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Theorem hvmul0or 22558
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 2607 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 oveq2 6118 . . . . . . . 8  |-  ( ( A  .h  B )  =  0h  ->  (
( 1  /  A
)  .h  ( A  .h  B ) )  =  ( ( 1  /  A )  .h 
0h ) )
32ad2antlr 709 . . . . . . 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 9724 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
54oveq1d 6125 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( 1  /  A )  x.  A )  .h  B
)  =  ( 1  .h  B ) )
65adantlr 697 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
( 1  /  A
)  x.  A )  .h  B )  =  ( 1  .h  B
) )
7 reccl 9716 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
87adantlr 697 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( 1  /  A )  e.  CC )
9 simpll 732 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  A  e.  CC )
10 simplr 733 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  B  e.  ~H )
11 ax-hvmulass 22541 . . . . . . . . . 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 1185 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
( 1  /  A
)  x.  A )  .h  B )  =  ( ( 1  /  A )  .h  ( A  .h  B )
) )
13 ax-hvmulid 22540 . . . . . . . . . 10  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
1413ad2antlr 709 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( 1  .h  B )  =  B )
156, 12, 143eqtr3d 2482 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
1  /  A )  .h  ( A  .h  B ) )  =  B )
1615adantlr 697 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e. 
~H )  /\  ( A  .h  B )  =  0h )  /\  A  =/=  0 )  ->  (
( 1  /  A
)  .h  ( A  .h  B ) )  =  B )
17 hvmul0 22557 . . . . . . . . . 10  |-  ( ( 1  /  A )  e.  CC  ->  (
( 1  /  A
)  .h  0h )  =  0h )
187, 17syl 16 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  .h  0h )  =  0h )
1918adantlr 697 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
1  /  A )  .h  0h )  =  0h )
2019adantlr 697 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e. 
~H )  /\  ( A  .h  B )  =  0h )  /\  A  =/=  0 )  ->  (
( 1  /  A
)  .h  0h )  =  0h )
213, 16, 203eqtr3d 2482 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  B  e. 
~H )  /\  ( A  .h  B )  =  0h )  /\  A  =/=  0 )  ->  B  =  0h )
2221ex 425 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( A  =/=  0  ->  B  =  0h )
)
231, 22syl5bir 211 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( -.  A  =  0  ->  B  =  0h ) )
2423orrd 369 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( A  =  0  \/  B  =  0h ) )
2524ex 425 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  =  0h  ->  ( A  =  0  \/  B  =  0h ) ) )
26 ax-hvmul0 22544 . . . . 5  |-  ( B  e.  ~H  ->  (
0  .h  B )  =  0h )
27 oveq1 6117 . . . . . 6  |-  ( A  =  0  ->  ( A  .h  B )  =  ( 0  .h  B ) )
2827eqeq1d 2450 . . . . 5  |-  ( A  =  0  ->  (
( A  .h  B
)  =  0h  <->  ( 0  .h  B )  =  0h ) )
2926, 28syl5ibrcom 215 . . . 4  |-  ( B  e.  ~H  ->  ( A  =  0  ->  ( A  .h  B )  =  0h ) )
3029adantl 454 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  =  0  ->  ( A  .h  B )  =  0h ) )
31 hvmul0 22557 . . . . 5  |-  ( A  e.  CC  ->  ( A  .h  0h )  =  0h )
32 oveq2 6118 . . . . . 6  |-  ( B  =  0h  ->  ( A  .h  B )  =  ( A  .h  0h ) )
3332eqeq1d 2450 . . . . 5  |-  ( B  =  0h  ->  (
( A  .h  B
)  =  0h  <->  ( A  .h  0h )  =  0h ) )
3431, 33syl5ibrcom 215 . . . 4  |-  ( A  e.  CC  ->  ( B  =  0h  ->  ( A  .h  B )  =  0h ) )
3534adantr 453 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( B  =  0h  ->  ( A  .h  B
)  =  0h )
)
3630, 35jaod 371 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  =  0  \/  B  =  0h )  ->  ( A  .h  B )  =  0h ) )
3725, 36impbid 185 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605  (class class class)co 6110   CCcc 9019   0cc0 9021   1c1 9022    x. cmul 9026    / cdiv 9708   ~Hchil 22453    .h csm 22455   0hc0v 22458
This theorem is referenced by:  hvmulcan  22605  hvmulcan2  22606  nmlnop0iALT  23529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-hv0cl 22537  ax-hvmulid 22540  ax-hvmulass 22541  ax-hvmul0 22544
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709
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