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Theorem bcsiALT 21588
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 5377 . . 3  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  =  ( abs `  0 ) )
2 abs0 11647 . . . 4  |-  ( abs `  0 )  =  0
3 bcs.1 . . . . . 6  |-  A  e. 
~H
4 normge0 21535 . . . . . 6  |-  ( A  e.  ~H  ->  0  <_  ( normh `  A )
)
53, 4ax-mp 10 . . . . 5  |-  0  <_  ( normh `  A )
6 bcs.2 . . . . . 6  |-  B  e. 
~H
7 normge0 21535 . . . . . 6  |-  ( B  e.  ~H  ->  0  <_  ( normh `  B )
)
86, 7ax-mp 10 . . . . 5  |-  0  <_  ( normh `  B )
93normcli 21540 . . . . . 6  |-  ( normh `  A )  e.  RR
106normcli 21540 . . . . . 6  |-  ( normh `  B )  e.  RR
119, 10mulge0i 9200 . . . . 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 3939 . . 3  |-  ( abs `  0 )  <_ 
( ( normh `  A
)  x.  ( normh `  B ) )
141, 13syl6eqbr 3957 . 2  |-  ( ( A  .ih  B )  =  0  ->  ( abs `  ( A  .ih  B ) )  <_  (
( normh `  A )  x.  ( normh `  B )
) )
15 df-ne 2414 . . . 4  |-  ( ( A  .ih  B )  =/=  0  <->  -.  ( A  .ih  B )  =  0 )
166, 3his1i 21509 . . . . . . . 8  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
1716oveq2i 5721 . . . . . . 7  |-  ( ( ( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  x.  ( B  .ih  A
) )  =  ( ( ( A  .ih  B )  /  ( abs `  ( A  .ih  B
) ) )  x.  ( * `  ( A  .ih  B ) ) )
1817oveq2i 5721 . . . . . 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 21490 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
20 abslem2 11700 . . . . . . 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 2298 . . . . 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 11758 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =  0  <->  ( A  .ih  B )  =  0 )
2423necon3bii 2444 . . . . . . 7  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  <->  ( A  .ih  B )  =/=  0
)
2519abscli 11755 . . . . . . . . . 10  |-  ( abs `  ( A  .ih  B
) )  e.  RR
2625recni 8729 . . . . . . . . 9  |-  ( abs `  ( A  .ih  B
) )  e.  CC
2719, 26divclzi 9375 . . . . . . . 8  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
2819, 26divreczi 9378 . . . . . . . . . 10  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( A  .ih  B
)  /  ( abs `  ( A  .ih  B
) ) )  =  ( ( A  .ih  B )  x.  ( 1  /  ( abs `  ( A  .ih  B ) ) ) ) )
2928fveq2d 5381 . . . . . . . . 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 9365 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  CC )
31 absmul 11656 . . . . . . . . . . 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 9404 . . . . . . . . . . . 12  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
1  /  ( abs `  ( A  .ih  B
) ) )  e.  RR )
34 0re 8718 . . . . . . . . . . . . . 14  |-  0  e.  RR
3533, 34jctil 525 . . . . . . . . . . . . 13  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
0  e.  RR  /\  ( 1  /  ( abs `  ( A  .ih  B ) ) )  e.  RR ) )
3619absgt0i 11759 . . . . . . . . . . . . . . 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 9541 . . . . . . . . . . . . . 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 8790 . . . . . . . . . . . . 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 11782 . . . . . . . . . . 11  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  ( abs `  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  ( 1  /  ( abs `  ( A  .ih  B
) ) ) )
4342oveq2d 5726 . . . . . . . . . 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 2285 . . . . . . . . 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 9367 . . . . . . . . 9  |-  ( ( abs `  ( A 
.ih  B ) )  =/=  0  ->  (
( abs `  ( A  .ih  B ) )  x.  ( 1  / 
( abs `  ( A  .ih  B ) ) ) )  =  1 )
4629, 44, 453eqtrd 2289 . . . . . . . 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 21528 . . . . . 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 3940 . . . 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 8729 . . . . . 6  |-  ( normh `  B )  e.  CC
549recni 8729 . . . . . 6  |-  ( normh `  A )  e.  CC
55 normval 21533 . . . . . . . 8  |-  ( B  e.  ~H  ->  ( normh `  B )  =  ( sqr `  ( B  .ih  B ) ) )
566, 55ax-mp 10 . . . . . . 7  |-  ( normh `  B )  =  ( sqr `  ( B 
.ih  B ) )
57 normval 21533 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
583, 57ax-mp 10 . . . . . . 7  |-  ( normh `  A )  =  ( sqr `  ( A 
.ih  A ) )
5956, 58oveq12i 5722 . . . . . 6  |-  ( (
normh `  B )  x.  ( normh `  A )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6053, 54, 59mulcomli 8724 . . . . 5  |-  ( (
normh `  A )  x.  ( normh `  B )
)  =  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )
6160breq2i 3928 . . . 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 9708 . . . . 5  |-  0  <  2
63 hiidge0 21507 . . . . . . . 8  |-  ( B  e.  ~H  ->  0  <_  ( B  .ih  B
) )
64 hiidrcl 21504 . . . . . . . . . 10  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
656, 64ax-mp 10 . . . . . . . . 9  |-  ( B 
.ih  B )  e.  RR
6665sqrcli 11732 . . . . . . . 8  |-  ( 0  <_  ( B  .ih  B )  ->  ( sqr `  ( B  .ih  B
) )  e.  RR )
676, 63, 66mp2b 11 . . . . . . 7  |-  ( sqr `  ( B  .ih  B
) )  e.  RR
68 hiidge0 21507 . . . . . . . 8  |-  ( A  e.  ~H  ->  0  <_  ( A  .ih  A
) )
69 hiidrcl 21504 . . . . . . . . . 10  |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
703, 69ax-mp 10 . . . . . . . . 9  |-  ( A 
.ih  A )  e.  RR
7170sqrcli 11732 . . . . . . . 8  |-  ( 0  <_  ( A  .ih  A )  ->  ( sqr `  ( A  .ih  A
) )  e.  RR )
723, 68, 71mp2b 11 . . . . . . 7  |-  ( sqr `  ( A  .ih  A
) )  e.  RR
7367, 72remulcli 8731 . . . . . 6  |-  ( ( sqr `  ( B 
.ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) )  e.  RR
74 2re 9695 . . . . . 6  |-  2  e.  RR
7525, 73, 74lemul2i 9560 . . . . 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 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    < clt 8747    <_ cle 8748    / cdiv 9303   2c2 9675   *ccj 11458   sqrcsqr 11595   abscabs 11596   ~Hchil 21329    .ih csp 21332   normhcno 21333
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-hfvadd 21410  ax-hv0cl 21413  ax-hfvmul 21415  ax-hvmulass 21417  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his2 21492  ax-his3 21493  ax-his4 21494
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-hnorm 21378  df-hvsub 21381
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