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Theorem hiidge0 22592
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 pm2.1 407 . . 3  |-  ( -.  A  =  0h  \/  A  =  0h )
2 df-ne 2600 . . . . . 6  |-  ( A  =/=  0h  <->  -.  A  =  0h )
3 ax-his4 22579 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( A  .ih  A ) )
42, 3sylan2br 463 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  ( A  .ih  A ) )
54ex 424 . . . 4  |-  ( A  e.  ~H  ->  ( -.  A  =  0h  ->  0  <  ( A 
.ih  A ) ) )
6 oveq1 6080 . . . . . . 7  |-  ( A  =  0h  ->  ( A  .ih  A )  =  ( 0h  .ih  A
) )
7 hi01 22590 . . . . . . 7  |-  ( A  e.  ~H  ->  ( 0h  .ih  A )  =  0 )
86, 7sylan9eqr 2489 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( A  .ih  A
)  =  0 )
98eqcomd 2440 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  0  =  ( A 
.ih  A ) )
109ex 424 . . . 4  |-  ( A  e.  ~H  ->  ( A  =  0h  ->  0  =  ( A  .ih  A ) ) )
115, 10orim12d 812 . . 3  |-  ( A  e.  ~H  ->  (
( -.  A  =  0h  \/  A  =  0h )  ->  (
0  <  ( A  .ih  A )  \/  0  =  ( A  .ih  A ) ) ) )
121, 11mpi 17 . 2  |-  ( A  e.  ~H  ->  (
0  <  ( A  .ih  A )  \/  0  =  ( A  .ih  A ) ) )
13 0re 9083 . . 3  |-  0  e.  RR
14 hiidrcl 22589 . . 3  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
15 leloe 9153 . . 3  |-  ( ( 0  e.  RR  /\  ( A  .ih  A )  e.  RR )  -> 
( 0  <_  ( A  .ih  A )  <->  ( 0  <  ( A  .ih  A )  \/  0  =  ( A  .ih  A
) ) ) )
1613, 14, 15sylancr 645 . 2  |-  ( A  e.  ~H  ->  (
0  <_  ( A  .ih  A )  <->  ( 0  <  ( A  .ih  A )  \/  0  =  ( A  .ih  A
) ) ) )
1712, 16mpbird 224 1  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204  (class class class)co 6073   RRcr 8981   0cc0 8982    < clt 9112    <_ cle 9113   ~Hchil 22414    .ih csp 22417   0hc0v 22419
This theorem is referenced by:  normlem5  22608  normlem6  22609  normlem7  22610  normf  22617  normge0  22620  normgt0  22621  normsqi  22626  norm-ii-i  22631  norm-iii-i  22633  bcsiALT  22673  pjhthlem1  22885  cnlnadjlem7  23568  branmfn  23600  leopsq  23624  idleop  23626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hv0cl 22498  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898
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