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Theorem hvsubadd 22610
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 6117 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21eqeq1d 2450 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  -h  B
)  =  C  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  C ) )
3 eqeq2 2451 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( B  +h  C
)  =  A  <->  ( B  +h  C )  =  if ( A  e.  ~H ,  A ,  0h )
) )
42, 3bibi12d 314 . 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 6118 . . . 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 2450 . . 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 6117 . . . 4  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( B  +h  C )  =  ( if ( B  e.  ~H ,  B ,  0h )  +h  C
) )
87eqeq1d 2450 . . 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 314 . 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 2451 . . 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 6118 . . . 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 2450 . . 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 314 . 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 22537 . . . 4  |-  0h  e.  ~H
1514elimel 3815 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
1614elimel 3815 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
1714elimel 3815 . . 3  |-  if ( C  e.  ~H ,  C ,  0h )  e.  ~H
1815, 16, 17hvsubaddi 22599 . 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 3806 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 178    /\ w3a 937    = wceq 1653    e. wcel 1727   ifcif 3763  (class class class)co 6110   ~Hchil 22453    +h cva 22454   0hc0v 22458    -h cmv 22459
This theorem is referenced by:  shmodsi  22922  pjop  22960  pjpo  22961  chscllem2  23171  pjo  23204  hodsi  23309  pjimai  23710  superpos  23888  sumdmdii  23949  sumdmdlem  23952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-hfvadd 22534  ax-hvcom 22535  ax-hvass 22536  ax-hv0cl 22537  ax-hvaddid 22538  ax-hfvmul 22539  ax-hvmulid 22540  ax-hvdistr2 22543  ax-hvmul0 22544
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-ltxr 9156  df-sub 9324  df-neg 9325  df-hvsub 22505
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