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Theorem bcsiALT 21758
Description: Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
bcs.1  |-  A  e. 
~H
bcs.2  |-  B  e. 
~H
Assertion
Ref Expression
bcsiALT  |-  ( abs `  ( A  .ih  B
) )  <_  (
( normh `  A )  x.  ( normh `  B )
)

Proof of Theorem bcsiALT
StepHypRef Expression
1 fveq2 5525 . . 3  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  =  ( abs `  0 ) )
2 abs0 11770 . . . 4  |-  ( abs `  0 )  =  0
3 bcs.1 . . . . . 6  |-  A  e. 
~H
4 normge0 21705 . . . . . 6  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
53, 4ax-mp 8 . . . . 5  |-  0  <_  ( normh `  A )
6 bcs.2 . . . . . 6  |-  B  e. 
~H
7 normge0 21705 . . . . . 6  |-  ( B  e.  ~H  ->  0  <_  ( normh `  B )
)
86, 7ax-mp 8 . . . . 5  |-  0  <_  ( normh `  B )
93normcli 21710 . . . . . 6  |-  ( normh `  A )  e.  RR
106normcli 21710 . . . . . 6  |-  ( normh `  B )  e.  RR
119, 10mulge0i 9320 . . . . 5  |-  ( ( 0  <_  ( normh `  A )  /\  0  <_  ( normh `  B )
)  ->  0  <_  ( ( normh `  A )  x.  ( normh `  B )
) )
125, 8, 11mp2an 653 . . . 4  |-  0  <_  ( ( normh `  A
)  x.  ( normh `  B ) )
132, 12eqbrtri 4042 . . 3  |-  ( abs `  0 )  <_ 
( ( normh `  A
)  x.  ( normh `  B ) )
141, 13syl6eqbr 4060 . 2  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  <_  (
( normh `  A )  x.  ( normh `  B )
) )
15 df-ne 2448 . . . 4  |-  ( ( A  .ih  B )  =/=  0  <->  -.  ( A  .ih  B )  =  0 )
166, 3his1i 21679 . . . . . . . 8  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
1716oveq2i 5869 . . . . . . 7  |-  ( ( ( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  x.  ( B  .ih  A
) )  =  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  x.  ( * `  ( A  .ih  B ) ) )
1817oveq2i 5869 . . . . . 6  |-  ( ( ( * `  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) ) )  x.  ( A  .ih  B ) )  +  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  x.  ( B  .ih  A
) ) )  =  ( ( ( * `
 ( ( A 
.ih  B )  / 
( abs `  ( A  .ih  B ) ) ) )  x.  ( A  .ih  B ) )  +  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B ) ) )  x.  ( * `
 ( A  .ih  B ) ) ) )
193, 6hicli 21660 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
20 abslem2 11823 . . . . . . 7  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( A  .ih  B )  =/=  0 )  -> 
( ( ( * `
 ( ( A 
.ih  B )  / 
( abs `  ( A  .ih  B ) ) ) )  x.  ( A  .ih  B ) )  +  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B ) ) )  x.  ( * `
 ( A  .ih  B ) ) ) )  =  ( 2  x.  ( abs `  ( A  .ih  B ) ) ) )
2119, 20mpan 651 . . . . . 6  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) ) )  x.  ( A  .ih  B ) )  +  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  x.  ( * `  ( A  .ih  B ) ) ) )  =  ( 2  x.  ( abs `  ( A  .ih  B
) ) ) )
2218, 21syl5req 2328 . . . . 5  |-  ( ( A  .ih  B )  =/=  0  ->  (
2  x.  ( abs `  ( A  .ih  B
) ) )  =  ( ( ( * `
 ( ( A 
.ih  B )  / 
( abs `  ( A  .ih  B ) ) ) )  x.  ( A  .ih  B ) )  +  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B ) ) )  x.  ( B 
.ih  A ) ) ) )
2319abs00i 11881 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =  0  <->  ( A  .ih  B )  =  0 )
2423necon3bii 2478 . . . . . . 7  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  <->  ( A  .ih  B )  =/=  0
)
2519abscli 11878 . . . . . . . . . 10  |-  ( abs `  ( A  .ih  B
) )  e.  RR
2625recni 8849 . . . . . . . . 9  |-  ( abs `  ( A  .ih  B
) )  e.  CC
2719, 26divclzi 9495 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
2819, 26divreczi 9498 . . . . . . . . . 10  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  =  ( ( A  .ih  B )  x.  ( 1  /  ( abs `  ( A  .ih  B ) ) ) ) )
2928fveq2d 5529 . . . . . . . . 9  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( ( A 
.ih  B )  / 
