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Theorem bcsiALT 28958
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 6658 . . 3 ((𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) = (abs‘0))
2 abs0 14641 . . . 4 (abs‘0) = 0
3 bcs.1 . . . . . 6 𝐴 ∈ ℋ
4 normge0 28905 . . . . . 6 (𝐴 ∈ ℋ → 0 ≤ (norm𝐴))
53, 4ax-mp 5 . . . . 5 0 ≤ (norm𝐴)
6 bcs.2 . . . . . 6 𝐵 ∈ ℋ
7 normge0 28905 . . . . . 6 (𝐵 ∈ ℋ → 0 ≤ (norm𝐵))
86, 7ax-mp 5 . . . . 5 0 ≤ (norm𝐵)
93normcli 28910 . . . . . 6 (norm𝐴) ∈ ℝ
106normcli 28910 . . . . . 6 (norm𝐵) ∈ ℝ
119, 10mulge0i 11179 . . . . 5 ((0 ≤ (norm𝐴) ∧ 0 ≤ (norm𝐵)) → 0 ≤ ((norm𝐴) · (norm𝐵)))
125, 8, 11mp2an 691 . . . 4 0 ≤ ((norm𝐴) · (norm𝐵))
132, 12eqbrtri 5073 . . 3 (abs‘0) ≤ ((norm𝐴) · (norm𝐵))
141, 13eqbrtrdi 5091 . 2 ((𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)))
15 df-ne 3015 . . . 4 ((𝐴 ·ih 𝐵) ≠ 0 ↔ ¬ (𝐴 ·ih 𝐵) = 0)
166, 3his1i 28879 . . . . . . . 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 28860 . . . . . . 7 (𝐴 ·ih 𝐵) ∈ ℂ
20 abslem2 14695 . . . . . . 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 689 . . . . . 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 2872 . . . . 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 14754 . . . . . . . 8 ((abs‘(𝐴 ·ih 𝐵)) = 0 ↔ (𝐴 ·ih 𝐵) = 0)
2423necon3bii 3066 . . . . . . 7 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 ↔ (𝐴 ·ih 𝐵) ≠ 0)
2519abscli 14751 . . . . . . . . . 10 (abs‘(𝐴 ·ih 𝐵)) ∈ ℝ
2625recni 10647 . . . . . . . . 9 (abs‘(𝐴 ·ih 𝐵)) ∈ ℂ
2719, 26divclzi 11367 . . . . . . . 8 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ)
2819, 26divreczi 11370 . . . . . . . . . 10 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) = ((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵)))))
2928fveq2d 6662 . . . . . . . . 9 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = (abs‘((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵))))))
3026recclzi 11357 . . . . . . . . . . 11 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ)
31 absmul 14650 . . . . . . . . . . 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 590 . . . . . . . . . 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 11396 . . . . . . . . . . . 12 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℝ)
34 0re 10635 . . . . . . . . . . . . . 14 0 ∈ ℝ
3533, 34jctil 523 . . . . . . . . . . . . 13 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (0 ∈ ℝ ∧ (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℝ))
3619absgt0i 14755 . . . . . . . . . . . . . . 15 ((𝐴 ·ih 𝐵) ≠ 0 ↔ 0 < (abs‘(𝐴 ·ih 𝐵)))
3724, 36bitri 278 . . . . . . . . . . . . . 14 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 ↔ 0 < (abs‘(𝐴 ·ih 𝐵)))
3825recgt0i 11537 . . . . . . . . . . . . . 14 (0 < (abs‘(𝐴 ·ih 𝐵)) → 0 < (1 / (abs‘(𝐴 ·ih 𝐵))))
3937, 38sylbi 220 . . . . . . . . . . . . 13 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → 0 < (1 / (abs‘(𝐴 ·ih 𝐵))))
40 ltle 10721 . . . . . . . . . . . . 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 14778 . . . . . . . . . . 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 2859 . . . . . . . . 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 11359 . . . . . . . . 9 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·ih 𝐵)) · (1 / (abs‘(𝐴 ·ih 𝐵)))) = 1)
4629, 44, 453eqtrd 2863 . . . . . . . 8 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = 1)
4727, 46jca 515 . . . . . . 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 238 . . . . . 6 ((𝐴 ·ih 𝐵) ≠ 0 → (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ ∧ (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = 1))
493, 6normlem7tALT 28898 . . . . . 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 5074 . . . 4 ((𝐴 ·ih 𝐵) ≠ 0 → (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
5215, 51sylbir 238 . . 3 (¬ (𝐴 ·ih 𝐵) = 0 → (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
5310recni 10647 . . . . . 6 (norm𝐵) ∈ ℂ
549recni 10647 . . . . . 6 (norm𝐴) ∈ ℂ
55 normval 28903 . . . . . . . 8 (𝐵 ∈ ℋ → (norm𝐵) = (√‘(𝐵 ·ih 𝐵)))
566, 55ax-mp 5 . . . . . . 7 (norm𝐵) = (√‘(𝐵 ·ih 𝐵))
57 normval 28903 . . . . . . . 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 10642 . . . . 5 ((norm𝐴) · (norm𝐵)) = ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))
6160breq2i 5060 . . . 4 ((abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)) ↔ (abs‘(𝐴 ·ih 𝐵)) ≤ ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))))
62 2pos 11733 . . . . 5 0 < 2
63 hiidge0 28877 . . . . . . . 8 (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·ih 𝐵))
64 hiidrcl 28874 . . . . . . . . . 10 (𝐵 ∈ ℋ → (𝐵 ·ih 𝐵) ∈ ℝ)
656, 64ax-mp 5 . . . . . . . . 9 (𝐵 ·ih 𝐵) ∈ ℝ
6665sqrtcli 14727 . . . . . . . 8 (0 ≤ (𝐵 ·ih 𝐵) → (√‘(𝐵 ·ih 𝐵)) ∈ ℝ)
676, 63, 66mp2b 10 . . . . . . 7 (√‘(𝐵 ·ih 𝐵)) ∈ ℝ
68 hiidge0 28877 . . . . . . . 8 (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
69 hiidrcl 28874 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
703, 69ax-mp 5 . . . . . . . . 9 (𝐴 ·ih 𝐴) ∈ ℝ
7170sqrtcli 14727 . . . . . . . 8 (0 ≤ (𝐴 ·ih 𝐴) → (√‘(𝐴 ·ih 𝐴)) ∈ ℝ)
723, 68, 71mp2b 10 . . . . . . 7 (√‘(𝐴 ·ih 𝐴)) ∈ ℝ
7367, 72remulcli 10649 . . . . . 6 ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))) ∈ ℝ
74 2re 11704 . . . . . 6 2 ∈ ℝ
7525, 73, 74lemul2i 11555 . . . . 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 278 . . 3 ((abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)) ↔ (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
7852, 77sylibr 237 . 2 (¬ (𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)))
7914, 78pm2.61i 185 1 (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7145  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534   < clt 10667  cle 10668   / cdiv 11289  2c2 11685  ccj 14451  csqrt 14588  abscabs 14589  chba 28698   ·ih csp 28701  normcno 28702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-hfvadd 28779  ax-hv0cl 28782  ax-hfvmul 28784  ax-hvmulass 28786  ax-hvmul0 28789  ax-hfi 28858  ax-his1 28861  ax-his2 28862  ax-his3 28863  ax-his4 28864
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-sup 8897  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11693  df-3 11694  df-4 11695  df-n0 11891  df-z 11975  df-uz 12237  df-rp 12383  df-seq 13370  df-exp 13431  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-hnorm 28747  df-hvsub 28750
This theorem is referenced by: (None)
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