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Theorem bcsiALT 28883
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 ∈ ℋ
bcs.2 𝐵 ∈ ℋ
Assertion
Ref Expression
bcsiALT (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵))

Proof of Theorem bcsiALT
StepHypRef Expression
1 fveq2 6663 . . 3 ((𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) = (abs‘0))
2 abs0 14633 . . . 4 (abs‘0) = 0
3 bcs.1 . . . . . 6 𝐴 ∈ ℋ
4 normge0 28830 . . . . . 6 (𝐴 ∈ ℋ → 0 ≤ (norm𝐴))
53, 4ax-mp 5 . . . . 5 0 ≤ (norm𝐴)
6 bcs.2 . . . . . 6 𝐵 ∈ ℋ
7 normge0 28830 . . . . . 6 (𝐵 ∈ ℋ → 0 ≤ (norm𝐵))
86, 7ax-mp 5 . . . . 5 0 ≤ (norm𝐵)
93normcli 28835 . . . . . 6 (norm𝐴) ∈ ℝ
106normcli 28835 . . . . . 6 (norm𝐵) ∈ ℝ
119, 10mulge0i 11175 . . . . 5 ((0 ≤ (norm𝐴) ∧ 0 ≤ (norm𝐵)) → 0 ≤ ((norm𝐴) · (norm𝐵)))
125, 8, 11mp2an 688 . . . 4 0 ≤ ((norm𝐴) · (norm𝐵))
132, 12eqbrtri 5078 . . 3 (abs‘0) ≤ ((norm𝐴) · (norm𝐵))
141, 13eqbrtrdi 5096 . 2 ((𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)))
15 df-ne 3014 . . . 4 ((𝐴 ·ih 𝐵) ≠ 0 ↔ ¬ (𝐴 ·ih 𝐵) = 0)
166, 3his1i 28804 . . . . . . . 8 (𝐵 ·ih 𝐴) = (∗‘(𝐴 ·ih 𝐵))
1716oveq2i 7156 . . . . . . 7 (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (𝐵 ·ih 𝐴)) = (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (∗‘(𝐴 ·ih 𝐵)))
1817oveq2i 7156 . . . . . 6 (((∗‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) · (𝐴 ·ih 𝐵)) + (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (𝐵 ·ih 𝐴))) = (((∗‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) · (𝐴 ·ih 𝐵)) + (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (∗‘(𝐴 ·ih 𝐵))))
193, 6hicli 28785 . . . . . . 7 (𝐴 ·ih 𝐵) ∈ ℂ
20 abslem2 14687 . . . . . . 7 (((𝐴 ·ih 𝐵) ∈ ℂ ∧ (𝐴 ·ih 𝐵) ≠ 0) → (((∗‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) · (𝐴 ·ih 𝐵)) + (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (∗‘(𝐴 ·ih 𝐵)))) = (2 · (abs‘(𝐴 ·ih 𝐵))))
2119, 20mpan 686 . . . . . 6 ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) · (𝐴 ·ih 𝐵)) + (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (∗‘(𝐴 ·ih 𝐵)))) = (2 · (abs‘(𝐴 ·ih 𝐵))))
2218, 21syl5req 2866 . . . . 5 ((𝐴 ·ih 𝐵) ≠ 0 → (2 · (abs‘(𝐴 ·ih 𝐵))) = (((∗‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) · (𝐴 ·ih 𝐵)) + (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (𝐵 ·ih 𝐴))))
2319abs00i 14746 . . . . . . . 8 ((abs‘(𝐴 ·ih 𝐵)) = 0 ↔ (𝐴 ·ih 𝐵) = 0)
2423necon3bii 3065 . . . . . . 7 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 ↔ (𝐴 ·ih 𝐵) ≠ 0)
2519abscli 14743 . . . . . . . . . 10 (abs‘(𝐴 ·ih 𝐵)) ∈ ℝ
2625recni 10643 . . . . . . . . 9 (abs‘(𝐴 ·ih 𝐵)) ∈ ℂ
2719, 26divclzi 11363 . . . . . . . 8 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ)
2819, 26divreczi 11366 . . . . . . . . . 10 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) = ((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵)))))
2928fveq2d 6667 . . . . . . . . 9 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = (abs‘((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵))))))
