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Theorem bcsiALT 26817
Description: Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification 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 5877 . . 3  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  =  ( abs `  0 ) )
2 abs0 13336 . . . 4  |-  ( abs `  0 )  =  0
3 bcs.1 . . . . . 6  |-  A  e. 
~H
4 normge0 26764 . . . . . 6  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
53, 4ax-mp 5 . . . . 5  |-  0  <_  ( normh `  A )
6 bcs.2 . . . . . 6  |-  B  e. 
~H
7 normge0 26764 . . . . . 6  |-  ( B  e.  ~H  ->  0  <_  ( normh `  B )
)
86, 7ax-mp 5 . . . . 5  |-  0  <_  ( normh `  B )
93normcli 26769 . . . . . 6  |-  ( normh `  A )  e.  RR
106normcli 26769 . . . . . 6  |-  ( normh `  B )  e.  RR
119, 10mulge0i 10161 . . . . 5  |-  ( ( 0  <_  ( normh `  A )  /\  0  <_  ( normh `  B )
)  ->  0  <_  ( ( normh `  A )  x.  ( normh `  B )
) )
125, 8, 11mp2an 676 . . . 4  |-  0  <_  ( ( normh `  A
)  x.  ( normh `  B ) )
132, 12eqbrtri 4440 . . 3  |-  ( abs `  0 )  <_ 
( ( normh `  A
)  x.  ( normh `  B ) )
141, 13syl6eqbr 4458 . 2  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  <_  (
( normh `  A )  x.  ( normh `  B )
) )
15 df-ne 2620 . . . 4  |-  ( ( A  .ih  B )  =/=  0  <->  -.  ( A  .ih  B )  =  0 )
166, 3his1i 26738 . . . . . . . 8  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
1716oveq2i 6312 . . . . . . 7  |-  ( ( ( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  x.  ( B  .ih  A
) )  =  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  x.  ( * `  ( A  .ih  B ) ) )
1817oveq2i 6312 . . . . . 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 26719 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
20 abslem2 13390 . . . . . . 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 674 . . . . . 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 2476 . . . . 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 13448 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =  0  <->  ( A  .ih  B )  =  0 )
2423necon3bii 2692 . . . . . . 7  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  <->  ( A  .ih  B )  =/=  0
)
2519abscli 13445 . . . . . . . . . 10  |-  ( abs `  ( A  .ih  B
) )  e.  RR
2625recni 9655 . . . . . . . . 9  |-  ( abs `  ( A  .ih  B
) )  e.  CC
2719, 26divclzi 10342 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
2819, 26divreczi 10345 . . . . . . . . . 10  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  =  ( ( A  .ih  B )  x.  ( 1  /  ( abs `  ( A  .ih  B ) ) ) ) )
2928fveq2d 5881 . . . . . . . . 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 10332 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
31 absmul 13345 . . . . . . . . . . 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 667 . . . . . . . . . 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 10371 . . . . . . . . . . . 12  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  RR )
34 0re 9643 . . . . . . . . . . . . . 14  |-  0  e.  RR
3533, 34jctil 539 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
0  e.  RR  /\  ( 1  /  ( abs `  ( A  .ih  B ) ) )  e.  RR ) )
3619absgt0i 13449 . . . . . . . . . . . . . . 15  |-  ( ( A  .ih  B )  =/=  0  <->  0  <  ( abs `  ( A 
.ih  B ) ) )
3724, 36bitri 252 . . . . . . . . . . . . . 14  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  <->  0  <  ( abs `  ( A 
.ih  B ) ) )
3825recgt0i 10511 . . . . . . . . . . . . . 14  |-  ( 0  <  ( abs `  ( A  .ih  B ) )  ->  0  <  (
1  /  ( abs `  ( A  .ih  B
) ) ) )
3937, 38sylbi 198 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  0  <  ( 1  /  ( abs `  ( A  .ih  B ) ) ) )
40 ltle 9722 . . . . . . . . . . . . 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 62 . . . . . . . . . . . 12  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  0  <_  ( 1  /  ( abs `  ( A  .ih  B ) ) ) )
4233, 41absidd 13472 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  ( 1  /  ( abs `  ( A  .ih  B
) ) ) )
4342oveq2d 6317 . . . . . . . . . 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 2463 . . . . . . . . 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 10334 . . . . . . . . 9  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( abs `  ( A  .ih  B ) )  x.  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  1 )
4629, 44, 453eqtrd 2467 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( ( A 
.ih  B )  / 
( abs `  ( A  .ih  B ) ) ) )  =  1 )
4727, 46jca 534 . . . . . . 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 216 . . . . . 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 26757 . . . . . 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 17 . . . . 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 4441 . . . 4  |-  ( ( A  .ih  B )  =/=  0  ->  (
2  x.  ( abs `  ( A  .ih  B
) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
5215, 51sylbir 216 . . 3  |-  ( -.  ( A  .ih  B
)  =  0  -> 
( 2  x.  ( abs `  ( A  .ih  B ) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
5310recni 9655 . . . . . 6  |-  ( normh `  B )  e.  CC
549recni 9655 . . . . . 6  |-  ( normh `  A )  e.  CC
55 normval 26762 . . . . . . . 8  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
566, 55ax-mp 5 . . . . . . 7  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
57 normval 26762 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
583, 57ax-mp 5 . . . . . . 7  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
5956, 58oveq12i 6313 . . . . . 6  |-  ( (
normh `  B )  x.  ( normh `  A )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6053, 54, 59mulcomli 9650 . . . . 5  |-  ( (
normh `  A )  x.  ( normh `  B )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6160breq2i 4428 . . . 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 10701 . . . . 5  |-  0  <  2
63 hiidge0 26736 . . . . . . . 8  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
64 hiidrcl 26733 . . . . . . . . . 10  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
656, 64ax-mp 5 . . . . . . . . 9  |-  ( B 
.ih  B )  e.  RR
6665sqrtcli 13422 . . . . . . . 8  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
676, 63, 66mp2b 10 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
68 hiidge0 26736 . . . . . . . 8  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
69 hiidrcl 26733 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
703, 69ax-mp 5 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  RR
7170sqrtcli 13422 . . . . . . . 8  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
723, 68, 71mp2b 10 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
7367, 72remulcli 9657 . . . . . 6  |-  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )  e.  RR
74 2re 10679 . . . . . 6  |-  2  e.  RR
7525, 73, 74lemul2i 10530 . . . . 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 5 . . . 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 252 . . 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 215 . 2  |-  ( -.  ( A  .ih  B
)  =  0  -> 
( abs `  ( A  .ih  B ) )  <_  ( ( normh `  A )  x.  ( normh `  B ) ) )
7914, 78pm2.61i 167 1  |-  ( abs `  ( A  .ih  B
) )  <_  (
( normh `  A )  x.  ( normh `  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   class class class wbr 4420   ` cfv 5597  (class class class)co 6301   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    / cdiv 10269   2c2 10659   *ccj 13147   sqrcsqrt 13284   abscabs 13285   ~Hchil 26557    .ih csp 26560   normhcno 26561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-hfvadd 26638  ax-hv0cl 26641  ax-hfvmul 26643  ax-hvmulass 26645  ax-hvmul0 26648  ax-hfi 26717  ax-his1 26720  ax-his2 26721  ax-his3 26722  ax-his4 26723
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-sup 7958  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12213  df-exp 12272  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-hnorm 26606  df-hvsub 26609
This theorem is referenced by: (None)
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