( abs `  ( A  .ih  B ) ) ) )  =  ( abs `  ( ( A  .ih  B )  x.  ( 1  / 
( abs `  ( A  .ih  B ) ) ) ) ) )
3026recclzi 9485 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
31 absmul 11779 . . . . . . . . . . 11  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( 1  /  ( abs `  ( A  .ih  B ) ) )  e.  CC )  ->  ( abs `  ( ( A 
.ih  B )  x.  ( 1  /  ( abs `  ( A  .ih  B ) ) ) ) )  =  ( ( abs `  ( A 
.ih  B ) )  x.  ( abs `  (
1  /  ( abs `  ( A  .ih  B
) ) ) ) ) )
3219, 30, 31sylancr 644 . . . . . . . . . 10  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( ( A 
.ih  B )  x.  ( 1  /  ( abs `  ( A  .ih  B ) ) ) ) )  =  ( ( abs `  ( A 
.ih  B ) )  x.  ( abs `  (
1  /  ( abs `  ( A  .ih  B
) ) ) ) ) )
3325rerecclzi 9524 . . . . . . . . . . . 12  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  RR )
34 0re 8838 . . . . . . . . . . . . . 14  |-  0  e.  RR
3533, 34jctil 523 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
0  e.  RR  /\  ( 1  /  ( abs `  ( A  .ih  B ) ) )  e.  RR ) )
3619absgt0i 11882 . . . . . . . . . . . . . . 15  |-  ( ( A  .ih  B )  =/=  0  <->  0  <  ( abs `  ( A 
.ih  B ) ) )
3724, 36bitri 240 . . . . . . . . . . . . . 14  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  <->  0  <  ( abs `  ( A 
.ih  B ) ) )
3825recgt0i 9661 . . . . . . . . . . . . . 14  |-  ( 0  <  ( abs `  ( A  .ih  B ) )  ->  0  <  (
1  /  ( abs `  ( A  .ih  B
) ) ) )
3937, 38sylbi 187 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  0  <  ( 1  /  ( abs `  ( A  .ih  B ) ) ) )
40 ltle 8910 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  ( 1  /  ( abs `  ( A  .ih  B ) ) )  e.  RR )  ->  (
0  <  ( 1  /  ( abs `  ( A  .ih  B ) ) )  ->  0  <_  ( 1  /  ( abs `  ( A  .ih  B
) ) ) ) )
4135, 39, 40sylc 56 . . . . . . . . . . . 12  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  0  <_  ( 1  /  ( abs `  ( A  .ih  B ) ) ) )
4233, 41absidd 11905 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  ( 1  /  ( abs `  ( A  .ih  B
) ) ) )
4342oveq2d 5874 . . . . . . . . . 10  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( abs `  ( A  .ih  B ) )  x.  ( abs `  (
1  /  ( abs `  ( A  .ih  B
) ) ) ) )  =  ( ( abs `  ( A 
.ih  B ) )  x.  ( 1  / 
( abs `  ( A  .ih  B ) ) ) ) )
4432, 43eqtrd 2315 . . . . . . . . 9  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( ( A 
.ih  B )  x.  ( 1  /  ( abs `  ( A  .ih  B ) ) ) ) )  =  ( ( abs `  ( A 
.ih  B ) )  x.  ( 1  / 
( abs `  ( A  .ih  B ) ) ) ) )
4526recidzi 9487 . . . . . . . . 9  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( abs `  ( A  .ih  B ) )  x.  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  1 )
4629, 44, 453eqtrd 2319 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( ( A 
.ih  B )  / 
( abs `  ( A  .ih  B ) ) ) )  =  1 )
4727, 46jca 518 . . . . . . 7  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  e.  CC  /\  ( abs `  ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) ) )  =  1 ) )
4824, 47sylbir 204 . . . . . 6  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  e.  CC  /\  ( abs `  ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) ) )  =  1 ) )
493, 6normlem7tALT 21698 . . . . . 6  |-  ( ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  e.  CC  /\  ( abs `  ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) ) )  =  1 )  -> 
( ( ( * `
 ( ( A 
.ih  B )  / 
( abs `  ( A  .ih  B ) ) ) )  x.  ( A  .ih  B ) )  +  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B ) ) )  x.  ( B 
.ih  A ) ) )  <_  ( 2  x.  ( ( sqr `  ( B  .ih  B
) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
5048, 49syl 15 . . . . 5  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) ) )  x.  ( A  .ih  B ) )  +  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  x.  ( B  .ih  A
) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