3026recclzi 11353 . . . . . . . . . . 11 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ)
31 absmul 14642 . . . . . . . . . . 11 (((𝐴 ·ih 𝐵) ∈ ℂ ∧ (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ) → (abs‘((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵))))) = ((abs‘(𝐴 ·ih 𝐵)) · (abs‘(1 / (abs‘(𝐴 ·ih 𝐵))))))
3219, 30, 31sylancr 587 . . . . . . . . . 10 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵))))) = ((abs‘(𝐴 ·ih 𝐵)) · (abs‘(1 / (abs‘(𝐴 ·ih 𝐵))))))
3325rerecclzi 11392 . . . . . . . . . . . 12 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℝ)
34 0re 10631 . . . . . . . . . . . . . 14 0 ∈ ℝ
3533, 34jctil 520 . . . . . . . . . . . . 13 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (0 ∈ ℝ ∧ (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℝ))
3619absgt0i 14747 . . . . . . . . . . . . . . 15 ((𝐴 ·ih 𝐵) ≠ 0 ↔ 0 < (abs‘(𝐴 ·ih 𝐵)))
3724, 36bitri 276 . . . . . . . . . . . . . 14 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 ↔ 0 < (abs‘(𝐴 ·ih 𝐵)))
3825recgt0i 11533 . . . . . . . . . . . . . 14 (0 < (abs‘(𝐴 ·ih 𝐵)) → 0 < (1 / (abs‘(𝐴 ·ih 𝐵))))
3937, 38sylbi 218 . . . . . . . . . . . . 13 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → 0 < (1 / (abs‘(𝐴 ·ih 𝐵))))
40 ltle 10717 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℝ) → (0 < (1 / (abs‘(𝐴 ·ih 𝐵))) → 0 ≤ (1 / (abs‘(𝐴 ·ih 𝐵)))))
4135, 39, 40sylc 65 . . . . . . . . . . . 12 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → 0 ≤ (1 / (abs‘(𝐴 ·ih 𝐵))))
4233, 41absidd 14770 . . . . . . . . . . 11 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘(1 / (abs‘(𝐴 ·ih 𝐵)))) = (1 / (abs‘(𝐴 ·ih 𝐵))))
4342oveq2d 7161 . . . . . . . . . 10 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·ih 𝐵)) · (abs‘(1 / (abs‘(𝐴 ·ih 𝐵))))) = ((abs‘(𝐴 ·ih 𝐵)) · (1 / (abs‘(𝐴 ·ih 𝐵)))))
4432, 43eqtrd 2853 . . . . . . . . 9 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵))))) = ((abs‘(𝐴 ·ih 𝐵)) · (1 / (abs‘(𝐴 ·ih 𝐵)))))
4526recidzi 11355 . . . . . . . . 9 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·ih 𝐵)) · (1 / (abs‘(𝐴 ·ih 𝐵)))) = 1)
4629, 44, 453eqtrd 2857 . . . . . . . 8 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = 1)
4727, 46jca 512 . . . . . . 7 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ ∧ (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = 1))
4824, 47sylbir 236 . . . . . 6 ((𝐴 ·ih 𝐵) ≠ 0 → (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ ∧ (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = 1))
493, 6normlem7tALT 28823 . . . . . 6 ((((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ ∧ (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = 1) → (((∗‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) · (𝐴 ·ih 𝐵)) + (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (𝐵 ·ih 𝐴))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
5048, 49syl 17 . . . . 5 ((𝐴 ·ih 𝐵) ≠ 0 → (((∗‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) · (𝐴 ·ih 𝐵)) + (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (𝐵 ·ih 𝐴))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
