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Theorem nvmul0or 21210
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 2448 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 oveq2 5866 . . . . . . . 8  |-  ( ( A S B )  =  Z  ->  (
( 1  /  A
) S ( A S B ) )  =  ( ( 1  /  A ) S Z ) )
32ad2antlr 707 . . . . . . 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 9439 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
54oveq1d 5873 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( 1  /  A )  x.  A ) S B )  =  ( 1 S B ) )
653ad2antl2 1118 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( ( 1  /  A )  x.  A
) S B )  =  ( 1 S B ) )
7 simpl1 958 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  U  e.  NrmCVec )
8 reccl 9431 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
983ad2antl2 1118 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
1  /  A )  e.  CC )
10 simpl2 959 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  A  e.  CC )
11 simpl3 960 . . . . . . . . . 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 21186 . . . . . . . . . 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 1184 . . . . . . . . 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 21185 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
1 S B )  =  B )
17163adant2 974 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
1 S B )  =  B )
1817adantr 451 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
1 S B )  =  B )
196, 15, 183eqtr3d 2323 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S ( A S B ) )  =  B )
2019adantlr 695 . . . . . . 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 21196 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  (
1  /  A )  e.  CC )  -> 
( ( 1  /  A ) S Z )  =  Z )
238, 22sylan2 460 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  CC  /\  A  =/=  0 ) )  -> 
( ( 1  /  A ) S Z )  =  Z )
2423anassrs 629 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC )  /\  A  =/=  0
)  ->  ( (
1  /  A ) S Z )  =  Z )
25243adantl3 1113 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S Z )  =  Z )
2625adantlr 695 . . . . . . 7  |-  ( ( ( ( U  e.  NrmCVec 
/\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  /\  A  =/=  0 )  ->  (
( 1  /  A
) S Z )  =  Z )
273, 20, 263eqtr3d 2323 . . . . . 6  |-  ( ( ( ( U  e.  NrmCVec 
/\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  /\  A  =/=  0 )  ->  B  =  Z )
2827ex 423 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( A  =/=  0  ->  B  =  Z ) )
291, 28syl5bir 209 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( -.  A  =  0  ->  B  =  Z ) )
3029orrd 367 . . 3  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  /\  ( A S B )  =  Z )  ->  ( A  =  0  \/  B  =  Z )
)
3130ex 423 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A S B )  =  Z  -> 
( A  =  0  \/  B  =  Z ) ) )
3212, 13, 21nv0 21195 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
0 S B )  =  Z )
33 oveq1 5865 . . . . . 6  |-  ( A  =  0  ->  ( A S B )  =  ( 0 S B ) )
3433eqeq1d 2291 . . . . 5  |-  ( A  =  0  ->  (
( A S B )  =  Z  <->  ( 0 S B )  =  Z ) )
3532, 34syl5ibrcom 213 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  ( A  =  0  ->  ( A S B )  =  Z ) )
36353adant2 974 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A  =  0  ->  ( A S B )  =  Z ) )
3713, 21nvsz 21196 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
38 oveq2 5866 . . . . . 6  |-  ( B  =  Z  ->  ( A S B )  =  ( A S Z ) )
3938eqeq1d 2291 . . . . 5  |-  ( B  =  Z  ->  (
( A S B )  =  Z  <->  ( A S Z )  =  Z ) )
4037, 39syl5ibrcom 213 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( B  =  Z  ->  ( A S B )  =  Z ) )
41403adant3 975 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( B  =  Z  ->  ( A S B )  =  Z ) )
4236, 41jaod 369 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  (
( A  =  0  \/  B  =  Z )  ->  ( A S B )  =  Z ) )
4331, 42impbid 183 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423   NrmCVeccnv 21140   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144
This theorem is referenced by:  nmlno0lem  21371
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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