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Theorem hvmulcan 22576
Description: Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvmulcan  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  =  ( A  .h  C )  <-> 
B  =  C ) )

Proof of Theorem hvmulcan
StepHypRef Expression
1 df-ne 2603 . . . . 5  |-  ( A  =/=  0  <->  -.  A  =  0 )
2 biorf 396 . . . . 5  |-  ( -.  A  =  0  -> 
( ( B  -h  C )  =  0h  <->  ( A  =  0  \/  ( B  -h  C
)  =  0h )
) )
31, 2sylbi 189 . . . 4  |-  ( A  =/=  0  ->  (
( B  -h  C
)  =  0h  <->  ( A  =  0  \/  ( B  -h  C )  =  0h ) ) )
43ad2antlr 709 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  B  e.  ~H )  ->  ( ( B  -h  C )  =  0h  <->  ( A  =  0  \/  ( B  -h  C )  =  0h ) ) )
543adant3 978 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( B  -h  C )  =  0h  <->  ( A  =  0  \/  ( B  -h  C
)  =  0h )
) )
6 hvsubeq0 22572 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( B  -h  C )  =  0h  <->  B  =  C ) )
763adant1 976 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( B  -h  C )  =  0h  <->  B  =  C ) )
8 hvsubdistr1 22553 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  -h  C ) )  =  ( ( A  .h  B )  -h  ( A  .h  C )
) )
98eqeq1d 2446 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  ( B  -h  C ) )  =  0h  <->  ( ( A  .h  B )  -h  ( A  .h  C
) )  =  0h ) )
10 hvsubcl 22522 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  -h  C
)  e.  ~H )
11 hvmul0or 22529 . . . . . 6  |-  ( ( A  e.  CC  /\  ( B  -h  C
)  e.  ~H )  ->  ( ( A  .h  ( B  -h  C
) )  =  0h  <->  ( A  =  0  \/  ( B  -h  C
)  =  0h )
) )
1210, 11sylan2 462 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( ( A  .h  ( B  -h  C ) )  =  0h  <->  ( A  =  0  \/  ( B  -h  C )  =  0h ) ) )
13123impb 1150 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  ( B  -h  C ) )  =  0h  <->  ( A  =  0  \/  ( B  -h  C )  =  0h ) ) )
14 hvmulcl 22518 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
15143adant3 978 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  B )  e.  ~H )
16 hvmulcl 22518 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
17163adant2 977 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  C )  e.  ~H )
18 hvsubeq0 22572 . . . . 5  |-  ( ( ( A  .h  B
)  e.  ~H  /\  ( A  .h  C
)  e.  ~H )  ->  ( ( ( A  .h  B )  -h  ( A  .h  C
) )  =  0h  <->  ( A  .h  B )  =  ( A  .h  C ) ) )
1915, 17, 18syl2anc 644 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( A  .h  B )  -h  ( A  .h  C )
)  =  0h  <->  ( A  .h  B )  =  ( A  .h  C ) ) )
209, 13, 193bitr3d 276 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  =  0  \/  ( B  -h  C )  =  0h ) 
<->  ( A  .h  B
)  =  ( A  .h  C ) ) )
21203adant1r 1178 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  =  0  \/  ( B  -h  C )  =  0h )  <->  ( A  .h  B )  =  ( A  .h  C ) ) )
225, 7, 213bitr3rd 277 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  =  ( A  .h  C )  <-> 
B  =  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601  (class class class)co 6083   CCcc 8990   0cc0 8992   ~Hchil 22424    .h csm 22426   0hc0v 22429    -h cmv 22430
This theorem is referenced by:  hvsubcan  22578  hvsubcan2  22579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-hfvadd 22505  ax-hvcom 22506  ax-hvass 22507  ax-hv0cl 22508  ax-hvaddid 22509  ax-hfvmul 22510  ax-hvmulid 22511  ax-hvmulass 22512  ax-hvdistr1 22513  ax-hvdistr2 22514  ax-hvmul0 22515
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-hvsub 22476
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