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Theorem hvaddcan 22572
Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvaddcan  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  =  ( A  +h  C )  <->  B  =  C ) )

Proof of Theorem hvaddcan
StepHypRef Expression
1 oveq1 6088 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  B
) )
2 oveq1 6088 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  +h  C )  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  C
) )
31, 2eqeq12d 2450 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  +h  B
)  =  ( A  +h  C )  <->  ( if ( A  e.  ~H ,  A ,  0h )  +h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  C
) ) )
43bibi1d 311 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( A  +h  B )  =  ( A  +h  C )  <-> 
B  =  C )  <-> 
( ( if ( A  e.  ~H ,  A ,  0h )  +h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  C
)  <->  B  =  C
) ) )
5 oveq2 6089 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  +h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
) )
65eqeq1d 2444 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  +h  B
)  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  C )  <->  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  C
) ) )
7 eqeq1 2442 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  =  C  <->  if ( B  e.  ~H ,  B ,  0h )  =  C ) )
86, 7bibi12d 313 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  +h  B )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  C
)  <->  B  =  C
)  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  C
)  <->  if ( B  e. 
~H ,  B ,  0h )  =  C
) ) )
9 oveq2 6089 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  +h  C )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  if ( C  e.  ~H ,  C ,  0h )
) )
109eqeq2d 2447 . . 3  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
)  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  C )  <->  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  if ( C  e.  ~H ,  C ,  0h )
) ) )
11 eqeq2 2445 . . 3  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( if ( B  e.  ~H ,  B ,  0h )  =  C  <->  if ( B  e. 
~H ,  B ,  0h )  =  if ( C  e.  ~H ,  C ,  0h )
) )
1210, 11bibi12d 313 . 2  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  C
)  <->  if ( B  e. 
~H ,  B ,  0h )  =  C
)  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( B  e. 
~H ,  B ,  0h ) )  =  ( if ( A  e. 
~H ,  A ,  0h )  +h  if ( C  e.  ~H ,  C ,  0h )
)  <->  if ( B  e. 
~H ,  B ,  0h )  =  if ( C  e.  ~H ,  C ,  0h )
) ) )
13 ax-hv0cl 22506 . . . 4  |-  0h  e.  ~H
1413elimel 3791 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
1513elimel 3791 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
1613elimel 3791 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
1714, 15, 16hvaddcani 22567 . 2  |-  ( ( if ( A  e. 
~H ,  A ,  0h )  +h  if ( B  e.  ~H ,  B ,  0h )
)  =  ( if ( A  e.  ~H ,  A ,  0h )  +h  if ( C  e. 
~H ,  C ,  0h ) )  <->  if ( B  e.  ~H ,  B ,  0h )  =  if ( C  e.  ~H ,  C ,  0h )
)
184, 8, 12, 17dedth3h 3782 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    /\ w3a 936    = wceq 1652    e. wcel 1725   ifcif 3739  (class class class)co 6081   ~Hchil 22422    +h cva 22423   0hc0v 22427
This theorem is referenced by:  hvaddcan2  22573  hvsubcan  22576
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvdistr2 22512  ax-hvmul0 22513
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-hvsub 22474
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