1810, 9, 6, 2, 17uvcvv1 18323 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  L ) `  L
)  =  ( 1r
`  R ) )
1918oveq2d 6206 . . . . . . . 8  |-  ( ph  ->  ( X ( .r
`  R ) ( ( U `  L
) `  L )
)  =  ( X ( .r `  R
) ( 1r `  R ) ) )
205, 15, 17rngridm 16775 . . . . . . . . 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 2496 . . . . . . 7  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =  X )
23 eldifsni 4099 . . . . . . . 8  |-  ( X  e.  ( K  \  {  .0.  } )  ->  X  =/=  .0.  )
247, 23syl 16 . . . . . . 7  |-  ( ph  ->  X  =/=  .0.  )
2522, 24eqnetrd 2741 . . . . . 6  |-  ( ph  ->  ( ( X  .x.  ( U `  L ) ) `  L )  =/=  .0.  )
26 fveq2 5789 . . . . . . . 8  |-  ( x  =  L  ->  (
( X  .x.  ( U `  L )
) `  x )  =  ( ( X 
.x.  ( U `  L ) ) `  L ) )
2726neeq1d 2725 . . . . . . 7  |-  ( x  =  L  ->  (
( ( X  .x.  ( U `  L ) ) `  x )  =/=  .0.  <->  ( ( X  .x.  ( U `  L ) ) `  L )  =/=  .0.  ) )
2827elrab 3214 . . . . . 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 3439 . . . . 5  |-  ( ph  ->  -.  L  e.  J
)
31 nelss 3513 . . . . 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 18283 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  F  e.  LMod )
349, 6, 33syl2anc 661 . . . . . . . 8  |-  ( ph  ->  F  e.  LMod )
353frlmsca 18287 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  I  e.  V )  ->  R  =  (Scalar `  F )
)
369, 6, 35syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  R  =  (Scalar `  F ) )
3736fveq2d 5793 . . . . . . . . . 10  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  F )
) )
385, 37syl5eq 2504 . . . . . . . . 9  |-  ( ph  ->  K  =  ( Base `  (Scalar `  F )
) )
398, 38eleqtrd 2541 . . . . . . . 8  |-  ( ph  ->  X  e.  ( Base `  (Scalar `  F )
) )
40 eqid 2451 . . . . . . . . 9  |-  (Scalar `  F )  =  (Scalar `  F )
41 eqid 2451 . . . . . . . . 9  |-  ( Base `  (Scalar `  F )
)  =  ( Base `  (Scalar `  F )
)
424, 40, 14, 41lmodvscl 17071 . . . . . . . 8  |-  ( ( F  e.  LMod  /\  X  e.  ( Base `  (Scalar `  F ) )  /\  ( U `  L )  e.  B )  -> 
( X  .x.  ( U `  L )
)  e.  B )
4334, 39, 13, 42syl3anc 1219 . . . . . . 7  |-  ( ph  ->  ( X  .x.  ( U `  L )
)  e.  B )
443, 5, 4frlmbasf 18297 . . . . . . 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 5657 . . . . . 6  |-  ( ( X  .x.  ( U `
 L ) ) : I --> K  -> 
( X  .x.  ( U `  L )
)  Fn  I )
47 fnniniseg2OLD 5926 . . . . . 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 3481 . . . 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 907 . 2  |-  ( ph  ->  -.  ( ( X 
.x.  ( U `  L ) )  e.  B  /\  ( `' ( X  .x.  ( U `  L )
) " ( _V 
\  {  .0.  }
) )  C_  J
) )
52 cnveq 5111 . . . . 5  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  `' x  =  `' ( X  .x.  ( U `  L ) ) )
5352imaeq1d 5266 . . . 4  |-  ( x  =  ( X  .x.  ( U `  L ) )  ->  ( `' x " ( _V  \  {  .0.  } ) )  =  ( `' ( X  .x.  ( U `
 L ) )
" ( _V  \  {  .0.  } ) ) )
5453sseq1d 3481 . . 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 3216 . 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 1370    e. wcel 1758    =/= wne 2644   {crab 2799   _Vcvv 3068    \ cdif 3423    C_ wss 3426   {csn 3975   `'ccnv 4937   "cima 4941    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   Basecbs 14276   .rcmulr 14341  Scalarcsca 14343   .scvsca 14344   0gc0g 14480   1rcur 16708   Ringcrg 16751   LModclmod 17054   freeLMod cfrlm 18280   unitVec cuvc 18316
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-hom 14364  df-cco 14365  df-0g 14482  df-prds 14488  df-pws 14490  df-mnd 15517  df-grp 15647  df-minusg 15648  df-sbg 15649  df-subg 15780  df-mgp 16697  df-ur 16709  df-rng 16753  df-subrg 16969  df-lmod 17056  df-lss 17120  df-sra 17359  df-rgmod 17360  df-dsmm 18266  df-frlm 18281  df-uvc 18317
This theorem is referenced by: (None)
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