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Theorem lvecvs0or 17554
Description: If a scalar product is zero, one of its factors must be zero. (hvmul0or 25646 analog.) (Contributed by NM, 2-Jul-2014.)
Hypotheses
Ref Expression
lvecmul0or.v  |-  V  =  ( Base `  W
)
lvecmul0or.s  |-  .x.  =  ( .s `  W )
lvecmul0or.f  |-  F  =  (Scalar `  W )
lvecmul0or.k  |-  K  =  ( Base `  F
)
lvecmul0or.o  |-  O  =  ( 0g `  F
)
lvecmul0or.z  |-  .0.  =  ( 0g `  W )
lvecmul0or.w  |-  ( ph  ->  W  e.  LVec )
lvecmul0or.a  |-  ( ph  ->  A  e.  K )
lvecmul0or.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
lvecvs0or  |-  ( ph  ->  ( ( A  .x.  X )  =  .0.  <->  ( A  =  O  \/  X  =  .0.  )
) )

Proof of Theorem lvecvs0or
StepHypRef Expression
1 df-ne 2664 . . . . 5  |-  ( A  =/=  O  <->  -.  A  =  O )
2 oveq2 6292 . . . . . . . 8  |-  ( ( A  .x.  X )  =  .0.  ->  (
( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) )  =  ( ( ( invr `  F
) `  A )  .x.  .0.  ) )
32ad2antlr 726 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  (
( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) )  =  ( ( ( invr `  F
) `  A )  .x.  .0.  ) )
4 lvecmul0or.w . . . . . . . . . . . . 13  |-  ( ph  ->  W  e.  LVec )
54adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  O )  ->  W  e.  LVec )
6 lvecmul0or.f . . . . . . . . . . . . 13  |-  F  =  (Scalar `  W )
76lvecdrng 17551 . . . . . . . . . . . 12  |-  ( W  e.  LVec  ->  F  e.  DivRing )
85, 7syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  O )  ->  F  e.  DivRing )
9 lvecmul0or.a . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  K )
109adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  O )  ->  A  e.  K )
11 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  O )  ->  A  =/=  O )
12 lvecmul0or.k . . . . . . . . . . . 12  |-  K  =  ( Base `  F
)
13 lvecmul0or.o . . . . . . . . . . . 12  |-  O  =  ( 0g `  F
)
14 eqid 2467 . . . . . . . . . . . 12  |-  ( .r
`  F )  =  ( .r `  F
)
15 eqid 2467 . . . . . . . . . . . 12  |-  ( 1r
`  F )  =  ( 1r `  F
)
16 eqid 2467 . . . . . . . . . . . 12  |-  ( invr `  F )  =  (
invr `  F )
1712, 13, 14, 15, 16drnginvrl 17215 . . . . . . . . . . 11  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
O )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
188, 10, 11, 17syl3anc 1228 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( invr `  F ) `  A ) ( .r
`  F ) A )  =  ( 1r
`  F ) )
1918oveq1d 6299 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( ( invr `  F
) `  A )
( .r `  F
) A )  .x.  X )  =  ( ( 1r `  F
)  .x.  X )
)
20 lveclmod 17552 . . . . . . . . . . . 12  |-  ( W  e.  LVec  ->  W  e. 
