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Theorem hvsubeq0 21477
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 5717 . . . 4  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  ( A  -h  B )  =  ( if ( A  e.  ~H ,  A ,  0h )  -h  B
) )
21eqeq1d 2261 . . 3  |-  ( A  =  if ( A  e.  ~H ,  A ,  0h )  ->  (
( A  -h  B
)  =  0h  <->  ( if ( A  e.  ~H ,  A ,  0h )  -h  B )  =  0h ) )
3 eqeq1 2259 . . 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 5718 . . . 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 2261 . . 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 2262 . . 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 21413 . . . 4  |-  0h  e.  ~H
109elimel 3522 . . 3  |-  if ( A  e.  ~H ,  A ,  0h )  e.  ~H
119elimel 3522 . . 3  |-  if ( B  e.  ~H ,  B ,  0h )  e.  ~H
1210, 11hvsubeq0i 21472 . 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 3512 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 1619    e. wcel 1621   ifcif 3470  (class class class)co 5710   ~Hchil 21329   0hc0v 21334    -h cmv 21335
This theorem is referenced by:  hvaddeq0  21478  hvmulcan  21481  hvmulcan2  21482  hi2eq  21514  shuni  21709  unopf1o  22326  riesz4i  22473  hmopidmchi  22561  cdjreui  22842
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvdistr2 21419  ax-hvmul0 21420
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752  df-sub 8919  df-neg 8920  df-hvsub 21381
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