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Theorem lvecvs0or 16180
Description: If a scalar product is zero, one of its factors must be zero. (hvmul0or 22527 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 2601 . . . . 5  |-  ( A  =/=  O  <->  -.  A  =  O )
2 oveq2 6089 . . . . . . . 8  |-  ( ( A  .x.  X )  =  .0.  ->  (
( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) )  =  ( ( ( invr `  F
) `  A )  .x.  .0.  ) )
32ad2antlr 708 . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  O )  ->  W  e.  LVec )
6 lvecmul0or.f . . . . . . . . . . . . 13  |-  F  =  (Scalar `  W )
76lvecdrng 16177 . . . . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  O )  ->  A  e.  K )
11 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  O )  ->  A  =/=  O )
12 lvecmul0or.k . . . . . . . . . . . 12  |-  K  =  ( Base `  F
)
13 lvecmul0or.o . . . . . . . . . . . 12  |-  O  =  ( 0g `  F
)
14 eqid 2436 . . . . . . . . . . . 12  |-  ( .r
`  F )  =  ( .r `  F
)
15 eqid 2436 . . . . . . . . . . . 12  |-  ( 1r
`  F )  =  ( 1r `  F
)
16 eqid 2436 . . . . . . . . . . . 12  |-  ( invr `  F )  =  (
invr `  F )
1712, 13, 14, 15, 16drnginvrl 15854 . . . . . . . . . . 11  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
O )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
188, 10, 11, 17syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( invr `  F ) `  A ) ( .r
`  F ) A )  =  ( 1r
`  F ) )
1918oveq1d 6096 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( ( invr `  F
) `  A )
( .r `  F
) A )  .x.  X )  =  ( ( 1r `  F
)  .x.  X )
)
20 lveclmod 16178 . . . . . . . . . . . 12  |-  ( W  e.  LVec  ->  W  e. 
LMod )
214, 20syl 16 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LMod )
2221adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  O )  ->  W  e.  LMod )
2312, 13, 16drnginvrcl 15852 . . . . . . . . . . 11  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
O )  ->  (
( invr `  F ) `  A )  e.  K
)
248, 10, 11, 23syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  O )  ->  ( ( invr `  F ) `  A )  e.  K
)
25 lvecmul0or.x . . . . . . . . . . 11  |-  ( ph  ->  X  e.  V )
2625adantr 452 . . . . . . . . . 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 15975 . . . . . . . . . 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 1186 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( ( invr `  F
) `  A )
( .r `  F
) A )  .x.  X )  =  ( ( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) ) )
3127, 6, 28, 15lmodvs1 15978 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 1r `  F
)  .x.  X )  =  X )
3221, 25, 31syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1r `  F )  .x.  X
)  =  X )
3332adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  O )  ->  ( ( 1r `  F )  .x.  X )  =  X )
3419, 30, 333eqtr3d 2476 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( invr `  F ) `  A )  .x.  ( A  .x.  X ) )  =  X )
3534adantlr 696 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  (
( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) )  =  X )
3621adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( A  .x.  X )  =  .0.  )  ->  W  e.  LMod )
3736adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  W  e.  LMod )
3824adantlr 696 . . . . . . . 8  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  (
( invr `  F ) `  A )  e.  K
)
39 lvecmul0or.z . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
406, 28, 12, 39lmodvs0 15984 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  (
( invr `  F ) `  A )  e.  K
)  ->  ( (
( invr `  F ) `  A )  .x.  .0.  )  =  .0.  )
4137, 38, 40syl2anc 643 . . . . . . 7  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  (
( ( invr `  F
) `  A )  .x.  .0.  )  =  .0.  )
423, 35, 413eqtr3d 2476 . . . . . 6  |-  ( ( ( ph  /\  ( A  .x.  X )  =  .0.  )  /\  A  =/=  O )  ->  X  =  .0.  )
4342ex 424 . . . . 5  |-  ( (
ph  /\  ( A  .x.  X )  =  .0.  )  ->  ( A  =/=  O  ->  X  =  .0.  ) )
441, 43syl5bir 210 . . . 4  |-  ( (
ph  /\  ( A  .x.  X )  =  .0.  )  ->  ( -.  A  =  O  ->  X  =  .0.  ) )
4544orrd 368 . . 3  |-  ( (
ph  /\  ( A  .x.  X )  =  .0.  )  ->  ( A  =  O  \/  X  =  .0.  ) )
4645ex 424 . 2  |-  ( ph  ->  ( ( A  .x.  X )  =  .0. 
->  ( A  =  O  \/  X  =  .0.  ) ) )
4727, 6, 28, 13, 39lmod0vs 15983 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
4821, 25, 47syl2anc 643 . . . 4  |-  ( ph  ->  ( O  .x.  X
)  =  .0.  )
49 oveq1 6088 . . . . 5  |-  ( A  =  O  ->  ( A  .x.  X )  =  ( O  .x.  X
) )
5049eqeq1d 2444 . . . 4  |-  ( A  =  O  ->  (
( A  .x.  X
)  =  .0.  <->  ( O  .x.  X )  =  .0.  ) )
5148, 50syl5ibrcom 214 . . 3  |-  ( ph  ->  ( A  =  O  ->  ( A  .x.  X )  =  .0.  ) )
526, 28, 12, 39lmodvs0 15984 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  K )  ->  ( A  .x.  .0.  )  =  .0.  )
5321, 9, 52syl2anc 643 . . . 4  |-  ( ph  ->  ( A  .x.  .0.  )  =  .0.  )
54 oveq2 6089 . . . . 5  |-  ( X  =  .0.  ->  ( A  .x.  X )  =  ( A  .x.  .0.  ) )
5554eqeq1d 2444 . . . 4  |-  ( X  =  .0.  ->  (
( A  .x.  X
)  =  .0.  <->  ( A  .x.  .0.  )  =  .0.  ) )
5653, 55syl5ibrcom 214 . . 3  |-  ( ph  ->  ( X  =  .0. 
->  ( A  .x.  X
)  =  .0.  )
)
5751, 56jaod 370 . 2  |-  ( ph  ->  ( ( A  =  O  \/  X  =  .0.  )  ->  ( A  .x.  X )  =  .0.  ) )
5846, 57impbid 184 1  |-  ( ph  ->  ( ( A  .x.  X )  =  .0.  <->  ( A  =  O  \/  X  =  .0.  )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   Basecbs 13469   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   0gc0g 13723   1rcur 15662   invrcinvr 15776   DivRingcdr 15835   LModclmod 15950   LVecclvec 16174
This theorem is referenced by:  lvecvsn0  16181  lvecvscan  16183  lvecvscan2  16184  lkreqN  29968  lkrlspeqN  29969  hdmap14lem6  32674  hgmapval0  32693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lvec 16175
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