5122, 50eqbrtrd 5079 . . . 4 ((𝐴 ·ih 𝐵) ≠ 0 → (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
5215, 51sylbir 236 . . 3 (¬ (𝐴 ·ih 𝐵) = 0 → (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
5310recni 10643 . . . . . 6 (norm𝐵) ∈ ℂ
549recni 10643 . . . . . 6 (norm𝐴) ∈ ℂ
55 normval 28828 . . . . . . . 8 (𝐵 ∈ ℋ → (norm𝐵) = (√‘(𝐵 ·ih 𝐵)))
566, 55ax-mp 5 . . . . . . 7 (norm𝐵) = (√‘(𝐵 ·ih 𝐵))
57 normval 28828 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) = (√‘(𝐴 ·ih 𝐴)))
583, 57ax-mp 5 . . . . . . 7 (norm𝐴) = (√‘(𝐴 ·ih 𝐴))
5956, 58oveq12i 7157 . . . . . 6 ((norm𝐵) · (norm𝐴)) = ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))
6053, 54, 59mulcomli 10638 . . . . 5 ((norm𝐴) · (norm𝐵)) = ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))
6160breq2i 5065 . . . 4 ((abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)) ↔ (abs‘(𝐴 ·ih 𝐵)) ≤ ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))))
62 2pos 11728 . . . . 5 0 < 2
63 hiidge0 28802 . . . . . . . 8 (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·ih 𝐵))
64 hiidrcl 28799 . . . . . . . . . 10 (𝐵 ∈ ℋ → (𝐵 ·ih 𝐵) ∈ ℝ)
656, 64ax-mp 5 . . . . . . . . 9 (𝐵 ·ih 𝐵) ∈ ℝ
6665sqrtcli 14719 . . . . . . . 8 (0 ≤ (𝐵 ·ih 𝐵) → (√‘(𝐵 ·ih 𝐵)) ∈ ℝ)
676, 63, 66mp2b 10 . . . . . . 7 (√‘(𝐵 ·ih 𝐵)) ∈ ℝ
68 hiidge0 28802 . . . . . . . 8 (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
69 hiidrcl 28799 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
703, 69ax-mp 5 . . . . . . . . 9 (𝐴 ·ih 𝐴) ∈ ℝ
7170sqrtcli 14719 . . . . . . . 8 (0 ≤ (𝐴 ·ih 𝐴) → (√‘(𝐴 ·ih 𝐴)) ∈ ℝ)
723, 68, 71mp2b 10 . . . . . . 7 (√‘(𝐴 ·ih 𝐴)) ∈ ℝ
7367, 72remulcli 10645 . . . . . 6 ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))) ∈ ℝ
74 2re 11699 . . . . . 6 2 ∈ ℝ
7525, 73, 74lemul2i 11551 . . . . 5 (0 < 2 → ((abs‘(𝐴 ·ih 𝐵)) ≤ ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))) ↔ (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))))))
7662, 75ax-mp 5 . . . 4 ((abs‘(𝐴 ·ih 𝐵)) ≤ ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))) ↔ (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
7761, 76bitri 276 . . 3 ((abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)) ↔ (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
7852, 77sylibr 235 . 2 (¬ (𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)))
7914, 78pm2.61i 183 1 (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  cfv 6348  (class class class)co 7145  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664   / cdiv 11285  2c2 11680  ccj 14443  csqrt 14580  abscabs 14581  chba 28623   ·ih csp 28626  normcno 28627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-hfvadd 28704  ax-hv0cl 28707  ax-hfvmul 28709  ax-hvmulass 28711  ax-hvmul0 28714  ax-hfi 28783  ax-his1 28786  ax-his2 28787  ax-his3 28788  ax-his4 28789
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-hnorm 28672  df-hvsub 28675
This theorem is referenced by: (None)
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