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Theorem hvsubcan2 22530
Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubcan2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  C
)  =  ( B  -h  C )  <->  A  =  B ) )

Proof of Theorem hvsubcan2
StepHypRef Expression
1 hvsubcl 22473 . . . . 5  |-  ( ( C  e.  ~H  /\  A  e.  ~H )  ->  ( C  -h  A
)  e.  ~H )
213adant3 977 . . . 4  |-  ( ( C  e.  ~H  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( C  -h  A )  e. 
~H )
3 hvsubcl 22473 . . . . 5  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( C  -h  B
)  e.  ~H )
433adant2 976 . . . 4  |-  ( ( C  e.  ~H  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( C  -h  B )  e. 
~H )
5 neg1cn 10023 . . . . . 6  |-  -u 1  e.  CC
6 ax-1cn 9004 . . . . . . 7  |-  1  e.  CC
7 ax-1ne0 9015 . . . . . . 7  |-  1  =/=  0
86, 7negne0i 9331 . . . . . 6  |-  -u 1  =/=  0
95, 8pm3.2i 442 . . . . 5  |-  ( -u
1  e.  CC  /\  -u 1  =/=  0 )
10 hvmulcan 22527 . . . . 5  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  ( C  -h  A )  e. 
~H  /\  ( C  -h  B )  e.  ~H )  ->  ( ( -u
1  .h  ( C  -h  A ) )  =  ( -u 1  .h  ( C  -h  B
) )  <->  ( C  -h  A )  =  ( C  -h  B ) ) )
119, 10mp3an1 1266 . . . 4  |-  ( ( ( C  -h  A
)  e.  ~H  /\  ( C  -h  B
)  e.  ~H )  ->  ( ( -u 1  .h  ( C  -h  A
) )  =  (
-u 1  .h  ( C  -h  B ) )  <-> 
( C  -h  A
)  =  ( C  -h  B ) ) )
122, 4, 11syl2anc 643 . . 3  |-  ( ( C  e.  ~H  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( -u 1  .h  ( C  -h  A ) )  =  ( -u 1  .h  ( C  -h  B
) )  <->  ( C  -h  A )  =  ( C  -h  B ) ) )
13 hvnegdi 22522 . . . . 5  |-  ( ( C  e.  ~H  /\  A  e.  ~H )  ->  ( -u 1  .h  ( C  -h  A
) )  =  ( A  -h  C ) )
14133adant3 977 . . . 4  |-  ( ( C  e.  ~H  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( -u 1  .h  ( C  -h  A ) )  =  ( A  -h  C ) )
15 hvnegdi 22522 . . . . 5  |-  ( ( C  e.  ~H  /\  B  e.  ~H )  ->  ( -u 1  .h  ( C  -h  B
) )  =  ( B  -h  C ) )
16153adant2 976 . . . 4  |-  ( ( C  e.  ~H  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( -u 1  .h  ( C  -h  B ) )  =  ( B  -h  C ) )
1714, 16eqeq12d 2418 . . 3  |-  ( ( C  e.  ~H  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( -u 1  .h  ( C  -h  A ) )  =  ( -u 1  .h  ( C  -h  B
) )  <->  ( A  -h  C )  =  ( B  -h  C ) ) )
18 hvsubcan 22529 . . 3  |-  ( ( C  e.  ~H  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( C  -h  A
)  =  ( C  -h  B )  <->  A  =  B ) )
1912, 17, 183bitr3d 275 . 2  |-  ( ( C  e.  ~H  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( A  -h  C
)  =  ( B  -h  C )  <->  A  =  B ) )
20193coml 1160 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  C
)  =  ( B  -h  C )  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947   -ucneg 9248   ~Hchil 22375    .h csm 22377    -h cmv 22381
This theorem is referenced by:  hvaddsub4  22533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-hvsub 22427
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