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Theorem nlmmul0or 21484
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 21481 . . . . . 6  |-  ( W  e. NrmMod  ->  F  e. NrmGrp )
323ad2ant1 1018 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  F  e. NrmGrp )
4 simp2 998 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  A  e.  K )
5 nlmmul0or.k . . . . . 6  |-  K  =  ( Base `  F
)
6 eqid 2402 . . . . . 6  |-  ( norm `  F )  =  (
norm `  F )
75, 6nmcl 21427 . . . . 5  |-  ( ( F  e. NrmGrp  /\  A  e.  K )  ->  (
( norm `  F ) `  A )  e.  RR )
83, 4, 7syl2anc 659 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  F ) `  A )  e.  RR )
98recnd 9652 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  F ) `  A )  e.  CC )
10 nlmngp 21478 . . . . . 6  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
11103ad2ant1 1018 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  W  e. NrmGrp )
12 simp3 999 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  B  e.  V )
13 nlmmul0or.v . . . . . 6  |-  V  =  ( Base `  W
)
14 eqid 2402 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
1513, 14nmcl 21427 . . . . 5  |-  ( ( W  e. NrmGrp  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  RR )
1611, 12, 15syl2anc 659 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  RR )
1716recnd 9652 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  CC )
189, 17mul0ord 10240 . 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 21477 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  ( A  .x.  B
) )  =  ( ( ( norm `  F
) `  A )  x.  ( ( norm `  W
) `  B )
) )
2120eqeq1d 2404 . . 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 21479 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
2313, 1, 19, 5lmodvscl 17849 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  K  /\  B  e.  V )  ->  ( A  .x.  B )  e.  V )
2422, 23syl3an1 1263 . . . 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 21429 . . . 4  |-  ( ( W  e. NrmGrp  /\  ( A  .x.  B )  e.  V )  ->  (
( ( norm `  W
) `  ( A  .x.  B ) )  =  0  <->  ( A  .x.  B )  =  .0.  ) )
2711, 24, 26syl2anc 659 . . 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 21429 . . . 4  |-  ( ( F  e. NrmGrp  /\  A  e.  K )  ->  (
( ( norm `  F
) `  A )  =  0  <->  A  =  O ) )
313, 4, 30syl2anc 659 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  F
) `  A )  =  0  <->  A  =  O ) )
3213, 14, 25nmeq0 21429 . . . 4  |-  ( ( W  e. NrmGrp  /\  B  e.  V )  ->  (
( ( norm `  W
) `  B )  =  0  <->  B  =  .0.  ) )
3311, 12, 32syl2anc 659 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  W
) `  B )  =  0  <->  B  =  .0.  ) )
3431, 33orbi12d 708 . 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 366    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   RRcr 9521   0cc0 9522    x. cmul 9527   Basecbs 14841  Scalarcsca 14912   .scvsca 14913   0gc0g 15054   LModclmod 17832   normcnm 21389  NrmGrpcngp 21390  NrmModcnlm 21393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-0g 15056  df-topgen 15058  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-lmod 17834  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-xms 21115  df-ms 21116  df-nm 21395  df-ngp 21396  df-nrg 21398  df-nlm 21399
This theorem is referenced by: (None)
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