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Theorem hvmul0or 22038
Description: If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvmul0or  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  =  0h  <->  ( A  =  0  \/  B  =  0h )
) )

Proof of Theorem hvmul0or
StepHypRef Expression
1 df-ne 2531 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 oveq2 5989 . . . . . . . 8  |-  ( ( A  .h  B )  =  0h  ->  (
( 1  /  A
)  .h  ( A  .h  B ) )  =  ( ( 1  /  A )  .h 
0h ) )
32ad2antlr 707 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e. 
~H )  /\  ( A  .h  B )  =  0h )  /\  A  =/=  0 )  ->  (
( 1  /  A
)  .h  ( A  .h  B ) )  =  ( ( 1  /  A )  .h 
0h ) )
4 recid2 9586 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
54oveq1d 5996 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( 1  /  A )  x.  A )  .h  B
)  =  ( 1  .h  B ) )
65adantlr 695 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
( 1  /  A
)  x.  A )  .h  B )  =  ( 1  .h  B
) )
7 reccl 9578 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
87adantlr 695 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( 1  /  A )  e.  CC )
9 simpll 730 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  A  e.  CC )
10 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  B  e.  ~H )
11 ax-hvmulass 22021 . . . . . . . . . 10  |-  ( ( ( 1  /  A
)  e.  CC  /\  A  e.  CC  /\  B  e.  ~H )  ->  (
( ( 1  /  A )  x.  A
)  .h  B )  =  ( ( 1  /  A )  .h  ( A  .h  B
) ) )
128, 9, 10, 11syl3anc 1183 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
( 1  /  A
)  x.  A )  .h  B )  =  ( ( 1  /  A )  .h  ( A  .h  B )
) )
13 ax-hvmulid 22020 . . . . . . . . . 10  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
1413ad2antlr 707 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( 1  .h  B )  =  B )
156, 12, 143eqtr3d 2406 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
1  /  A )  .h  ( A  .h  B ) )  =  B )
1615adantlr 695 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e. 
~H )  /\  ( A  .h  B )  =  0h )  /\  A  =/=  0 )  ->  (
( 1  /  A
)  .h  ( A  .h  B ) )  =  B )
17 hvmul0 22037 . . . . . . . . . 10  |-  ( ( 1  /  A )  e.  CC  ->  (
( 1  /  A
)  .h  0h )  =  0h )
187, 17syl 15 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  .h  0h )  =  0h )
1918adantlr 695 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
1  /  A )  .h  0h )  =  0h )
2019adantlr 695 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e. 
~H )  /\  ( A  .h  B )  =  0h )  /\  A  =/=  0 )  ->  (
( 1  /  A
)  .h  0h )  =  0h )
213, 16, 203eqtr3d 2406 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e. 
~H )  /\  ( A  .h  B )  =  0h )  /\  A  =/=  0 )  ->  B  =  0h )
2221ex 423 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( A  =/=  0  ->  B  =  0h )
)
231, 22syl5bir 209 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( -.  A  =  0  ->  B  =  0h ) )
2423orrd 367 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( A  =  0  \/  B  =  0h ) )
2524ex 423 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  =  0h  ->  ( A  =  0  \/  B  =  0h ) ) )
26 ax-hvmul0 22024 . . . . 5  |-  ( B  e.  ~H  ->  (
0  .h  B )  =  0h )
27 oveq1 5988 . . . . . 6  |-  ( A  =  0  ->  ( A  .h  B )  =  ( 0  .h  B ) )
2827eqeq1d 2374 . . . . 5  |-  ( A  =  0  ->  (
( A  .h  B
)  =  0h  <->  ( 0  .h  B )  =  0h ) )
2926, 28syl5ibrcom 213 . . . 4  |-  ( B  e.  ~H  ->  ( A  =  0  ->  ( A  .h  B )  =  0h ) )
3029adantl 452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  =  0  ->  ( A  .h  B )  =  0h ) )
31 hvmul0 22037 . . . . 5  |-  ( A  e.  CC  ->  ( A  .h  0h )  =  0h )
32 oveq2 5989 . . . . . 6  |-  ( B  =  0h  ->  ( A  .h  B )  =  ( A  .h  0h ) )
3332eqeq1d 2374 . . . . 5  |-  ( B  =  0h  ->  (
( A  .h  B
)  =  0h  <->  ( A  .h  0h )  =  0h ) )
3431, 33syl5ibrcom 213 . . . 4  |-  ( A  e.  CC  ->  ( B  =  0h  ->  ( A  .h  B )  =  0h ) )
3534adantr 451 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( B  =  0h  ->  ( A  .h  B
)  =  0h )
)
3630, 35jaod 369 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  =  0  \/  B  =  0h )  ->  ( A  .h  B )  =  0h ) )
3725, 36impbid 183 1  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  =  0h  <->  ( A  =  0  \/  B  =  0h )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529  (class class class)co 5981   CCcc 8882   0cc0 8884   1c1 8885    x. cmul 8889    / cdiv 9570   ~Hchil 21933    .h csm 21935   0hc0v 21938
This theorem is referenced by:  hvmulcan  22085  hvmulcan2  22086  nmlnop0iALT  23009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-hv0cl 22017  ax-hvmulid 22020  ax-hvmulass 22021  ax-hvmul0 22024
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571
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