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Theorem bcsiALT 21683
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 5423 . . 3  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  =  ( abs `  0 ) )
2 abs0 11700 . . . 4  |-  ( abs `  0 )  =  0
3 bcs.1 . . . . . 6  |-  A  e. 
~H
4 normge0 21630 . . . . . 6  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
53, 4ax-mp 10 . . . . 5  |-  0  <_  ( normh `  A )
6 bcs.2 . . . . . 6  |-  B  e. 
~H
7 normge0 21630 . . . . . 6  |-  ( B  e.  ~H  ->  0  <_  ( normh `  B )
)
86, 7ax-mp 10 . . . . 5  |-  0  <_  ( normh `  B )
93normcli 21635 . . . . . 6  |-  ( normh `  A )  e.  RR
106normcli 21635 . . . . . 6  |-  ( normh `  B )  e.  RR
119, 10mulge0i 9253 . . . . 5  |-  ( ( 0  <_  ( normh `  A )  /\  0  <_  ( normh `  B )
)  ->  0  <_  ( ( normh `  A )  x.  ( normh `  B )
) )
125, 8, 11mp2an 656 . . . 4  |-  0  <_  ( ( normh `  A
)  x.  ( normh `  B ) )
132, 12eqbrtri 3982 . . 3  |-  ( abs `  0 )  <_ 
( ( normh `  A
)  x.  ( normh `  B ) )
141, 13syl6eqbr 4000 . 2  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  <_  (
( normh `  A )  x.  ( normh `  B )
) )
15 df-ne 2421 . . . 4  |-  ( ( A  .ih  B )  =/=  0  <->  -.  ( A  .ih  B )  =  0 )
166, 3his1i 21604 . . . . . . . 8  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
1716oveq2i 5768 . . . . . . 7  |-  ( ( ( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  x.  ( B  .ih  A
) )  =  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  x.  ( * `  ( A  .ih  B ) ) )
1817oveq2i 5768 . . . . . 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 21585 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
20 abslem2 11753 . . . . . . 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 654 . . . . . 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 2301 . . . . 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 11811 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =  0  <->  ( A  .ih  B )  =  0 )
2423necon3bii 2451 . . . . . . 7  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  <->  ( A  .ih  B )  =/=  0
)
2519abscli 11808 . . . . . . . . . 10  |-  ( abs `  ( A  .ih  B
) )  e.  RR
2625recni 8782 . . . . . . . . 9  |-  ( abs `  ( A  .ih  B
) )  e.  CC
2719, 26divclzi 9428 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
2819, 26divreczi 9431 . . . . . . . . . 10  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  =  ( ( A  .ih  B )  x.  ( 1  /  ( abs `  ( A  .ih  B ) ) ) ) )
2928fveq2d 5427 . . . . . . . . 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 9418 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
31 absmul 11709 . . . . . . . . . . 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 647 . . . . . . . . . 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 9457 . . . . . . . . . . . 12  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  RR )
34 0re 8771 . . . . . . . . . . . . . 14  |-  0  e.  RR
3533, 34jctil 525 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
0  e.  RR  /\  ( 1  /  ( abs `  ( A  .ih  B ) ) )  e.  RR ) )
3619absgt0i 11812 . . . . . . . . . . . . . . 15  |-  ( ( A  .ih  B )  =/=  0  <->  0  <  ( abs `  ( A 
.ih  B ) ) )
3724, 36bitri 242 . . . . . . . . . . . . . 14  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  <->  0  <  ( abs `  ( A 
.ih  B ) ) )
3825recgt0i 9594 . . . . . . . . . . . . . 14  |-  ( 0  <  ( abs `  ( A  .ih  B ) )  ->  0  <  (
1  /  ( abs `  ( A  .ih  B
) ) ) )
3937, 38sylbi 189 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  0  <  ( 1  /  ( abs `  ( A  .ih  B ) ) ) )
40 ltle 8843 . . . . . . . . . . . . 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 58 . . . . . . . . . . . 12  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  0  <_  ( 1  /  ( abs `  ( A  .ih  B ) ) ) )
4233, 41absidd 11835 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  ( 1  /  ( abs `  ( A  .ih  B
) ) ) )
4342oveq2d 5773 . . . . . . . . . 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 2288 . . . . . . . . 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 9420 . . . . . . . . 9  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( abs `  ( A  .ih  B ) )  x.  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  1 )
4629, 44, 453eqtrd 2292 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( ( A 
.ih  B )  / 
( abs `  ( A  .ih  B ) ) ) )  =  1 )
4727, 46jca 520 . . . . . . 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 206 . . . . . 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 21623 . . . . . 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 3983 . . . 4  |-  ( ( A  .ih  B )  =/=  0  ->  (
2  x.  ( abs `  ( A  .ih  B
) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
5215, 51sylbir 206 . . 3  |-  ( -.  ( A  .ih  B
)  =  0  -> 
( 2  x.  ( abs `  ( A  .ih  B ) ) )  <_ 
( 2  x.  (
( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
5310recni 8782 . . . . . 6  |-  ( normh `  B )  e.  CC
549recni 8782 . . . . . 6  |-  ( normh `  A )  e.  CC
55 normval 21628 . . . . . . . 8  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
566, 55ax-mp 10 . . . . . . 7  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
57 normval 21628 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
583, 57ax-mp 10 . . . . . . 7  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
5956, 58oveq12i 5769 . . . . . 6  |-  ( (
normh `  B )  x.  ( normh `  A )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6053, 54, 59mulcomli 8777 . . . . 5  |-  ( (
normh `  A )  x.  ( normh `  B )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6160breq2i 3971 . . . 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 9761 . . . . 5  |-  0  <  2
63 hiidge0 21602 . . . . . . . 8  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
64 hiidrcl 21599 . . . . . . . . . 10  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
656, 64ax-mp 10 . . . . . . . . 9  |-  ( B 
.ih  B )  e.  RR
6665sqrcli 11785 . . . . . . . 8  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
676, 63, 66mp2b 11 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
68 hiidge0 21602 . . . . . . . 8  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
69 hiidrcl 21599 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
703, 69ax-mp 10 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  RR
7170sqrcli 11785 . . . . . . . 8  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
723, 68, 71mp2b 11 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
7367, 72remulcli 8784 . . . . . 6  |-  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )  e.  RR
74 2re 9748 . . . . . 6  |-  2  e.  RR
7525, 73, 74lemul2i 9613 . . . . 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 10 . . . 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 242 . . 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 205 . 2  |-  ( -.  ( A  .ih  B
)  =  0  -> 
( abs `  ( A  .ih  B ) )  <_  ( ( normh `  A )  x.  ( normh `  B ) ) )
7914, 78pm2.61i 158 1  |-  ( abs `  ( A  .ih  B
) )  <_  (
( normh `  A )  x.  ( normh `  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    < clt 8800    <_ cle 8801    / cdiv 9356   2c2 9728   *ccj 11511   sqrcsqr 11648   abscabs 11649   ~Hchil 21424    .ih csp 21427   normhcno 21428
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 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-hfvadd 21505  ax-hv0cl 21508  ax-hfvmul 21510  ax-hvmulass 21512  ax-hvmul0 21515  ax-hfi 21583  ax-his1 21586  ax-his2 21587  ax-his3 21588  ax-his4 21589
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-seq 10978  df-exp 11036  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-hnorm 21473  df-hvsub 21476
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