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Theorem lssvs0or 16184
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 16179 . . . . . . . . . . . 12  |-  ( W  e.  LVec  ->  F  e.  DivRing )
41, 3syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  DivRing )
54ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  F  e.  DivRing )
6 lssvs0or.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  K )
76ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  A  e.  K )
8 simpr 449 . . . . . . . . . 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 2438 . . . . . . . . . . 11  |-  ( .r
`  F )  =  ( .r `  F
)
12 eqid 2438 . . . . . . . . . . 11  |-  ( 1r
`  F )  =  ( 1r `  F
)
13 eqid 2438 . . . . . . . . . . 11  |-  ( invr `  F )  =  (
invr `  F )
149, 10, 11, 12, 13drnginvrl 15856 . . . . . . . . . 10  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
.0.  )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
155, 7, 8, 14syl3anc 1185 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
1615oveq1d 6098 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( ( invr `  F ) `  A
) ( .r `  F ) A ) 
.x.  X )  =  ( ( 1r `  F )  .x.  X
) )
17 lveclmod 16180 . . . . . . . . . . 11  |-  ( W  e.  LVec  ->  W  e. 
LMod )
181, 17syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  e.  LMod )
1918ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  W  e.  LMod )
209, 10, 13drnginvrcl 15854 . . . . . . . . . 10  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
.0.  )  ->  (
( invr `  F ) `  A )  e.  K
)
215, 7, 8, 20syl3anc 1185 . . . . . . . . 9  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( invr `  F ) `  A )  e.  K
)
22 lssvs0or.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  V )
2322ad2antrr 708 . . . . . . . . 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 15977 . . . . . . . . 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 1187 . . . . . . . 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 15980 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 1r `  F
)  .x.  X )  =  X )
2919, 23, 28syl2anc 644 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( 1r `  F
)  .x.  X )  =  X )
3016, 27, 293eqtr3rd 2479 . . . . . . 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 708 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  U  e.  S )
33 simplr 733 . . . . . . . 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 16033 . . . . . . . 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 1186 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  (
( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) )  e.  U
)
3730, 36eqeltrd 2512 . . . . . 6  |-  ( ( ( ph  /\  ( A  .x.  X )  e.  U )  /\  A  =/=  .0.  )  ->  X  e.  U )
3837ex 425 . . . . 5  |-  ( (
ph  /\  ( A  .x.  X )  e.  U
)  ->  ( A  =/=  .0.  ->  X  e.  U ) )
3938necon1bd 2674 . . . 4  |-  ( (
ph  /\  ( A  .x.  X )  e.  U
)  ->  ( -.  X  e.  U  ->  A  =  .0.  ) )
4039orrd 369 . . 3  |-  ( (
ph  /\  ( A  .x.  X )  e.  U
)  ->  ( X  e.  U  \/  A  =  .0.  ) )
4140orcomd 379 . 2  |-  ( (
ph  /\  ( A  .x.  X )  e.  U
)  ->  ( A  =  .0.  \/  X  e.  U ) )
42 oveq1 6090 . . . . 5  |-  ( A  =  .0.  ->  ( A  .x.  X )  =  (  .0.  .x.  X
) )
4342adantl 454 . . . 4  |-  ( (
ph  /\  A  =  .0.  )  ->  ( A 
.x.  X )  =  (  .0.  .x.  X
) )
44 eqid 2438 . . . . . . . 8  |-  ( 0g
`  W )  =  ( 0g `  W
)
4524, 2, 25, 10, 44lmod0vs 15985 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (  .0.  .x.  X )  =  ( 0g `  W
) )
4618, 22, 45syl2anc 644 . . . . . 6  |-  ( ph  ->  (  .0.  .x.  X
)  =  ( 0g
`  W ) )
4744, 34lss0cl 16025 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( 0g `  W )  e.  U )
4818, 31, 47syl2anc 644 . . . . . 6  |-  ( ph  ->  ( 0g `  W
)  e.  U )
4946, 48eqeltrd 2512 . . . . 5  |-  ( ph  ->  (  .0.  .x.  X
)  e.  U )
5049adantr 453 . . . 4  |-  ( (
ph  /\  A  =  .0.  )  ->  (  .0. 
.x.  X )  e.  U )
5143, 50eqeltrd 2512 . . 3  |-  ( (
ph  /\  A  =  .0.  )  ->  ( A 
.x.  X )  e.  U )
5218adantr 453 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  W  e.  LMod )
5331adantr 453 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  U  e.  S )
546adantr 453 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  A  e.  K )
55 simpr 449 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  X  e.  U )
562, 25, 9, 34lssvscl 16033 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( A  e.  K  /\  X  e.  U ) )  -> 
( A  .x.  X
)  e.  U )
5752, 53, 54, 55, 56syl22anc 1186 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( A  .x.  X )  e.  U )
5851, 57jaodan 762 . 2  |-  ( (
ph  /\  ( A  =  .0.  \/  X  e.  U ) )  -> 
( A  .x.  X
)  e.  U )
5941, 58impbida 807 1  |-  ( ph  ->  ( ( A  .x.  X )  e.  U  <->  ( A  =  .0.  \/  X  e.  U )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5456  (class class class)co 6083   Basecbs 13471   .rcmulr 13532  Scalarcsca 13534   .scvsca 13535   0gc0g 13725   1rcur 15664   invrcinvr 15778   DivRingcdr 15837   LModclmod 15952   LSubSpclss 16010   LVecclvec 16176
This theorem is referenced by:  lspdisj  16199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mgp 15651  df-rng 15665  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-drng 15839  df-lmod 15954  df-lss 16011  df-lvec 16177
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