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Theorem his6 22554
Description: Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
Assertion
Ref Expression
his6  |-  ( A  e.  ~H  ->  (
( A  .ih  A
)  =  0  <->  A  =  0h ) )

Proof of Theorem his6
StepHypRef Expression
1 ax-his4 22540 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
21gt0ne0d 9547 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( A  .ih  A
)  =/=  0 )
32ex 424 . . 3  |-  ( A  e.  ~H  ->  ( A  =/=  0h  ->  ( A  .ih  A )  =/=  0 ) )
43necon4d 2630 . 2  |-  ( A  e.  ~H  ->  (
( A  .ih  A
)  =  0  ->  A  =  0h )
)
5 hi01 22551 . . 3  |-  ( A  e.  ~H  ->  ( 0h  .ih  A )  =  0 )
6 oveq1 6047 . . . 4  |-  ( A  =  0h  ->  ( A  .ih  A )  =  ( 0h  .ih  A
) )
76eqeq1d 2412 . . 3  |-  ( A  =  0h  ->  (
( A  .ih  A
)  =  0  <->  ( 0h  .ih  A )  =  0 ) )
85, 7syl5ibrcom 214 . 2  |-  ( A  e.  ~H  ->  ( A  =  0h  ->  ( A  .ih  A )  =  0 ) )
94, 8impbid 184 1  |-  ( A  e.  ~H  ->  (
( A  .ih  A
)  =  0  <->  A  =  0h ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567  (class class class)co 6040   0cc0 8946   ~Hchil 22375    .ih csp 22378   0hc0v 22380
This theorem is referenced by:  hial0  22557  hial02  22558  hi2eq  22560  bcseqi  22575  ocin  22751  h1de2bi  23009  h1de2ctlem  23010  normcan  23031  unopf1o  23372  riesz3i  23518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hv0cl 22459  ax-hvmul0 22466  ax-hfi 22534  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081
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