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Theorem hvsubadd 21770
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 5952 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21eqeq1d 2366 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  -h  B
)  =  C  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  C ) )
3 eqeq2 2367 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( B  +h  C
)  =  A  <->  ( B  +h  C )  =  if ( A  e.  ~H ,  A ,  0h )
) )
42, 3bibi12d 312 . 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 5953 . . . 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 2366 . . 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 5952 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  +h  C )  =  ( if ( B  e.  ~H ,  B ,  0h )  +h  C
) )
87eqeq1d 2366 . . 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 312 . 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 2367 . . 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 5953 . . . 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 2366 . . 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 312 . 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 21697 . . . 4  |-  0h  e.  ~H
1514elimel 3693 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
1614elimel 3693 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
1714elimel 3693 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
1815, 16, 17hvsubaddi 21759 . 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 3684 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 176    /\ w3a 934    = wceq 1642    e. wcel 1710   ifcif 3641  (class class class)co 5945   ~Hchil 21613    +h cva 21614   0hc0v 21618    -h cmv 21619
This theorem is referenced by:  shmodsi  22082  pjop  22120  pjpo  22121  chscllem2  22331  pjo  22364  hodsi  22469  pjimai  22870  superpos  23048  sumdmdii  23109  sumdmdlem  23112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-hfvadd 21694  ax-hvcom 21695  ax-hvass 21696  ax-hv0cl 21697  ax-hvaddid 21698  ax-hfvmul 21699  ax-hvmulid 21700  ax-hvdistr2 21703  ax-hvmul0 21704
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-ltxr 8962  df-sub 9129  df-neg 9130  df-hvsub 21665
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