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Theorem nvmul0or 22133
Description: If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvmul0or.1  |-  X  =  ( BaseSet `  U )
nvmul0or.4  |-  S  =  ( .s OLD `  U
)
nvmul0or.6  |-  Z  =  ( 0vec `  U
)
Assertion
Ref Expression
nvmul0or  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A S B )  =  Z  <->  ( A  =  0  \/  B  =  Z ) ) )

Proof of Theorem nvmul0or
StepHypRef Expression
1 df-ne 2601 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 oveq2 6089 . . . . . . . 8  |-  ( ( A S B )  =  Z  ->  (
( 1  /  A
) S ( A S B ) )  =  ( ( 1  /  A ) S Z ) )
32ad2antlr 708 . . . . . . 7  |-  ( ( ( ( U  e.  NrmCVec 
/\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S ( A S B ) )  =  ( ( 1  /  A ) S Z ) )
4 recid2 9693 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
54oveq1d 6096 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( 1  /  A )  x.  A ) S B )  =  ( 1 S B ) )
653ad2antl2 1120 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( ( 1  /  A )  x.  A
) S B )  =  ( 1 S B ) )
7 simpl1 960 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  U  e.  NrmCVec )
8 reccl 9685 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
983ad2antl2 1120 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
1  /  A )  e.  CC )
10 simpl2 961 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  A  e.  CC )
11 simpl3 962 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  B  e.  X )
12 nvmul0or.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
13 nvmul0or.4 . . . . . . . . . . 11  |-  S  =  ( .s OLD `  U
)
1412, 13nvsass 22109 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  A
)  e.  CC  /\  A  e.  CC  /\  B  e.  X ) )  -> 
( ( ( 1  /  A )  x.  A ) S B )  =  ( ( 1  /  A ) S ( A S B ) ) )
157, 9, 10, 11, 14syl13anc 1186 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( ( 1  /  A )  x.  A
) S B )  =  ( ( 1  /  A ) S ( A S B ) ) )
1612, 13nvsid 22108 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
17163adant2 976 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
1 S B )  =  B )
1817adantr 452 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
1 S B )  =  B )
196, 15, 183eqtr3d 2476 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S ( A S B ) )  =  B )
2019adantlr 696 . . . . . . 7  |-  ( ( ( ( U  e.  NrmCVec 
/\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S ( A S B ) )  =  B )
21 nvmul0or.6 . . . . . . . . . . . 12  |-  Z  =  ( 0vec `  U
)
2213, 21nvsz 22119 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  (
1  /  A )  e.  CC )  -> 
( ( 1  /  A ) S Z )  =  Z )
238, 22sylan2 461 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( 1  /  A ) S Z )  =  Z )
2423anassrs 630 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC )  /\  A  =/=  0
)  ->  ( (
1  /  A ) S Z )  =  Z )
25243adantl3 1115 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S Z )  =  Z )
2625adantlr 696 . . . . . . 7  |-  ( ( ( ( U  e.  NrmCVec 
/\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S Z )  =  Z )
273, 20, 263eqtr3d 2476 . . . . . 6  |-  ( ( ( ( U  e.  NrmCVec 
/\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  /\  A  =/=  0 )  ->  B  =  Z )
2827ex 424 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( A  =/=  0  ->  B  =  Z ) )
291, 28syl5bir 210 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( -.  A  =  0  ->  B  =  Z ) )
3029orrd 368 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( A  =  0  \/  B  =  Z )
)
3130ex 424 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A S B )  =  Z  -> 
( A  =  0  \/  B  =  Z ) ) )
3212, 13, 21nv0 22118 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
0 S B )  =  Z )
33 oveq1 6088 . . . . . 6  |-  ( A  =  0  ->  ( A S B )  =  ( 0 S B ) )
3433eqeq1d 2444 . . . . 5  |-  ( A  =  0  ->  (
( A S B )  =  Z  <->  ( 0 S B )  =  Z ) )
3532, 34syl5ibrcom 214 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( A  =  0  ->  ( A S B )  =  Z ) )
36353adant2 976 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A  =  0  ->  ( A S B )  =  Z ) )
3713, 21nvsz 22119 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
38 oveq2 6089 . . . . . 6  |-  ( B  =  Z  ->  ( A S B )  =  ( A S Z ) )
3938eqeq1d 2444 . . . . 5  |-  ( B  =  Z  ->  (
( A S B )  =  Z  <->  ( A S Z )  =  Z ) )
4037, 39syl5ibrcom 214 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( B  =  Z  ->  ( A S B )  =  Z ) )
41403adant3 977 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( B  =  Z  ->  ( A S B )  =  Z ) )
4236, 41jaod 370 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A  =  0  \/  B  =  Z )  ->  ( A S B )  =  Z ) )
4331, 42impbid 184 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A S B )  =  Z  <->  ( A  =  0  \/  B  =  Z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    x. cmul 8995    / cdiv 9677   NrmCVeccnv 22063   BaseSetcba 22065   .s
OLDcns 22066   0veccn0v 22067
This theorem is referenced by:  nmlno0lem  22294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-grpo 21779  df-gid 21780  df-ginv 21781  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079
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