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Theorem nvmul0or 24030
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  =  ( .sOLD `  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 2606 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 oveq2 6097 . . . . . . . 8  |-  ( ( A S B )  =  Z  ->  (
( 1  /  A
) S ( A S B ) )  =  ( ( 1  /  A ) S Z ) )
32ad2antlr 726 . . . . . . 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 10007 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
54oveq1d 6104 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( 1  /  A )  x.  A ) S B )  =  ( 1 S B ) )
653ad2antl2 1151 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( ( 1  /  A )  x.  A
) S B )  =  ( 1 S B ) )
7 simpl1 991 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  U  e.  NrmCVec )
8 reccl 9999 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
983ad2antl2 1151 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
1  /  A )  e.  CC )
10 simpl2 992 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  A  e.  CC )
11 simpl3 993 . . . . . . . . . 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  =  ( .sOLD `  U )
1412, 13nvsass 24006 . . . . . . . . . 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 1220 . . . . . . . . 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 24005 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
17163adant2 1007 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
1 S B )  =  B )
1817adantr 465 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
1 S B )  =  B )
196, 15, 183eqtr3d 2481 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S ( A S B ) )  =  B )
2019adantlr 714 . . . . . . 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 24016 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  (
1  /  A )  e.  CC )  -> 
( ( 1  /  A ) S Z )  =  Z )
238, 22sylan2 474 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( 1  /  A ) S Z )  =  Z )
2423anassrs 648 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC )  /\  A  =/=  0
)  ->  ( (
1  /  A ) S Z )  =  Z )
25243adantl3 1146 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S Z )  =  Z )
2625adantlr 714 . . . . . . 7  |-  ( ( ( ( U  e.  NrmCVec 
/\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S Z )  =  Z )
273, 20, 263eqtr3d 2481 . . . . . 6  |-  ( ( ( ( U  e.  NrmCVec 
/\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  /\  A  =/=  0 )  ->  B  =  Z )
2827ex 434 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( A  =/=  0  ->  B  =  Z ) )
291, 28syl5bir 218 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( -.  A  =  0  ->  B  =  Z ) )
3029orrd 378 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( A  =  0  \/  B  =  Z )
)
3130ex 434 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A S B )  =  Z  -> 
( A  =  0  \/  B  =  Z ) ) )
3212, 13, 21nv0 24015 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
0 S B )  =  Z )
33 oveq1 6096 . . . . . 6  |-  ( A  =  0  ->  ( A S B )  =  ( 0 S B ) )
3433eqeq1d 2449 . . . . 5  |-  ( A  =  0  ->  (
( A S B )  =  Z  <->  ( 0 S B )  =  Z ) )
3532, 34syl5ibrcom 222 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( A  =  0  ->  ( A S B )  =  Z ) )
36353adant2 1007 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A  =  0  ->  ( A S B )  =  Z ) )
3713, 21nvsz 24016 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
38 oveq2 6097 . . . . . 6  |-  ( B  =  Z  ->  ( A S B )  =  ( A S Z ) )
3938eqeq1d 2449 . . . . 5  |-  ( B  =  Z  ->  (
( A S B )  =  Z  <->  ( A S Z )  =  Z ) )
4037, 39syl5ibrcom 222 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( B  =  Z  ->  ( A S B )  =  Z ) )
41403adant3 1008 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( B  =  Z  ->  ( A S B )  =  Z ) )
4236, 41jaod 380 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A  =  0  \/  B  =  Z )  ->  ( A S B )  =  Z ) )
4331, 42impbid 191 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A S B )  =  Z  <->  ( A  =  0  \/  B  =  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   ` cfv 5416  (class class class)co 6089   CCcc 9278   0cc0 9280   1c1 9281    x. cmul 9285    / cdiv 9991   NrmCVeccnv 23960   BaseSetcba 23962   .sOLDcns 23963   0veccn0v 23964
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-grpo 23676  df-gid 23677  df-ginv 23678  df-ablo 23767  df-vc 23922  df-nv 23968  df-va 23971  df-ba 23972  df-sm 23973  df-0v 23974  df-nmcv 23976
This theorem is referenced by:  nmlno0lem  24191
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