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Theorem lssvs0or 17189
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 17184 . . . . . . . . . . . 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 2441 . . . . . . . . . . 11  |-  ( .r
`  F )  =  ( .r `  F
)
12 eqid 2441 . . . . . . . . . . 11  |-  ( 1r
`  F )  =  ( 1r `  F
)
13 eqid 2441 . . . . . . . . . . 11  |-  ( invr `  F )  =  (
invr `  F )
149, 10, 11, 12, 13drnginvrl 16849 . . . . . . . . . 10  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
.0.  )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
155, 7, 8, 14syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
1615oveq1d 6104 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( ( invr `  F ) `  A
) ( .r `  F ) A ) 
.x.  X )  =  ( ( 1r `  F )  .x.  X
) )
17 lveclmod 17185 . . . . . . . . . . 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 16847 . . . . . . . . . 10  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
.0.  )  ->  (
( invr `  F ) `  A )  e.  K
)
215, 7, 8, 20syl3anc 1218 . . . . . . . . 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 16971 . . . . . . . . 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 1220 . . . . . . . 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 16974 . . . . . . . . 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 2482 . . . . . . 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 754 . . . . . . . 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 17034 . . . . . . . 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 1219 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) )  e.  U
)
3730, 36eqeltrd 2515 . . . . . 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 2677 . . . 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 6096 . . . . 5  |-  ( A  =  .0.  ->  ( A  .x.  X )  =  (  .0.  .x.  X
) )
4342adantl 466 . . . 4  |-  ( (
ph  /\  A  =  .0.  )  ->  ( A 
.x.  X )  =  (  .0.  .x.  X
) )
44 eqid 2441 . . . . . . . 8  |-  ( 0g
`  W )  =  ( 0g `  W
)
4524, 2, 25, 10, 44lmod0vs 16979 . . . . . . 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 17026 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 0g `  W )  e.  U )
4818, 31, 47syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 0g `  W
)  e.  U )
4946, 48eqeltrd 2515 . . . . 5  |-  ( ph  ->  (  .0.  .x.  X
)  e.  U )
5049adantr 465 . . . 4  |-  ( (
ph  /\  A  =  .0.  )  ->  (  .0. 
.x.  X )  e.  U )
5143, 50eqeltrd 2515 . . 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 17034 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( A  e.  K  /\  X  e.  U ) )  -> 
( A  .x.  X
)  e.  U )
5752, 53, 54, 55, 56syl22anc 1219 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( A  .x.  X )  e.  U )
5851, 57jaodan 783 . 2  |-  ( (
ph  /\  ( A  =  .0.  \/  X  e.  U ) )  -> 
( A  .x.  X
)  e.  U )
5941, 58impbida 828 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 1369    e. wcel 1756    =/= wne 2604   ` cfv 5416  (class class class)co 6089   Basecbs 14172   .rcmulr 14237  Scalarcsca 14239   .scvsca 14240   0gc0g 14376   1rcur 16601   invrcinvr 16761   DivRingcdr 16830   LModclmod 16946   LSubSpclss 17011   LVecclvec 17181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-tpos 6743  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mgp 16590  df-ur 16602  df-rng 16645  df-oppr 16713  df-dvdsr 16731  df-unit 16732  df-invr 16762  df-drng 16832  df-lmod 16948  df-lss 17012  df-lvec 17182
This theorem is referenced by:  lspdisj  17204
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