MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nlmmul0or Structured version   Unicode version

Theorem nlmmul0or 20924
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 20921 . . . . . 6  |-  ( W  e. NrmMod  ->  F  e. NrmGrp )
323ad2ant1 1017 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  F  e. NrmGrp )
4 simp2 997 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  A  e.  K )
5 nlmmul0or.k . . . . . 6  |-  K  =  ( Base `  F
)
6 eqid 2467 . . . . . 6  |-  ( norm `  F )  =  (
norm `  F )
75, 6nmcl 20867 . . . . 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 9618 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  F ) `  A )  e.  CC )
10 nlmngp 20918 . . . . . 6  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
11103ad2ant1 1017 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  W  e. NrmGrp )
12 simp3 998 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  B  e.  V )
13 nlmmul0or.v . . . . . 6  |-  V  =  ( Base `  W
)
14 eqid 2467 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
1513, 14nmcl 20867 . . . . 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 9618 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  CC )
189, 17mul0ord 10195 . 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 20917 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  ( A  .x.  B
) )  =  ( ( ( norm `  F
) `  A )  x.  ( ( norm `  W
) `  B )
) )
2120eqeq1d 2469 . . 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 20919 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
2313, 1, 19, 5lmodvscl 17309 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  K  /\  B  e.  V )  ->  ( A  .x.  B )  e.  V )
2422, 23syl3an1 1261 . . . 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 20869 . . . 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 20869 . . . 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 20869 . . . 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 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488    x. cmul 9493   Basecbs 14483  Scalarcsca 14551   .scvsca 14552   0gc0g 14688   LModclmod 17292   normcnm 20829  NrmGrpcngp 20830  NrmModcnlm 20833
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-0g 14690  df-topgen 14692  df-mnd 15725  df-grp 15855  df-lmod 17294  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-xms 20555  df-ms 20556  df-nm 20835  df-ngp 20836  df-nrg 20838  df-nlm 20839
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator