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Theorem nlmmul0or 20269
Description: If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
nlmmul0or.v  |-  V  =  ( Base `  W
)
nlmmul0or.s  |-  .x.  =  ( .s `  W )
nlmmul0or.z  |-  .0.  =  ( 0g `  W )
nlmmul0or.f  |-  F  =  (Scalar `  W )
nlmmul0or.k  |-  K  =  ( Base `  F
)
nlmmul0or.o  |-  O  =  ( 0g `  F
)
Assertion
Ref Expression
nlmmul0or  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( A  .x.  B
)  =  .0.  <->  ( A  =  O  \/  B  =  .0.  ) ) )

Proof of Theorem nlmmul0or
StepHypRef Expression
1 nlmmul0or.f . . . . . . 7  |-  F  =  (Scalar `  W )
21nlmngp2 20266 . . . . . 6  |-  ( W  e. NrmMod  ->  F  e. NrmGrp )
323ad2ant1 1009 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  F  e. NrmGrp )
4 simp2 989 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  A  e.  K )
5 nlmmul0or.k . . . . . 6  |-  K  =  ( Base `  F
)
6 eqid 2443 . . . . . 6  |-  ( norm `  F )  =  (
norm `  F )
75, 6nmcl 20212 . . . . 5  |-  ( ( F  e. NrmGrp  /\  A  e.  K )  ->  (
( norm `  F ) `  A )  e.  RR )
83, 4, 7syl2anc 661 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  F ) `  A )  e.  RR )
98recnd 9417 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  F ) `  A )  e.  CC )
10 nlmngp 20263 . . . . . 6  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
11103ad2ant1 1009 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  W  e. NrmGrp )
12 simp3 990 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  B  e.  V )
13 nlmmul0or.v . . . . . 6  |-  V  =  ( Base `  W
)
14 eqid 2443 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
1513, 14nmcl 20212 . . . . 5  |-  ( ( W  e. NrmGrp  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  RR )
1611, 12, 15syl2anc 661 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  RR )
1716recnd 9417 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  CC )
189, 17mul0ord 9991 . 2  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( ( norm `  F ) `  A
)  x.  ( (
norm `  W ) `  B ) )  =  0  <->  ( ( (
norm `  F ) `  A )  =  0  \/  ( ( norm `  W ) `  B
)  =  0 ) ) )
19 nlmmul0or.s . . . . 5  |-  .x.  =  ( .s `  W )
2013, 14, 19, 1, 5, 6nmvs 20262 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  ( A  .x.  B
) )  =  ( ( ( norm `  F
) `  A )  x.  ( ( norm `  W
) `  B )
) )
2120eqeq1d 2451 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  W
) `  ( A  .x.  B ) )  =  0  <->  ( ( (
norm `  F ) `  A )  x.  (
( norm `  W ) `  B ) )  =  0 ) )
22 nlmlmod 20264 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
2313, 1, 19, 5lmodvscl 16970 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  K  /\  B  e.  V )  ->  ( A  .x.  B )  e.  V )
2422, 23syl3an1 1251 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  ( A  .x.  B )  e.  V )
25 nlmmul0or.z . . . . 5  |-  .0.  =  ( 0g `  W )
2613, 14, 25nmeq0 20214 . . . 4  |-  ( ( W  e. NrmGrp  /\  ( A  .x.  B )  e.  V )  ->  (
( ( norm `  W
) `  ( A  .x.  B ) )  =  0  <->  ( A  .x.  B )  =  .0.  ) )
2711, 24, 26syl2anc 661 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  W
) `  ( A  .x.  B ) )  =  0  <->  ( A  .x.  B )  =  .0.  ) )
2821, 27bitr3d 255 . 2  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( ( norm `  F ) `  A
)  x.  ( (
norm `  W ) `  B ) )  =  0  <->  ( A  .x.  B )  =  .0.  ) )
29 nlmmul0or.o . . . . 5  |-  O  =  ( 0g `  F
)
305, 6, 29nmeq0 20214 . . . 4  |-  ( ( F  e. NrmGrp  /\  A  e.  K )  ->  (
( ( norm `  F
) `  A )  =  0  <->  A  =  O ) )
313, 4, 30syl2anc 661 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  F
) `  A )  =  0  <->  A  =  O ) )
3213, 14, 25nmeq0 20214 . . . 4  |-  ( ( W  e. NrmGrp  /\  B  e.  V )  ->  (
( ( norm `  W
) `  B )  =  0  <->  B  =  .0.  ) )
3311, 12, 32syl2anc 661 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  W
) `  B )  =  0  <->  B  =  .0.  ) )
3431, 33orbi12d 709 . 2  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( ( norm `  F ) `  A
)  =  0  \/  ( ( norm `  W
) `  B )  =  0 )  <->  ( A  =  O  \/  B  =  .0.  ) ) )
3518, 28, 343bitr3d 283 1  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( A  .x.  B
)  =  .0.  <->  ( A  =  O  \/  B  =  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   RRcr 9286   0cc0 9287    x. cmul 9292   Basecbs 14179  Scalarcsca 14246   .scvsca 14247   0gc0g 14383   LModclmod 16953   normcnm 20174  NrmGrpcngp 20175  NrmModcnlm 20178
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-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-0g 14385  df-topgen 14387  df-mnd 15420  df-grp 15550  df-lmod 16955  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-xms 19900  df-ms 19901  df-nm 20180  df-ngp 20181  df-nrg 20183  df-nlm 20184
This theorem is referenced by: (None)
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