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Theorem his6 21678
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 21664 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
21gt0ne0d 9337 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( A  .ih  A
)  =/=  0 )
32ex 423 . . 3  |-  ( A  e.  ~H  ->  ( A  =/=  0h  ->  ( A  .ih  A )  =/=  0 ) )
43necon4d 2509 . 2  |-  ( A  e.  ~H  ->  (
( A  .ih  A
)  =  0  ->  A  =  0h )
)
5 hi01 21675 . . 3  |-  ( A  e.  ~H  ->  ( 0h  .ih  A )  =  0 )
6 oveq1 5865 . . . 4  |-  ( A  =  0h  ->  ( A  .ih  A )  =  ( 0h  .ih  A
) )
76eqeq1d 2291 . . 3  |-  ( A  =  0h  ->  (
( A  .ih  A
)  =  0  <->  ( 0h  .ih  A )  =  0 ) )
85, 7syl5ibrcom 213 . 2  |-  ( A  e.  ~H  ->  ( A  =  0h  ->  ( A  .ih  A )  =  0 ) )
94, 8impbid 183 1  |-  ( A  e.  ~H  ->  (
( A  .ih  A
)  =  0  <->  A  =  0h ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   0cc0 8737   ~Hchil 21499    .ih csp 21502   0hc0v 21504
This theorem is referenced by:  hial0  21681  hial02  21682  hi2eq  21684  bcseqi  21699  ocin  21875  h1de2bi  22133  h1de2ctlem  22134  normcan  22155  unopf1o  22496  riesz3i  22642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hv0cl 21583  ax-hvmul0 21590  ax-hfi 21658  ax-his3 21663  ax-his4 21664
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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