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Theorem hvsubadd 22532
Description: Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubadd  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  =  C  <->  ( B  +h  C )  =  A ) )

Proof of Theorem hvsubadd
StepHypRef Expression
1 oveq1 6047 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21eqeq1d 2412 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  -h  B
)  =  C  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  C ) )
3 eqeq2 2413 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( B  +h  C
)  =  A  <->  ( B  +h  C )  =  if ( A  e.  ~H ,  A ,  0h )
) )
42, 3bibi12d 313 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( A  -h  B )  =  C  <-> 
( B  +h  C
)  =  A )  <-> 
( ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  C  <-> 
( B  +h  C
)  =  if ( A  e.  ~H ,  A ,  0h )
) ) )
5 oveq2 6048 . . . 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 2412 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  B
)  =  C  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  =  C ) )
7 oveq1 6047 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  +h  C )  =  ( if ( B  e.  ~H ,  B ,  0h )  +h  C
) )
87eqeq1d 2412 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( B  +h  C
)  =  if ( A  e.  ~H ,  A ,  0h )  <->  ( if ( B  e. 
~H ,  B ,  0h )  +h  C
)  =  if ( A  e.  ~H ,  A ,  0h )
) )
96, 8bibi12d 313 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  C  <-> 
( B  +h  C
)  =  if ( A  e.  ~H ,  A ,  0h )
)  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  =  C  <-> 
( if ( B  e.  ~H ,  B ,  0h )  +h  C
)  =  if ( A  e.  ~H ,  A ,  0h )
) ) )
10 eqeq2 2413 . . 3  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
)  =  C  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  =  if ( C  e.  ~H ,  C ,  0h )
) )
11 oveq2 6048 . . . 4  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  ( if ( B  e.  ~H ,  B ,  0h )  +h  C )  =  ( if ( B  e. 
~H ,  B ,  0h )  +h  if ( C  e.  ~H ,  C ,  0h )
) )
1211eqeq1d 2412 . . 3  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( if ( B  e.  ~H ,  B ,  0h )  +h  C
)  =  if ( A  e.  ~H ,  A ,  0h )  <->  ( if ( B  e. 
~H ,  B ,  0h )  +h  if ( C  e.  ~H ,  C ,  0h )
)  =  if ( A  e.  ~H ,  A ,  0h )
) )
1310, 12bibi12d 313 . 2  |-  ( C  =  if ( C  e.  ~H ,  C ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  =  C  <-> 
( if ( B  e.  ~H ,  B ,  0h )  +h  C
)  =  if ( A  e.  ~H ,  A ,  0h )
)  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  =  if ( C  e.  ~H ,  C ,  0h )  <->  ( if ( B  e. 
~H ,  B ,  0h )  +h  if ( C  e.  ~H ,  C ,  0h )
)  =  if ( A  e.  ~H ,  A ,  0h )
) ) )
14 ax-hv0cl 22459 . . . 4  |-  0h  e.  ~H
1514elimel 3751 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
1614elimel 3751 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
1714elimel 3751 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
1815, 16, 17hvsubaddi 22521 . 2  |-  ( ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
)  =  if ( C  e.  ~H ,  C ,  0h )  <->  ( if ( B  e. 
~H ,  B ,  0h )  +h  if ( C  e.  ~H ,  C ,  0h )
)  =  if ( A  e.  ~H ,  A ,  0h )
)
194, 9, 13, 18dedth3h 3742 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  =  C  <->  ( B  +h  C )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1721   ifcif 3699  (class class class)co 6040   ~Hchil 22375    +h cva 22376   0hc0v 22380    -h cmv 22381
This theorem is referenced by:  shmodsi  22844  pjop  22882  pjpo  22883  chscllem2  23093  pjo  23126  hodsi  23231  pjimai  23632  superpos  23810  sumdmdii  23871  sumdmdlem  23874
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-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  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-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-ltxr 9081  df-sub 9249  df-neg 9250  df-hvsub 22427
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