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Theorem lvecvs0or 17201
Description: If a scalar product is zero, one of its factors must be zero. (hvmul0or 24439 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 2620 . . . . 5  |-  ( A  =/=  O  <->  -.  A  =  O )
2 oveq2 6111 . . . . . . . 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 17198 . . . . . . . . . . . 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 2443 . . . . . . . . . . . 12  |-  ( .r
`  F )  =  ( .r `  F
)
15 eqid 2443 . . . . . . . . . . . 12  |-  ( 1r
`  F )  =  ( 1r `  F
)
16 eqid 2443 . . . . . . . . . . . 12  |-  ( invr `  F )  =  (
invr `  F )
1712, 13, 14, 15, 16drnginvrl 16863 . . . . . . . . . . 11  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
O )  ->  (
( ( invr `  F
) `  A )
( .r `  F
) A )  =  ( 1r `  F
) )
188, 10, 11, 17syl3anc 1218 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( invr `  F ) `  A ) ( .r
`  F ) A )  =  ( 1r
`  F ) )
1918oveq1d 6118 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( ( invr `  F
) `  A )
( .r `  F
) A )  .x.  X )  =  ( ( 1r `  F
)  .x.  X )
)
20 lveclmod 17199 . . . . . . . . . . . 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 16861 . . . . . . . . . . 11  |-  ( ( F  e.  DivRing  /\  A  e.  K  /\  A  =/= 
O )  ->  (
( invr `  F ) `  A )  e.  K
)
248, 10, 11, 23syl3anc 1218 . . . . . . . . . 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 16985 . . . . . . . . . 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 1220 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  O )  ->  ( (
( ( invr `  F
) `  A )
( .r `  F
) A )  .x.  X )  =  ( ( ( invr `  F
) `  A )  .x.  ( A  .x.  X
) ) )
3127, 6, 28, 15lmodvs1 16988 . . . . . . . . . . 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 2483 . . . . . . . 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 16994 . . . . . . . 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 2483 . . . . . 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 16993 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( O  .x.  X )  =  .0.  )
4821, 25, 47syl2anc 661 . . . 4  |-  ( ph  ->  ( O  .x.  X
)  =  .0.  )
49 oveq1 6110 . . . . 5  |-  ( A  =  O  ->  ( A  .x.  X )  =  ( O  .x.  X
) )
5049eqeq1d 2451 . . . 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 16994 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  K )  ->  ( A  .x.  .0.  )  =  .0.  )
5321, 9, 52syl2anc 661 . . . 4  |-  ( ph  ->  ( A  .x.  .0.  )  =  .0.  )
54 oveq2 6111 . . . . 5  |-  ( X  =  .0.  ->  ( A  .x.  X )  =  ( A  .x.  .0.  ) )
5554eqeq1d 2451 . . . 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 1369    e. wcel 1756    =/= wne 2618   ` cfv 5430  (class class class)co 6103   Basecbs 14186   .rcmulr 14251  Scalarcsca 14253   .scvsca 14254   0gc0g 14390   1rcur 16615   invrcinvr 16775   DivRingcdr 16844   LModclmod 16960   LVecclvec 17195
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 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-tpos 6757  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-drng 16846  df-lmod 16962  df-lvec 17196
This theorem is referenced by:  lvecvsn0  17202  lvecvscan  17204  lvecvscan2  17205  lkreqN  32827  lkrlspeqN  32828  hdmap14lem6  35533  hgmapval0  35552
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