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Theorem hvsubadd 22536
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 6051 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21eqeq1d 2416 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  -h  B
)  =  C  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  C ) )
3 eqeq2 2417 . . 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 6052 . . . 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 2416 . . 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 6051 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  +h  C )  =  ( if ( B  e.  ~H ,  B ,  0h )  +h  C
) )
87eqeq1d 2416 . . 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 2417 . . 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 6052 . . . 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 2416 . . 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 22463 . . . 4  |-  0h  e.  ~H
1514elimel 3755 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
1614elimel 3755 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
1714elimel 3755 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
1815, 16, 17hvsubaddi 22525 . 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 3746 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 3703  (class class class)co 6044   ~Hchil 22379    +h cva 22380   0hc0v 22384    -h cmv 22385
This theorem is referenced by:  shmodsi  22848  pjop  22886  pjpo  22887  chscllem2  23097  pjo  23130  hodsi  23235  pjimai  23636  superpos  23814  sumdmdii  23875  sumdmdlem  23878
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-hfvadd 22460  ax-hvcom 22461  ax-hvass 22462  ax-hv0cl 22463  ax-hvaddid 22464  ax-hfvmul 22465  ax-hvmulid 22466  ax-hvdistr2 22469  ax-hvmul0 22470
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-ltxr 9085  df-sub 9253  df-neg 9254  df-hvsub 22431
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