LMod )
214, 20syl 16 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LMod )
2221adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  O )  ->  W  e.  LMod )
2312, 13, 16drnginvrcl 17213 . . . . . . . . . . 11  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
O )  ->  (
( invr `  F ) `  A )  e.  K
)
248, 10, 11, 23syl3anc 1228 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  O )  ->  ( ( invr `  F ) `  A )  e.  K
)
25 lvecmul0or.x . . . . . . . . . . 11  |-  ( ph  ->  X  e.  V )
2625adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  O )  ->  X  e.  V )
27 lvecmul0or.v . . . . . . . . . . 11  |-  V  =  ( Base `  W
)
28 lvecmul0or.s . . . . . . . . . . 11  |-  .x.  =  ( .s `  W )
2927, 6, 28, 12, 14lmodvsass 17337 . . . . . . . . . 10  |-  ( ( 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 ) ) )
3022, 24, 10, 26, 29syl13anc 1230 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( ( invr `  F
) `  A )
( .r `  F
) A )  .x.  X )  =  ( ( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) ) )
3127, 6, 28, 15lmodvs1 17340 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 1r `  F
)  .x.  X )  =  X )
3221, 25, 31syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  F )  .x.  X
)  =  X )
3332adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  O )  ->  ( ( 1r `  F )  .x.  X )  =  X )
3419, 30, 333eqtr3d 2516 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( invr `  F ) `  A )  .x.  ( A  .x.  X ) )  =  X )
3534adantlr 714 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  (
( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) )  =  X )
3621adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( A  .x.  X )  =  .0.  )  ->  W  e.  LMod )
3736adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  W  e.  LMod )
3824adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  (
( invr `  F ) `  A )  e.  K
)
39 lvecmul0or.z . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
406, 28, 12, 39lmodvs0 17346 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  (
( invr `  F ) `  A )  e.  K
)  ->  ( (
( invr `  F ) `  A )  .x.  .0.  )  =  .0.  )
4137, 38, 40syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  (
( ( invr `  F
) `  A )  .x.  .0.  )  =  .0.  )
423, 35, 413eqtr3d 2516 . . . . . 6  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  X  =  .0.  )
4342ex 434 . . . . 5  |-  ( (
ph  /\  ( A  .x.  X )  =  .0.  )  ->  ( A  =/=  O  ->  X  =  .0.  ) )
441, 43syl5bir 218 . . . 4  |-  ( (
ph  /\  ( A  .x.  X )  =  .0.  )  ->  ( -.  A  =  O  ->  X  =  .0.  ) )
4544orrd 378 . . 3  |-  ( (
ph  /\  ( A  .x.  X )  =  .0.  )  ->  ( A  =  O  \/  X  =  .0.  ) )
4645ex 434 . 2  |-  ( ph  ->  ( ( A  .x.  X )  =  .0. 
->  ( A  =  O  \/  X  =  .0.  ) ) )
4727, 6, 28, 13, 39lmod0vs 17345 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
4821, 25, 47syl2anc 661 . . . 4  |-  ( ph  ->  ( O  .x.  X
)  =  .0.  )
49 oveq1 6291 . . . . 5  |-  ( A  =  O  ->  ( A  .x.  X )  =  ( O  .x.  X
) )
5049eqeq1d 2469 . . . 4  |-  ( A  =  O  ->  (
( A  .x.  X
)  =  .0.  <->  ( O  .x.  X )  =  .0.  ) )
5148, 50syl5ibrcom 222 . . 3  |-  ( ph  ->  ( A  =  O  ->  ( A  .x.  X )  =  .0.  ) )
526, 28, 12, 39lmodvs0 17346 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  K )  ->  ( A  .x.  .0.  )  =  .0.  )
5321, 9, 52syl2anc 661 . . . 4  |-  ( ph  ->  ( A  .x.  .0.  )  =  .0.  )
54 oveq2 6292 . . . . 5  |-  ( X  =  .0.  ->  ( A  .x.  X )  =  ( A  .x.  .0.  ) )
5554eqeq1d 2469 . . . 4  |-  ( X  =  .0.  ->  (
( A  .x.  X
)  =  .0.  <->  ( A  .x.  .0.  )  =  .0.  ) )
5653, 55syl5ibrcom 222 . . 3  |-  ( ph  ->  ( X  =  .0. 
->  ( A  .x.  X
)  =  .0.  )
)
5751, 56jaod 380 . 2  |-  ( ph  ->  ( ( A  =  O  \/  X  =  .0.  )  ->  ( A  .x.  X )  =  .0.  ) )
5846, 57impbid 191 1  |-  ( ph  ->  ( ( A  .x.  X )  =  .0.  <->  ( A  =  O  \/  X  =  .0.  )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> 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   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-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-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-lvec 17549
This theorem is referenced by:  lvecvsn0  17555  lvecvscan  17557  lvecvscan2  17558  lkreqN  33985  lkrlspeqN  33986  hdmap14lem6  36691  hgmapval0  36710
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