MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lssvs0or Structured version   Unicode version

Theorem lssvs0or 17883
Description: If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs. (Contributed by NM, 5-Apr-2015.)
Hypotheses
Ref Expression
lssvs0or.v  |-  V  =  ( Base `  W
)
lssvs0or.t  |-  .x.  =  ( .s `  W )
lssvs0or.f  |-  F  =  (Scalar `  W )
lssvs0or.k  |-  K  =  ( Base `  F
)
lssvs0or.o  |-  .0.  =  ( 0g `  F )
lssvs0or.s  |-  S  =  ( LSubSp `  W )
lssvs0or.w  |-  ( ph  ->  W  e.  LVec )
lssvs0or.u  |-  ( ph  ->  U  e.  S )
lssvs0or.x  |-  ( ph  ->  X  e.  V )
lssvs0or.a  |-  ( ph  ->  A  e.  K )
Assertion
Ref Expression
lssvs0or  |-  ( ph  ->  ( ( A  .x.  X )  e.  U  <->  ( A  =  .0.  \/  X  e.  U )
) )

Proof of Theorem lssvs0or
StepHypRef Expression
1 lssvs0or.w . . . . . . . . . . . 12  |-  ( ph  ->  W  e.  LVec )
2 lssvs0or.f . . . . . . . . . . . . 13  |-  F  =  (Scalar `  W )
32lvecdrng 17878 . . . . . . . . . . . 12  |-  ( W  e.  LVec  ->  F  e.  DivRing )
41, 3syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  DivRing )
54ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  F  e.  DivRing )
6 lssvs0or.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  K )
76ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  A  e.  K )
8 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  A  =/=  .0.  )
9 lssvs0or.k . . . . . . . . . . 11  |-  K  =  ( Base `  F
)
10 lssvs0or.o . . . . . . . . . . 11  |-  .0.  =  ( 0g `  F )
11 eqid 2457 . . . . . . . . . . 11  |-  ( .r
`  F )  =  ( .r `  F
)
12 eqid 2457 . . . . . . . . . . 11  |-  ( 1r
`  F )  =  ( 1r `  F
)
13 eqid 2457 . . . . . . . . . . 11  |-  ( invr `  F )  =  (
invr `  F )
149, 10, 11, 12, 13drnginvrl 17542 . . . . . . . . . 10  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
.0.  )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
155, 7, 8, 14syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
1615oveq1d 6311 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( ( invr `  F ) `  A
) ( .r `  F ) A ) 
.x.  X )  =  ( ( 1r `  F )  .x.  X
) )
17 lveclmod 17879 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
181, 17syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
1918ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  W  e.  LMod )
209, 10, 13drnginvrcl 17540 . . . . . . . . . 10  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
.0.  )  ->  (
( invr `  F ) `  A )  e.  K
)
215, 7, 8, 20syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( invr `  F ) `  A )  e.  K
)
22 lssvs0or.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  V )
2322ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  X  e.  V )
24 lssvs0or.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
25 lssvs0or.t . . . . . . . . . 10  |-  .x.  =  ( .s `  W )
2624, 2, 25, 9, 11lmodvsass 17664 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  (
( ( invr `  F
) `  A )  e.  K  /\  A  e.  K  /\  X  e.  V ) )  -> 
( ( ( (
invr `  F ) `  A ) ( .r
`  F ) A )  .x.  X )  =  ( ( (
invr `  F ) `  A )  .x.  ( A  .x.  X ) ) )
2719, 21, 7, 23, 26syl13anc 1230 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( ( invr `  F ) `  A
) ( .r `  F ) A ) 
.x.  X )  =  ( ( ( invr `  F ) `  A
)  .x.  ( A  .x.  X ) ) )
2824, 2, 25, 12lmodvs1 17667 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 1r `  F
)  .x.  X )  =  X )
