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Theorem hiidge0 21693
Description: Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hiidge0  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )

Proof of Theorem hiidge0
StepHypRef Expression
1 pm2.1 406 . . 3  |-  ( -.  A  =  0h  \/  A  =  0h )
2 df-ne 2461 . . . . . 6  |-  ( A  =/=  0h  <->  -.  A  =  0h )
3 ax-his4 21680 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
42, 3sylan2br 462 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  ( A  .ih  A ) )
54ex 423 . . . 4  |-  ( A  e.  ~H  ->  ( -.  A  =  0h  ->  0  <  ( A 
.ih  A ) ) )
6 oveq1 5881 . . . . . . 7  |-  ( A  =  0h  ->  ( A  .ih  A )  =  ( 0h  .ih  A
) )
7 hi01 21691 . . . . . . 7  |-  ( A  e.  ~H  ->  ( 0h  .ih  A )  =  0 )
86, 7sylan9eqr 2350 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( A  .ih  A
)  =  0 )
98eqcomd 2301 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  0  =  ( A 
.ih  A ) )
109ex 423 . . . 4  |-  ( A  e.  ~H  ->  ( A  =  0h  ->  0  =  ( A  .ih  A ) ) )
115, 10orim12d 811 . . 3  |-  ( A  e.  ~H  ->  (
( -.  A  =  0h  \/  A  =  0h )  ->  (
0  <  ( A  .ih  A )  \/  0  =  ( A  .ih  A ) ) ) )
121, 11mpi 16 . 2  |-  ( A  e.  ~H  ->  (
0  <  ( A  .ih  A )  \/  0  =  ( A  .ih  A ) ) )
13 0re 8854 . . 3  |-  0  e.  RR
14 hiidrcl 21690 . . 3  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
15 leloe 8924 . . 3  |-  ( ( 0  e.  RR  /\  ( A  .ih  A )  e.  RR )  -> 
( 0  <_  ( A  .ih  A )  <->  ( 0  <  ( A  .ih  A )  \/  0  =  ( A  .ih  A
) ) ) )
1613, 14, 15sylancr 644 . 2  |-  ( A  e.  ~H  ->  (
0  <_  ( A  .ih  A )  <->  ( 0  <  ( A  .ih  A )  \/  0  =  ( A  .ih  A
) ) ) )
1712, 16mpbird 223 1  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753    < clt 8883    <_ cle 8884   ~Hchil 21515    .ih csp 21518   0hc0v 21520
This theorem is referenced by:  normlem5  21709  normlem6  21710  normlem7  21711  normf  21718  normge0  21721  normgt0  21722  normsqi  21727  norm-ii-i  21732  norm-iii-i  21734  bcsiALT  21774  pjhthlem1  21986  cnlnadjlem7  22669  branmfn  22701  leopsq  22725  idleop  22727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hv0cl 21599  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his3 21679  ax-his4 21680
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602
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