5122, 50eqbrtrd 4043 . . . 4  |-  ( ( A  .ih  B )  =/=  0  ->  (
2  x.  ( abs `  ( A  .ih  B
) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
5215, 51sylbir 204 . . 3  |-  ( -.  ( A  .ih  B
)  =  0  -> 
( 2  x.  ( abs `  ( A  .ih  B ) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
5310recni 8849 . . . . . 6  |-  ( normh `  B )  e.  CC
549recni 8849 . . . . . 6  |-  ( normh `  A )  e.  CC
55 normval 21703 . . . . . . . 8  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
566, 55ax-mp 8 . . . . . . 7  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
57 normval 21703 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
583, 57ax-mp 8 . . . . . . 7  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
5956, 58oveq12i 5870 . . . . . 6  |-  ( (
normh `  B )  x.  ( normh `  A )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6053, 54, 59mulcomli 8844 . . . . 5  |-  ( (
normh `  A )  x.  ( normh `  B )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6160breq2i 4031 . . . 4  |-  ( ( abs `  ( A 
.ih  B ) )  <_  ( ( normh `  A )  x.  ( normh `  B ) )  <-> 
( abs `  ( A  .ih  B ) )  <_  ( ( sqr `  ( B  .ih  B
) )  x.  ( sqr `  ( A  .ih  A ) ) ) )
62 2pos 9828 . . . . 5  |-  0  <  2
63 hiidge0 21677 . . . . . . . 8  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
64 hiidrcl 21674 . . . . . . . . . 10  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
656, 64ax-mp 8 . . . . . . . . 9  |-  ( B 
.ih  B )  e.  RR
6665sqrcli 11855 . . . . . . . 8  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
676, 63, 66mp2b 9 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
68 hiidge0 21677 . . . . . . . 8  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
69 hiidrcl 21674 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
703, 69ax-mp 8 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  RR
7170sqrcli 11855 . . . . . . . 8  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
723, 68, 71mp2b 9 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
7367, 72remulcli 8851 . . . . . 6  |-  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )  e.  RR
74 2re 9815 . . . . . 6  |-  2  e.  RR
7525, 73, 74lemul2i 9680 . . . . 5  |-  ( 0  <  2  ->  (
( abs `  ( A  .ih  B ) )  <_  ( ( sqr `  ( B  .ih  B
) )  x.  ( sqr `  ( A  .ih  A ) ) )  <->  ( 2  x.  ( abs `  ( A  .ih  B ) ) )  <_  ( 2  x.  ( ( sqr `  ( B  .ih  B
) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) ) )
7662, 75ax-mp 8 . . . 4  |-  ( ( abs `  ( A 
.ih  B ) )  <_  ( ( sqr `  ( B  .ih  B
) )  x.  ( sqr `  ( A  .ih  A ) ) )  <->  ( 2  x.  ( abs `  ( A  .ih  B ) ) )  <_  ( 2  x.  ( ( sqr `  ( B  .ih  B
) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
7761, 76bitri 240 . . 3  |-  ( ( abs `  ( A 
.ih  B ) )  <_  ( ( normh `  A )  x.  ( normh `  B ) )  <-> 
( 2  x.  ( abs `  ( A  .ih  B ) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
7852, 77sylibr 203 . 2  |-  ( -.  ( A  .ih  B
)  =  0  -> 
( abs `  ( A  .ih  B ) )  <_  ( ( normh `  A )  x.  ( normh `  B ) ) )
7914, 78pm2.61i 156 1  |-  ( abs `  ( A  .ih  B
) )  <_  (
( normh `  A )  x.  ( normh `  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   2c2 9795   *ccj 11581   sqrcsqr 11718   abscabs 11719   ~Hchil 21499    .ih csp 21502   normhcno 21503
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-cnex 8793  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-pre-mulgt0 8814  ax-pre-sup 8815  ax-hfvadd 21580  ax-hv0cl 21583  ax-hfvmul 21585  ax-hvmulass 21587  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  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-reu 2550  df-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-hnorm 21548  df-hvsub 21551
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