2919, 23, 28syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( 1r `  F
)  .x.  X )  =  X )
3016, 27, 293eqtr3rd 2507 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  X  =  ( ( (
invr `  F ) `  A )  .x.  ( A  .x.  X ) ) )
31 lssvs0or.u . . . . . . . . 9  |-  ( ph  ->  U  e.  S )
3231ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  U  e.  S )
33 simplr 755 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  ( A  .x.  X )  e.  U )
34 lssvs0or.s . . . . . . . . 9  |-  S  =  ( LSubSp `  W )
352, 25, 9, 34lssvscl 17728 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( ( (
invr `  F ) `  A )  e.  K  /\  ( A  .x.  X
)  e.  U ) )  ->  ( (
( invr `  F ) `  A )  .x.  ( A  .x.  X ) )  e.  U )
3619, 32, 21, 33, 35syl22anc 1229 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) )  e.  U
)
3730, 36eqeltrd 2545 . . . . . 6  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  X  e.  U )
3837ex 434 . . . . 5  |-  ( (
ph  /\  ( A  .x.  X )  e.  U
)  ->  ( A  =/=  .0.  ->  X  e.  U ) )
3938necon1bd 2675 . . . 4  |-  ( (
ph  /\  ( A  .x.  X )  e.  U
)  ->  ( -.  X  e.  U  ->  A  =  .0.  ) )
4039orrd 378 . . 3  |-  ( (
ph  /\  ( A  .x.  X )  e.  U
)  ->  ( X  e.  U  \/  A  =  .0.  ) )
4140orcomd 388 . 2  |-  ( (
ph  /\  ( A  .x.  X )  e.  U
)  ->  ( A  =  .0.  \/  X  e.  U ) )
42 oveq1 6303 . . . . 5  |-  ( A  =  .0.  ->  ( A  .x.  X )  =  (  .0.  .x.  X
) )
4342adantl 466 . . . 4  |-  ( (
ph  /\  A  =  .0.  )  ->  ( A 
.x.  X )  =  (  .0.  .x.  X
) )
44 eqid 2457 . . . . . . . 8  |-  ( 0g
`  W )  =  ( 0g `  W
)
4524, 2, 25, 10, 44lmod0vs 17672 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (  .0.  .x.  X )  =  ( 0g `  W
) )
4618, 22, 45syl2anc 661 . . . . . 6  |-  ( ph  ->  (  .0.  .x.  X
)  =  ( 0g
`  W ) )
4744, 34lss0cl 17720 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 0g `  W )  e.  U )
4818, 31, 47syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 0g `  W
)  e.  U )
4946, 48eqeltrd 2545 . . . . 5  |-  ( ph  ->  (  .0.  .x.  X
)  e.  U )
5049adantr 465 . . . 4  |-  ( (
ph  /\  A  =  .0.  )  ->  (  .0. 
.x.  X )  e.  U )
5143, 50eqeltrd 2545 . . 3  |-  ( (
ph  /\  A  =  .0.  )  ->  ( A 
.x.  X )  e.  U )
5218adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  W  e.  LMod )
5331adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  S )
546adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  A  e.  K )
55 simpr 461 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
562, 25, 9, 34lssvscl 17728 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( A  e.  K  /\  X  e.  U ) )  -> 
( A  .x.  X
)  e.  U )
5752, 53, 54, 55, 56syl22anc 1229 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( A  .x.  X )  e.  U )
5851, 57jaodan 785 . 2  |-  ( (
ph  /\  ( A  =  .0.  \/  X  e.  U ) )  -> 
( A  .x.  X
)  e.  U )
5941, 58impbida 832 1  |-  ( ph  ->  ( ( A  .x.  X )  e.  U  <->  ( A  =  .0.  \/  X  e.  U )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   ` cfv 5594  (class class class)co 6296   Basecbs 14644   .rcmulr 14713  Scalarcsca 14715   .scvsca 14716   0gc0g 14857   1rcur 17280   invrcinvr 17447   DivRingcdr 17523   LModclmod 17639   LSubSpclss 17705   LVecclvec 17875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-drng 17525  df-lmod 17641  df-lss 17706  df-lvec 17876
This theorem is referenced by:  lspdisj  17898
  Copyright terms: Public domain W3C validator