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Theorem his6 21638
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 21624 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
21gt0ne0d 9305 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( A  .ih  A
)  =/=  0 )
32ex 425 . . 3  |-  ( A  e.  ~H  ->  ( A  =/=  0h  ->  ( A  .ih  A )  =/=  0 ) )
43necon4d 2484 . 2  |-  ( A  e.  ~H  ->  (
( A  .ih  A
)  =  0  ->  A  =  0h )
)
5 hi01 21635 . . 3  |-  ( A  e.  ~H  ->  ( 0h  .ih  A )  =  0 )
6 oveq1 5799 . . . 4  |-  ( A  =  0h  ->  ( A  .ih  A )  =  ( 0h  .ih  A
) )
76eqeq1d 2266 . . 3  |-  ( A  =  0h  ->  (
( A  .ih  A
)  =  0  <->  ( 0h  .ih  A )  =  0 ) )
85, 7syl5ibrcom 215 . 2  |-  ( A  e.  ~H  ->  ( A  =  0h  ->  ( A  .ih  A )  =  0 ) )
94, 8impbid 185 1  |-  ( A  e.  ~H  ->  (
( A  .ih  A
)  =  0  <->  A  =  0h ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421  (class class class)co 5792   0cc0 8705   ~Hchil 21459    .ih csp 21462   0hc0v 21464
This theorem is referenced by:  hial0  21641  hial02  21642  hi2eq  21644  bcseqi  21659  ocin  21835  h1de2bi  22093  h1de2ctlem  22094  normcan  22115  unopf1o  22456  riesz3i  22602
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-hv0cl 21543  ax-hvmul0 21550  ax-hfi 21618  ax-his3 21623  ax-his4 21624
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-ltxr 8840
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