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Theorem nlmmul0or 18194
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 18191 . . . . . 6  |-  ( W  e. NrmMod  ->  F  e. NrmGrp )
323ad2ant1 976 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  F  e. NrmGrp )
4 simp2 956 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  A  e.  K )
5 nlmmul0or.k . . . . . 6  |-  K  =  ( Base `  F
)
6 eqid 2283 . . . . . 6  |-  ( norm `  F )  =  (
norm `  F )
75, 6nmcl 18137 . . . . 5  |-  ( ( F  e. NrmGrp  /\  A  e.  K )  ->  (
( norm `  F ) `  A )  e.  RR )
83, 4, 7syl2anc 642 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  F ) `  A )  e.  RR )
98recnd 8861 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  F ) `  A )  e.  CC )
10 nlmngp 18188 . . . . . 6  |-  ( W  e. NrmMod  ->  W  e. NrmGrp )
11103ad2ant1 976 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  W  e. NrmGrp )
12 simp3 957 . . . . 5  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  B  e.  V )
13 nlmmul0or.v . . . . . 6  |-  V  =  ( Base `  W
)
14 eqid 2283 . . . . . 6  |-  ( norm `  W )  =  (
norm `  W )
1513, 14nmcl 18137 . . . . 5  |-  ( ( W  e. NrmGrp  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  RR )
1611, 12, 15syl2anc 642 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  RR )
1716recnd 8861 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  B )  e.  CC )
189, 17mul0ord 9418 . 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 18187 . . . 4  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( norm `  W ) `  ( A  .x.  B
) )  =  ( ( ( norm `  F
) `  A )  x.  ( ( norm `  W
) `  B )
) )
2120eqeq1d 2291 . . 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 18189 . . . . 5  |-  ( W  e. NrmMod  ->  W  e.  LMod )
2313, 1, 19, 5lmodvscl 15644 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  K  /\  B  e.  V )  ->  ( A  .x.  B )  e.  V )
2422, 23syl3an1 1215 . . . 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 18139 . . . 4  |-  ( ( W  e. NrmGrp  /\  ( A  .x.  B )  e.  V )  ->  (
( ( norm `  W
) `  ( A  .x.  B ) )  =  0  <->  ( A  .x.  B )  =  .0.  ) )
2711, 24, 26syl2anc 642 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  W
) `  ( A  .x.  B ) )  =  0  <->  ( A  .x.  B )  =  .0.  ) )
2821, 27bitr3d 246 . 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 18139 . . . 4  |-  ( ( F  e. NrmGrp  /\  A  e.  K )  ->  (
( ( norm `  F
) `  A )  =  0  <->  A  =  O ) )
313, 4, 30syl2anc 642 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  F
) `  A )  =  0  <->  A  =  O ) )
3213, 14, 25nmeq0 18139 . . . 4  |-  ( ( W  e. NrmGrp  /\  B  e.  V )  ->  (
( ( norm `  W
) `  B )  =  0  <->  B  =  .0.  ) )
3311, 12, 32syl2anc 642 . . 3  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( ( norm `  W
) `  B )  =  0  <->  B  =  .0.  ) )
3431, 33orbi12d 690 . 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 274 1  |-  ( ( W  e. NrmMod  /\  A  e.  K  /\  B  e.  V )  ->  (
( A  .x.  B
)  =  .0.  <->  ( A  =  O  \/  B  =  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    x. cmul 8742   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LModclmod 15627   normcnm 18099  NrmGrpcngp 18100  NrmModcnlm 18103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-topgen 13344  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106  df-nrg 18108  df-nlm 18109
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