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Theorem hvaddcani 21957
Description: Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvnegdi.1  |-  A  e. 
~H
hvnegdi.2  |-  B  e. 
~H
hvaddcan.3  |-  C  e. 
~H
Assertion
Ref Expression
hvaddcani  |-  ( ( A  +h  B )  =  ( A  +h  C )  <->  B  =  C )

Proof of Theorem hvaddcani
StepHypRef Expression
1 oveq1 5988 . . 3  |-  ( ( A  +h  B )  =  ( A  +h  C )  ->  (
( A  +h  B
)  +h  ( -u
1  .h  A ) )  =  ( ( A  +h  C )  +h  ( -u 1  .h  A ) ) )
2 hvnegdi.1 . . . . 5  |-  A  e. 
~H
3 hvnegdi.2 . . . . 5  |-  B  e. 
~H
4 neg1cn 9960 . . . . . 6  |-  -u 1  e.  CC
54, 2hvmulcli 21907 . . . . 5  |-  ( -u
1  .h  A )  e.  ~H
62, 3, 5hvadd32i 21946 . . . 4  |-  ( ( A  +h  B )  +h  ( -u 1  .h  A ) )  =  ( ( A  +h  ( -u 1  .h  A
) )  +h  B
)
72hvnegidi 21922 . . . . 5  |-  ( A  +h  ( -u 1  .h  A ) )  =  0h
87oveq1i 5991 . . . 4  |-  ( ( A  +h  ( -u
1  .h  A ) )  +h  B )  =  ( 0h  +h  B )
93hvaddid2i 21921 . . . 4  |-  ( 0h 
+h  B )  =  B
106, 8, 93eqtri 2390 . . 3  |-  ( ( A  +h  B )  +h  ( -u 1  .h  A ) )  =  B
11 hvaddcan.3 . . . . 5  |-  C  e. 
~H
122, 11, 5hvadd32i 21946 . . . 4  |-  ( ( A  +h  C )  +h  ( -u 1  .h  A ) )  =  ( ( A  +h  ( -u 1  .h  A
) )  +h  C
)
137oveq1i 5991 . . . 4  |-  ( ( A  +h  ( -u
1  .h  A ) )  +h  C )  =  ( 0h  +h  C )
1411hvaddid2i 21921 . . . 4  |-  ( 0h 
+h  C )  =  C
1512, 13, 143eqtri 2390 . . 3  |-  ( ( A  +h  C )  +h  ( -u 1  .h  A ) )  =  C
161, 10, 153eqtr3g 2421 . 2  |-  ( ( A  +h  B )  =  ( A  +h  C )  ->  B  =  C )
17 oveq2 5989 . 2  |-  ( B  =  C  ->  ( A  +h  B )  =  ( A  +h  C
) )
1816, 17impbii 180 1  |-  ( ( A  +h  B )  =  ( A  +h  C )  <->  B  =  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1647    e. wcel 1715  (class class class)co 5981   1c1 8885   -ucneg 9185   ~Hchil 21812    +h cva 21813    .h csm 21814   0hc0v 21817
This theorem is referenced by:  hvsubaddi  21958  hvaddcan  21962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-hvcom 21894  ax-hvass 21895  ax-hv0cl 21896  ax-hvaddid 21897  ax-hfvmul 21898  ax-hvmulid 21899  ax-hvdistr2 21902  ax-hvmul0 21903
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-ltxr 9019  df-sub 9186  df-neg 9187  df-hvsub 21864
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