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Theorem bcsiALT 22071
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 5632 . . 3  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  =  ( abs `  0 ) )
2 abs0 11977 . . . 4  |-  ( abs `  0 )  =  0
3 bcs.1 . . . . . 6  |-  A  e. 
~H
4 normge0 22018 . . . . . 6  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
53, 4ax-mp 8 . . . . 5  |-  0  <_  ( normh `  A )
6 bcs.2 . . . . . 6  |-  B  e. 
~H
7 normge0 22018 . . . . . 6  |-  ( B  e.  ~H  ->  0  <_  ( normh `  B )
)
86, 7ax-mp 8 . . . . 5  |-  0  <_  ( normh `  B )
93normcli 22023 . . . . . 6  |-  ( normh `  A )  e.  RR
106normcli 22023 . . . . . 6  |-  ( normh `  B )  e.  RR
119, 10mulge0i 9467 . . . . 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 4144 . . 3  |-  ( abs `  0 )  <_ 
( ( normh `  A
)  x.  ( normh `  B ) )
141, 13syl6eqbr 4162 . 2  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  <_  (
( normh `  A )  x.  ( normh `  B )
) )
15 df-ne 2531 . . . 4  |-  ( ( A  .ih  B )  =/=  0  <->  -.  ( A  .ih  B )  =  0 )
166, 3his1i 21992 . . . . . . . 8  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
1716oveq2i 5992 . . . . . . 7  |-  ( ( ( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  x.  ( B  .ih  A
) )  =  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  x.  ( * `  ( A  .ih  B ) ) )
1817oveq2i 5992 . . . . . 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 21973 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
20 abslem2 12030 . . . . . . 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 2411 . . . . 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 12088 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =  0  <->  ( A  .ih  B )  =  0 )
2423necon3bii 2561 . . . . . . 7  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  <->  ( A  .ih  B )  =/=  0
)
2519abscli 12085 . . . . . . . . . 10  |-  ( abs `  ( A  .ih  B
) )  e.  RR
2625recni 8996 . . . . . . . . 9  |-  ( abs `  ( A  .ih  B
) )  e.  CC
2719, 26divclzi 9642 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
2819, 26divreczi 9645 . . . . . . . . . 10  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  =  ( ( A  .ih  B )  x.  ( 1  /  ( abs `  ( A  .ih  B ) ) ) ) )
2928fveq2d 5636 . . . . . . . . 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 9632 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
31 absmul 11986 . . . . . . . . . . 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 9671 . . . . . . . . . . . 12  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  RR )
34 0re 8985 . . . . . . . . . . . . . 14  |-  0  e.  RR
3533, 34jctil 523 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
0  e.  RR  /\  ( 1  /  ( abs `  ( A  .ih  B ) ) )  e.  RR ) )
3619absgt0i 12089 . . . . . . . . . . . . . . 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 9808 . . . . . . . . . . . . . 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 9057 . . . . . . . . . . . . 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 12112 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  ( 1  /  ( abs `  ( A  .ih  B
) ) ) )
4342oveq2d 5997 . . . . . . . . . 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 2398 . . . . . . . . 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 9634 . . . . . . . . 9  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( abs `  ( A  .ih  B ) )  x.  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  1 )
4629, 44, 453eqtrd 2402 . . . . . . . 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 22011 . . . . . 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 4145 . . . 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 8996 . . . . . 6  |-  ( normh `  B )  e.  CC
549recni 8996 . . . . . 6  |-  ( normh `  A )  e.  CC
55 normval 22016 . . . . . . . 8  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
566, 55ax-mp 8 . . . . . . 7  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
57 normval 22016 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
583, 57ax-mp 8 . . . . . . 7  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
5956, 58oveq12i 5993 . . . . . 6  |-  ( (
normh `  B )  x.  ( normh `  A )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6053, 54, 59mulcomli 8991 . . . . 5  |-  ( (
normh `  A )  x.  ( normh `  B )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6160breq2i 4133 . . . 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 9975 . . . . 5  |-  0  <  2
63 hiidge0 21990 . . . . . . . 8  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
64 hiidrcl 21987 . . . . . . . . . 10  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
656, 64ax-mp 8 . . . . . . . . 9  |-  ( B 
.ih  B )  e.  RR
6665sqrcli 12062 . . . . . . . 8  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
676, 63, 66mp2b 9 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
68 hiidge0 21990 . . . . . . . 8  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
69 hiidrcl 21987 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
703, 69ax-mp 8 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  RR
7170sqrcli 12062 . . . . . . . 8  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
723, 68, 71mp2b 9 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
7367, 72remulcli 8998 . . . . . 6  |-  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )  e.  RR
74 2re 9962 . . . . . 6  |-  2  e.  RR
7525, 73, 74lemul2i 9827 . . . . 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 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    < clt 9014    <_ cle 9015    / cdiv 9570   2c2 9942   *ccj 11788   sqrcsqr 11925   abscabs 11926   ~Hchil 21812    .ih csp 21815   normhcno 21816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-hfvadd 21893  ax-hv0cl 21896  ax-hfvmul 21898  ax-hvmulass 21900  ax-hvmul0 21903  ax-hfi 21971  ax-his1 21974  ax-his2 21975  ax-his3 21976  ax-his4 21977
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-hnorm 21861  df-hvsub 21864
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