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Theorem hvsubeq0 21641
Description: If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubeq0  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  =  0h  <->  A  =  B ) )

Proof of Theorem hvsubeq0
StepHypRef Expression
1 oveq1 5828 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21eqeq1d 2294 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  -h  B
)  =  0h  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  0h ) )
3 eqeq1 2292 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  =  B  <->  if ( A  e.  ~H ,  A ,  0h )  =  B ) )
42, 3bibi12d 314 . 2  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( ( A  -h  B )  =  0h  <->  A  =  B )  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  0h  <->  if ( A  e.  ~H ,  A ,  0h )  =  B ) ) )
5 oveq2 5829 . . . 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 2294 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( if ( A  e.  ~H ,  A ,  0h )  -h  B
)  =  0h  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  =  0h ) )
7 eqeq2 2295 . . 3  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  ( if ( A  e.  ~H ,  A ,  0h )  =  B  <->  if ( A  e. 
~H ,  A ,  0h )  =  if ( B  e.  ~H ,  B ,  0h )
) )
86, 7bibi12d 314 . 2  |-  ( B  =  if ( B  e.  ~H ,  B ,  0h )  ->  (
( ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  0h  <->  if ( A  e.  ~H ,  A ,  0h )  =  B )  <->  ( ( if ( A  e.  ~H ,  A ,  0h )  -h  if ( B  e. 
~H ,  B ,  0h ) )  =  0h  <->  if ( A  e.  ~H ,  A ,  0h )  =  if ( B  e. 
~H ,  B ,  0h ) ) ) )
9 ax-hv0cl 21577 . . . 4  |-  0h  e.  ~H
109elimel 3620 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
119elimel 3620 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
1210, 11hvsubeq0i 21636 . 2  |-  ( ( if ( A  e. 
~H ,  A ,  0h )  -h  if ( B  e.  ~H ,  B ,  0h )
)  =  0h  <->  if ( A  e.  ~H ,  A ,  0h )  =  if ( B  e.  ~H ,  B ,  0h )
)
134, 8, 12dedth2h 3610 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  =  0h  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687   ifcif 3568  (class class class)co 5821   ~Hchil 21493   0hc0v 21498    -h cmv 21499
This theorem is referenced by:  hvaddeq0  21642  hvmulcan  21645  hvmulcan2  21646  hi2eq  21678  shuni  21873  unopf1o  22490  riesz4i  22637  hmopidmchi  22725  cdjreui  23006
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-hvcom 21575  ax-hvass 21576  ax-hv0cl 21577  ax-hvaddid 21578  ax-hfvmul 21579  ax-hvmulid 21580  ax-hvdistr2 21583  ax-hvmul0 21584
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-po 4315  df-so 4316  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-iota 6254  df-riota 6301  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-pnf 8866  df-mnf 8867  df-ltxr 8869  df-sub 9036  df-neg 9037  df-hvsub 21545
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