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Theorem hvsubaddi 22529
Description: Relationship between vector subtraction and 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
hvsubaddi  |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )

Proof of Theorem hvsubaddi
StepHypRef Expression
1 hvnegdi.1 . . . 4  |-  A  e. 
~H
2 hvnegdi.2 . . . 4  |-  B  e. 
~H
31, 2hvsubvali 22484 . . 3  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
43eqeq1i 2419 . 2  |-  ( ( A  -h  B )  =  C  <->  ( A  +h  ( -u 1  .h  B ) )  =  C )
5 neg1cn 10031 . . . . . . 7  |-  -u 1  e.  CC
65, 2hvmulcli 22478 . . . . . 6  |-  ( -u
1  .h  B )  e.  ~H
72, 1, 6hvadd12i 22520 . . . . 5  |-  ( B  +h  ( A  +h  ( -u 1  .h  B
) ) )  =  ( A  +h  ( B  +h  ( -u 1  .h  B ) ) )
82hvnegidi 22493 . . . . . 6  |-  ( B  +h  ( -u 1  .h  B ) )  =  0h
98oveq2i 6059 . . . . 5  |-  ( A  +h  ( B  +h  ( -u 1  .h  B
) ) )  =  ( A  +h  0h )
10 ax-hvaddid 22468 . . . . . 6  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
111, 10ax-mp 8 . . . . 5  |-  ( A  +h  0h )  =  A
127, 9, 113eqtri 2436 . . . 4  |-  ( B  +h  ( A  +h  ( -u 1  .h  B
) ) )  =  A
1312eqeq1i 2419 . . 3  |-  ( ( B  +h  ( A  +h  ( -u 1  .h  B ) ) )  =  ( B  +h  C )  <->  A  =  ( B  +h  C
) )
141, 6hvaddcli 22482 . . . 4  |-  ( A  +h  ( -u 1  .h  B ) )  e. 
~H
15 hvaddcan.3 . . . 4  |-  C  e. 
~H
162, 14, 15hvaddcani 22528 . . 3  |-  ( ( B  +h  ( A  +h  ( -u 1  .h  B ) ) )  =  ( B  +h  C )  <->  ( A  +h  ( -u 1  .h  B ) )  =  C )
17 eqcom 2414 . . 3  |-  ( A  =  ( B  +h  C )  <->  ( B  +h  C )  =  A )
1813, 16, 173bitr3i 267 . 2  |-  ( ( A  +h  ( -u
1  .h  B ) )  =  C  <->  ( B  +h  C )  =  A )
194, 18bitri 241 1  |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721  (class class class)co 6048   1c1 8955   -ucneg 9256   ~Hchil 22383    +h cva 22384    .h csm 22385   0hc0v 22388    -h cmv 22389
This theorem is referenced by:  hvsubadd  22540  omlsilem  22865
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-hfvadd 22464  ax-hvcom 22465  ax-hvass 22466  ax-hv0cl 22467  ax-hvaddid 22468  ax-hfvmul 22469  ax-hvmulid 22470  ax-hvdistr2 22473  ax-hvmul0 22474
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-ltxr 9089  df-sub 9257  df-neg 9258  df-hvsub 22435
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