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Theorem hvmul0or 26068
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 2654 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 oveq2 6304 . . . . . . . 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 10243 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  A )  x.  A
)  =  1 )
54oveq1d 6311 . . . . . . . . . 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 10235 . . . . . . . . . . 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 755 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  B  e.  ~H )
11 ax-hvmulass 26050 . . . . . . . . . 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 1228 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  A  =/=  0
)  ->  ( (
( 1  /  A
)  x.  A )  .h  B )  =  ( ( 1  /  A )  .h  ( A  .h  B )
) )
13 ax-hvmulid 26049 . . . . . . . . . 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 2506 . . . . . . . 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 26067 . . . . . . . . . 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 2506 . . . . . 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 26053 . . . . 5  |-  ( B  e.  ~H  ->  (
0  .h  B )  =  0h )
27 oveq1 6303 . . . . . 6  |-  ( A  =  0  ->  ( A  .h  B )  =  ( 0  .h  B ) )
2827eqeq1d 2459 . . . . 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 26067 . . . . 5  |-  ( A  e.  CC  ->  ( A  .h  0h )  =  0h )
32 oveq2 6304 . . . . . 6  |-  ( B  =  0h  ->  ( A  .h  B )  =  ( A  .h  0h ) )
3332eqeq1d 2459 . . . . 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 1395    e. wcel 1819    =/= wne 2652  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    x. cmul 9514    / cdiv 10227   ~Hchil 25962    .h csm 25964   0hc0v 25967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-hv0cl 26046  ax-hvmulid 26049  ax-hvmulass 26050  ax-hvmul0 26053
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228
This theorem is referenced by:  hvmulcan  26115  hvmulcan2  26116  nmlnop0iALT  27040
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