HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvsubeq0i Structured version   Unicode version

Theorem hvsubeq0i 22570
Description: If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvnegdi.1  |-  A  e. 
~H
hvnegdi.2  |-  B  e. 
~H
Assertion
Ref Expression
hvsubeq0i  |-  ( ( A  -h  B )  =  0h  <->  A  =  B )

Proof of Theorem hvsubeq0i
StepHypRef Expression
1 hvnegdi.1 . . . . . 6  |-  A  e. 
~H
2 hvnegdi.2 . . . . . 6  |-  B  e. 
~H
31, 2hvsubvali 22528 . . . . 5  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
43eqeq1i 2445 . . . 4  |-  ( ( A  -h  B )  =  0h  <->  ( A  +h  ( -u 1  .h  B ) )  =  0h )
5 oveq1 6091 . . . 4  |-  ( ( A  +h  ( -u
1  .h  B ) )  =  0h  ->  ( ( A  +h  ( -u 1  .h  B ) )  +h  B )  =  ( 0h  +h  B ) )
64, 5sylbi 189 . . 3  |-  ( ( A  -h  B )  =  0h  ->  (
( A  +h  ( -u 1  .h  B ) )  +h  B )  =  ( 0h  +h  B ) )
7 neg1cn 10072 . . . . . 6  |-  -u 1  e.  CC
87, 2hvmulcli 22522 . . . . 5  |-  ( -u
1  .h  B )  e.  ~H
91, 8, 2hvadd32i 22561 . . . 4  |-  ( ( A  +h  ( -u
1  .h  B ) )  +h  B )  =  ( ( A  +h  B )  +h  ( -u 1  .h  B ) )
101, 2, 8hvassi 22560 . . . . 5  |-  ( ( A  +h  B )  +h  ( -u 1  .h  B ) )  =  ( A  +h  ( B  +h  ( -u 1  .h  B ) ) )
112hvnegidi 22537 . . . . . . 7  |-  ( B  +h  ( -u 1  .h  B ) )  =  0h
1211oveq2i 6095 . . . . . 6  |-  ( A  +h  ( B  +h  ( -u 1  .h  B
) ) )  =  ( A  +h  0h )
13 ax-hvaddid 22512 . . . . . . 7  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
141, 13ax-mp 5 . . . . . 6  |-  ( A  +h  0h )  =  A
1512, 14eqtri 2458 . . . . 5  |-  ( A  +h  ( B  +h  ( -u 1  .h  B
) ) )  =  A
1610, 15eqtri 2458 . . . 4  |-  ( ( A  +h  B )  +h  ( -u 1  .h  B ) )  =  A
179, 16eqtri 2458 . . 3  |-  ( ( A  +h  ( -u
1  .h  B ) )  +h  B )  =  A
182hvaddid2i 22536 . . 3  |-  ( 0h 
+h  B )  =  B
196, 17, 183eqtr3g 2493 . 2  |-  ( ( A  -h  B )  =  0h  ->  A  =  B )
20 oveq1 6091 . . 3  |-  ( A  =  B  ->  ( A  -h  B )  =  ( B  -h  B
) )
21 hvsubid 22533 . . . 4  |-  ( B  e.  ~H  ->  ( B  -h  B )  =  0h )
222, 21ax-mp 5 . . 3  |-  ( B  -h  B )  =  0h
2320, 22syl6eq 2486 . 2  |-  ( A  =  B  ->  ( A  -h  B )  =  0h )
2419, 23impbii 182 1  |-  ( ( A  -h  B )  =  0h  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726  (class class class)co 6084   1c1 8996   -ucneg 9297   ~Hchil 22427    +h cva 22428    .h csm 22429   0hc0v 22432    -h cmv 22433
This theorem is referenced by:  hvsubeq0  22575  bcseqi  22627  normsub0i  22642  pjss2i  23187
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 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-hvcom 22509  ax-hvass 22510  ax-hv0cl 22511  ax-hvaddid 22512  ax-hfvmul 22513  ax-hvmulid 22514  ax-hvdistr2 22517  ax-hvmul0 22518
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-ltxr 9130  df-sub 9298  df-neg 9299  df-hvsub 22479
  Copyright terms: Public domain W3C validator