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Theorem hi2eq 22638
Description: Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hi2eq  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  ( A  -h  B
) )  =  ( B  .ih  ( A  -h  B ) )  <-> 
A  =  B ) )

Proof of Theorem hi2eq
StepHypRef Expression
1 hvsubcl 22551 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  e.  ~H )
2 his2sub 22625 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  ( A  -h  B )  e. 
~H )  ->  (
( A  -h  B
)  .ih  ( A  -h  B ) )  =  ( ( A  .ih  ( A  -h  B
) )  -  ( B  .ih  ( A  -h  B ) ) ) )
31, 2mpd3an3 1281 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  .ih  ( A  -h  B ) )  =  ( ( A 
.ih  ( A  -h  B ) )  -  ( B  .ih  ( A  -h  B ) ) ) )
43eqeq1d 2450 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( A  -h  B )  .ih  ( A  -h  B
) )  =  0  <-> 
( ( A  .ih  ( A  -h  B
) )  -  ( B  .ih  ( A  -h  B ) ) )  =  0 ) )
5 his6 22632 . . . 4  |-  ( ( A  -h  B )  e.  ~H  ->  (
( ( A  -h  B )  .ih  ( A  -h  B ) )  =  0  <->  ( A  -h  B )  =  0h ) )
61, 5syl 16 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( A  -h  B )  .ih  ( A  -h  B
) )  =  0  <-> 
( A  -h  B
)  =  0h )
)
74, 6bitr3d 248 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( A 
.ih  ( A  -h  B ) )  -  ( B  .ih  ( A  -h  B ) ) )  =  0  <->  ( A  -h  B )  =  0h ) )
8 hicl 22613 . . . 4  |-  ( ( A  e.  ~H  /\  ( A  -h  B
)  e.  ~H )  ->  ( A  .ih  ( A  -h  B ) )  e.  CC )
91, 8syldan 458 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( A  -h  B ) )  e.  CC )
10 simpr 449 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  B  e.  ~H )
11 hicl 22613 . . . 4  |-  ( ( B  e.  ~H  /\  ( A  -h  B
)  e.  ~H )  ->  ( B  .ih  ( A  -h  B ) )  e.  CC )
1210, 1, 11syl2anc 644 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( B  .ih  ( A  -h  B ) )  e.  CC )
139, 12subeq0ad 9452 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( A 
.ih  ( A  -h  B ) )  -  ( B  .ih  ( A  -h  B ) ) )  =  0  <->  ( A  .ih  ( A  -h  B ) )  =  ( B  .ih  ( A  -h  B ) ) ) )
14 hvsubeq0 22601 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  =  0h  <->  A  =  B ) )
157, 13, 143bitr3d 276 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  ( A  -h  B
) )  =  ( B  .ih  ( A  -h  B ) )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727  (class class class)co 6110   CCcc 9019   0cc0 9021    - cmin 9322   ~Hchil 22453    .ih csp 22456   0hc0v 22458    -h cmv 22459
This theorem is referenced by:  hial2eq  22639
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  ax-hfi 22612  ax-his2 22616  ax-his3 22617  ax-his4 22618
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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