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Theorem lssvs0or 17556
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 17551 . . . . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( .r
`  F )  =  ( .r `  F
)
12 eqid 2467 . . . . . . . . . . 11  |-  ( 1r
`  F )  =  ( 1r `  F
)
13 eqid 2467 . . . . . . . . . . 11  |-  ( invr `  F )  =  (
invr `  F )
149, 10, 11, 12, 13drnginvrl 17215 . . . . . . . . . 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 6299 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( ( invr `  F ) `  A
) ( .r `  F ) A ) 
.x.  X )  =  ( ( 1r `  F )  .x.  X
) )
17 lveclmod 17552 . . . . . . . . . . 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 17213 . . . . . . . . . 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 17337 . . . . . . . . 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 17340 . . . . . . . . 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 2517 . . . . . . 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 17401 . . . . . . . 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 2555 . . . . . 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 2685 . . . 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 6291 . . . . 5  |-  ( A  =  .0.  ->  ( A  .x.  X )  =  (  .0.  .x.  X
) )
4342adantl 466 . . . 4  |-  ( (
ph  /\  A  =  .0.  )  ->  ( A 
.x.  X )  =  (  .0.  .x.  X
) )
44 eqid 2467 . . . . . . . 8  |-  ( 0g
`  W )  =  ( 0g `  W
)
4524, 2, 25, 10, 44lmod0vs 17345 . . . . . . 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 17393 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 0g `  W )  e.  U )
4818, 31, 47syl2anc 661 . . . . . 6  |-  ( ph  ->  ( 0g `  W
)  e.  U )
4946, 48eqeltrd 2555 . . . . 5  |-  ( ph  ->  (  .0.  .x.  X
)  e.  U )
5049adantr 465 . . . 4  |-  ( (
ph  /\  A  =  .0.  )  ->  (  .0. 
.x.  X )  e.  U )
5143, 50eqeltrd 2555 . . 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 17401 . . . 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 783 . 2  |-  ( (
ph  /\  ( A  =  .0.  \/  X  e.  U ) )  -> 
( A  .x.  X
)  e.  U )
5941, 58impbida 830 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 1379    e. wcel 1767    =/= wne 2662   ` cfv 5588  (class class class)co 6284   Basecbs 14490   .rcmulr 14556  Scalarcsca 14558   .scvsca 14559   0gc0g 14695   1rcur 16955   invrcinvr 17121   DivRingcdr 17196   LModclmod 17312   LSubSpclss 17378   LVecclvec 17548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-drng 17198  df-lmod 17314  df-lss 17379  df-lvec 17549
This theorem is referenced by:  lspdisj  17571
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