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Theorem nvmul0or 21226
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 2461 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 oveq2 5882 . . . . . . . 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 9455 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
54oveq1d 5889 . . . . . . . . . 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 9447 . . . . . . . . . . 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 21202 . . . . . . . . . 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 21201 . . . . . . . . . . 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 2336 . . . . . . . 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 21212 . . . . . . . . . . 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 2336 . . . . . 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 21211 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
0 S B )  =  Z )
33 oveq1 5881 . . . . . 6  |-  ( A  =  0  ->  ( A S B )  =  ( 0 S B ) )
3433eqeq1d 2304 . . . . 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 21212 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC )  ->  ( A S Z )  =  Z )
38 oveq2 5882 . . . . . 6  |-  ( B  =  Z  ->  ( A S B )  =  ( A S Z ) )
3938eqeq1d 2304 . . . . 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 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758    / cdiv 9439   NrmCVeccnv 21156   BaseSetcba 21158   .s
OLDcns 21159   0veccn0v 21160
This theorem is referenced by:  nmlno0lem  21387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172
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