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Theorem hvsubcan 22564
Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubcan  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  =  ( A  -h  C )  <->  B  =  C ) )

Proof of Theorem hvsubcan
StepHypRef Expression
1 hvsubval 22507 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
213adant3 977 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) ) )
3 hvsubval 22507 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  =  ( A  +h  ( -u 1  .h  C ) ) )
433adant2 976 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C )  =  ( A  +h  ( -u 1  .h  C ) ) )
52, 4eqeq12d 2449 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  =  ( A  -h  C )  <->  ( A  +h  ( -u 1  .h  B ) )  =  ( A  +h  ( -u 1  .h  C ) ) ) )
6 neg1cn 10056 . . . 4  |-  -u 1  e.  CC
7 hvmulcl 22504 . . . 4  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
86, 7mpan 652 . . 3  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
9 hvmulcl 22504 . . . . 5  |-  ( (
-u 1  e.  CC  /\  C  e.  ~H )  ->  ( -u 1  .h  C )  e.  ~H )
106, 9mpan 652 . . . 4  |-  ( C  e.  ~H  ->  ( -u 1  .h  C )  e.  ~H )
11 hvaddcan 22560 . . . 4  |-  ( ( A  e.  ~H  /\  ( -u 1  .h  B
)  e.  ~H  /\  ( -u 1  .h  C
)  e.  ~H )  ->  ( ( A  +h  ( -u 1  .h  B
) )  =  ( A  +h  ( -u
1  .h  C ) )  <->  ( -u 1  .h  B )  =  (
-u 1  .h  C
) ) )
1210, 11syl3an3 1219 . . 3  |-  ( ( A  e.  ~H  /\  ( -u 1  .h  B
)  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  ( -u 1  .h  B
) )  =  ( A  +h  ( -u
1  .h  C ) )  <->  ( -u 1  .h  B )  =  (
-u 1  .h  C
) ) )
138, 12syl3an2 1218 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  ( -u 1  .h  B ) )  =  ( A  +h  ( -u 1  .h  C ) )  <->  ( -u 1  .h  B )  =  (
-u 1  .h  C
) ) )
14 ax-1cn 9037 . . . . . 6  |-  1  e.  CC
15 ax-1ne0 9048 . . . . . 6  |-  1  =/=  0
1614, 15negne0i 9364 . . . . 5  |-  -u 1  =/=  0
176, 16pm3.2i 442 . . . 4  |-  ( -u
1  e.  CC  /\  -u 1  =/=  0 )
18 hvmulcan 22562 . . . 4  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  B  e.  ~H  /\  C  e. 
~H )  ->  (
( -u 1  .h  B
)  =  ( -u
1  .h  C )  <-> 
B  =  C ) )
1917, 18mp3an1 1266 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  =  (
-u 1  .h  C
)  <->  B  =  C
) )
20193adant1 975 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( -u 1  .h  B
)  =  ( -u
1  .h  C )  <-> 
B  =  C ) )
215, 13, 203bitrd 271 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  =  ( A  -h  C )  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6072   CCcc 8977   0cc0 8979   1c1 8980   -ucneg 9281   ~Hchil 22410    +h cva 22411    .h csm 22412    -h cmv 22416
This theorem is referenced by:  hvsubcan2  22565
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 4692  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-hfvadd 22491  ax-hvcom 22492  ax-hvass 22493  ax-hv0cl 22494  ax-hvaddid 22495  ax-hfvmul 22496  ax-hvmulid 22497  ax-hvmulass 22498  ax-hvdistr1 22499  ax-hvdistr2 22500  ax-hvmul0 22501
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-hvsub 22462
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