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Theorem hvmulcan2 22567
Description: Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvmulcan2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h ) )  -> 
( ( A  .h  C )  =  ( B  .h  C )  <-> 
A  =  B ) )

Proof of Theorem hvmulcan2
StepHypRef Expression
1 hvmulcl 22508 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
213adant2 976 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C )  e.  ~H )
3 hvmulcl 22508 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  ~H )  ->  ( B  .h  C
)  e.  ~H )
433adant1 975 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( B  .h  C )  e.  ~H )
5 hvsubeq0 22562 . . . 4  |-  ( ( ( A  .h  C
)  e.  ~H  /\  ( B  .h  C
)  e.  ~H )  ->  ( ( ( A  .h  C )  -h  ( B  .h  C
) )  =  0h  <->  ( A  .h  C )  =  ( B  .h  C ) ) )
62, 4, 5syl2anc 643 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( ( A  .h  C )  -h  ( B  .h  C )
)  =  0h  <->  ( A  .h  C )  =  ( B  .h  C ) ) )
763adant3r 1181 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h ) )  -> 
( ( ( A  .h  C )  -h  ( B  .h  C
) )  =  0h  <->  ( A  .h  C )  =  ( B  .h  C ) ) )
8 hvsubdistr2 22544 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  -  B
)  .h  C )  =  ( ( A  .h  C )  -h  ( B  .h  C
) ) )
98eqeq1d 2443 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( ( A  -  B )  .h  C
)  =  0h  <->  ( ( A  .h  C )  -h  ( B  .h  C
) )  =  0h ) )
10 subcl 9297 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
11 hvmul0or 22519 . . . . . . 7  |-  ( ( ( A  -  B
)  e.  CC  /\  C  e.  ~H )  ->  ( ( ( A  -  B )  .h  C )  =  0h  <->  ( ( A  -  B
)  =  0  \/  C  =  0h )
) )
1210, 11sylan 458 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  C  e.  ~H )  ->  ( ( ( A  -  B )  .h  C )  =  0h  <->  ( ( A  -  B )  =  0  \/  C  =  0h ) ) )
13123impa 1148 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( ( A  -  B )  .h  C
)  =  0h  <->  ( ( A  -  B )  =  0  \/  C  =  0h ) ) )
149, 13bitr3d 247 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( ( A  .h  C )  -h  ( B  .h  C )
)  =  0h  <->  ( ( A  -  B )  =  0  \/  C  =  0h ) ) )
15143adant3r 1181 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h ) )  -> 
( ( ( A  .h  C )  -h  ( B  .h  C
) )  =  0h  <->  ( ( A  -  B
)  =  0  \/  C  =  0h )
) )
16 df-ne 2600 . . . . . 6  |-  ( C  =/=  0h  <->  -.  C  =  0h )
17 biorf 395 . . . . . . 7  |-  ( -.  C  =  0h  ->  ( ( A  -  B
)  =  0  <->  ( C  =  0h  \/  ( A  -  B
)  =  0 ) ) )
18 orcom 377 . . . . . . 7  |-  ( ( C  =  0h  \/  ( A  -  B
)  =  0 )  <-> 
( ( A  -  B )  =  0  \/  C  =  0h ) )
1917, 18syl6bb 253 . . . . . 6  |-  ( -.  C  =  0h  ->  ( ( A  -  B
)  =  0  <->  (
( A  -  B
)  =  0  \/  C  =  0h )
) )
2016, 19sylbi 188 . . . . 5  |-  ( C  =/=  0h  ->  (
( A  -  B
)  =  0  <->  (
( A  -  B
)  =  0  \/  C  =  0h )
) )
2120ad2antll 710 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h )
)  ->  ( ( A  -  B )  =  0  <->  ( ( A  -  B )  =  0  \/  C  =  0h ) ) )
22213adant1 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h ) )  -> 
( ( A  -  B )  =  0  <-> 
( ( A  -  B )  =  0  \/  C  =  0h ) ) )
23 subeq0 9319 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  0  <-> 
A  =  B ) )
24233adant3 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h ) )  -> 
( ( A  -  B )  =  0  <-> 
A  =  B ) )
2515, 22, 243bitr2d 273 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h ) )  -> 
( ( ( A  .h  C )  -h  ( B  .h  C
) )  =  0h  <->  A  =  B ) )
267, 25bitr3d 247 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h ) )  -> 
( ( A  .h  C )  =  ( B  .h  C )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   0cc0 8982    - cmin 9283   ~Hchil 22414    .h csm 22416   0hc0v 22419    -h cmv 22420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr2 22504  ax-hvmul0 22505
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-hvsub 22466
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