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Theorem hvmul0or 24426
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 6098 . . . . . . . 8  |-  ( ( A  .h  B )  =  0h  ->  (
( 1  /  A
)  .h  ( A  .h  B ) )  =  ( ( 1  /  A )  .h 
0h ) )
32ad2antlr 726 . . . . . . 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 10008 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
54oveq1d 6105 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( ( 1  /  A )  x.  A )  .h  B
)  =  ( 1  .h  B ) )
65adantlr 714 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
( 1  /  A
)  x.  A )  .h  B )  =  ( 1  .h  B
) )
7 reccl 10000 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
87adantlr 714 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( 1  /  A )  e.  CC )
9 simpll 753 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  A  e.  CC )
10 simplr 754 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  B  e.  ~H )
11 ax-hvmulass 24408 . . . . . . . . . 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 1218 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
( 1  /  A
)  x.  A )  .h  B )  =  ( ( 1  /  A )  .h  ( A  .h  B )
) )
13 ax-hvmulid 24407 . . . . . . . . . 10  |-  ( B  e.  ~H  ->  (
1  .h  B )  =  B )
1413ad2antlr 726 . . . . . . . . 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 714 . . . . . . 7  |-  ( ( ( ( A  e.  CC  /\  B  e. 
~H )  /\  ( A  .h  B )  =  0h )  /\  A  =/=  0 )  ->  (
( 1  /  A
)  .h  ( A  .h  B ) )  =  B )
17 hvmul0 24425 . . . . . . . . . 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 714 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
1  /  A )  .h  0h )  =  0h )
2019adantlr 714 . . . . . . 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 434 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( A  =/=  0  ->  B  =  0h )
)
231, 22syl5bir 218 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( -.  A  =  0  ->  B  =  0h ) )
2423orrd 378 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( A  .h  B
)  =  0h )  ->  ( A  =  0  \/  B  =  0h ) )
2524ex 434 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  =  0h  ->  ( A  =  0  \/  B  =  0h ) ) )
26 ax-hvmul0 24411 . . . . 5  |-  ( B  e.  ~H  ->  (
0  .h  B )  =  0h )
27 oveq1 6097 . . . . . 6  |-  ( A  =  0  ->  ( A  .h  B )  =  ( 0  .h  B ) )
2827eqeq1d 2450 . . . . 5  |-  ( A  =  0  ->  (
( A  .h  B
)  =  0h  <->  ( 0  .h  B )  =  0h ) )
2926, 28syl5ibrcom 222 . . . 4  |-  ( B  e.  ~H  ->  ( A  =  0  ->  ( A  .h  B )  =  0h ) )
3029adantl 466 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  =  0  ->  ( A  .h  B )  =  0h ) )
31 hvmul0 24425 . . . . 5  |-  ( A  e.  CC  ->  ( A  .h  0h )  =  0h )
32 oveq2 6098 . . . . . 6  |-  ( B  =  0h  ->  ( A  .h  B )  =  ( A  .h  0h ) )
3332eqeq1d 2450 . . . . 5  |-  ( B  =  0h  ->  (
( A  .h  B
)  =  0h  <->  ( A  .h  0h )  =  0h ) )
3431, 33syl5ibrcom 222 . . . 4  |-  ( A  e.  CC  ->  ( B  =  0h  ->  ( A  .h  B )  =  0h ) )
3534adantr 465 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( B  =  0h  ->  ( A  .h  B
)  =  0h )
)
3630, 35jaod 380 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  =  0  \/  B  =  0h )  ->  ( A  .h  B )  =  0h ) )
3725, 36impbid 191 1  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( A  .h  B )  =  0h  <->  ( A  =  0  \/  B  =  0h )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605  (class class class)co 6090   CCcc 9279   0cc0 9281   1c1 9282    x. cmul 9286    / cdiv 9992   ~Hchil 24320    .h csm 24322   0hc0v 24325
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 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-hv0cl 24404  ax-hvmulid 24407  ax-hvmulass 24408  ax-hvmul0 24411
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993
This theorem is referenced by:  hvmulcan  24473  hvmulcan2  24474  nmlnop0iALT  25398
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