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Theorem bcsiALT 30687
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 6891 . . 3 ((𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) = (abs‘0))
2 abs0 15236 . . . 4 (abs‘0) = 0
3 bcs.1 . . . . . 6 𝐴 ∈ ℋ
4 normge0 30634 . . . . . 6 (𝐴 ∈ ℋ → 0 ≤ (norm𝐴))
53, 4ax-mp 5 . . . . 5 0 ≤ (norm𝐴)
6 bcs.2 . . . . . 6 𝐵 ∈ ℋ
7 normge0 30634 . . . . . 6 (𝐵 ∈ ℋ → 0 ≤ (norm𝐵))
86, 7ax-mp 5 . . . . 5 0 ≤ (norm𝐵)
93normcli 30639 . . . . . 6 (norm𝐴) ∈ ℝ
106normcli 30639 . . . . . 6 (norm𝐵) ∈ ℝ
119, 10mulge0i 11765 . . . . 5 ((0 ≤ (norm𝐴) ∧ 0 ≤ (norm𝐵)) → 0 ≤ ((norm𝐴) · (norm𝐵)))
125, 8, 11mp2an 690 . . . 4 0 ≤ ((norm𝐴) · (norm𝐵))
132, 12eqbrtri 5169 . . 3 (abs‘0) ≤ ((norm𝐴) · (norm𝐵))
141, 13eqbrtrdi 5187 . 2 ((𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)))
15 df-ne 2941 . . . 4 ((𝐴 ·ih 𝐵) ≠ 0 ↔ ¬ (𝐴 ·ih 𝐵) = 0)
166, 3his1i 30608 . . . . . . . 8 (𝐵 ·ih 𝐴) = (∗‘(𝐴 ·ih 𝐵))
1716oveq2i 7422 . . . . . . 7 (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (𝐵 ·ih 𝐴)) = (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) · (∗‘(𝐴 ·ih 𝐵)))
1817oveq2i 7422 . . . . . 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 30589 . . . . . . 7 (𝐴 ·ih 𝐵) ∈ ℂ
20 abslem2 15290 . . . . . . 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 688 . . . . . 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, 21eqtr2id 2785 . . . . 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 15349 . . . . . . . 8 ((abs‘(𝐴 ·ih 𝐵)) = 0 ↔ (𝐴 ·ih 𝐵) = 0)
2423necon3bii 2993 . . . . . . 7 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 ↔ (𝐴 ·ih 𝐵) ≠ 0)
2519abscli 15346 . . . . . . . . . 10 (abs‘(𝐴 ·ih 𝐵)) ∈ ℝ
2625recni 11232 . . . . . . . . 9 (abs‘(𝐴 ·ih 𝐵)) ∈ ℂ
2719, 26divclzi 11953 . . . . . . . 8 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ)
2819, 26divreczi 11956 . . . . . . . . . 10 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) = ((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵)))))
2928fveq2d 6895 . . . . . . . . 9 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = (abs‘((𝐴 ·ih 𝐵) · (1 / (abs‘(𝐴 ·ih 𝐵))))))
3026recclzi 11943 . . . . . . . . . . 11 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ)
31 absmul 15245 . . . . . . . . . . 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 11982 . . . . . . . . . . . 12 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℝ)
34 0re 11220 . . . . . . . . . . . . . 14 0 ∈ ℝ
3533, 34jctil 520 . . . . . . . . . . . . 13 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (0 ∈ ℝ ∧ (1 / (abs‘(𝐴 ·ih 𝐵))) ∈ ℝ))
3619absgt0i 15350 . . . . . . . . . . . . . . 15 ((𝐴 ·ih 𝐵) ≠ 0 ↔ 0 < (abs‘(𝐴 ·ih 𝐵)))
3724, 36bitri 274 . . . . . . . . . . . . . 14 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 ↔ 0 < (abs‘(𝐴 ·ih 𝐵)))
3825recgt0i 12123 . . . . . . . . . . . . . 14 (0 < (abs‘(𝐴 ·ih 𝐵)) → 0 < (1 / (abs‘(𝐴 ·ih 𝐵))))
3937, 38sylbi 216 . . . . . . . . . . . . 13 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → 0 < (1 / (abs‘(𝐴 ·ih 𝐵))))
40 ltle 11306 . . . . . . . . . . . . 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 15373 . . . . . . . . . . 11 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → (abs‘(1 / (abs‘(𝐴 ·ih 𝐵)))) = (1 / (abs‘(𝐴 ·ih 𝐵))))
4342oveq2d 7427 . . . . . . . . . 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 2772 . . . . . . . . 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 11945 . . . . . . . . 9 ((abs‘(𝐴 ·ih 𝐵)) ≠ 0 → ((abs‘(𝐴 ·ih 𝐵)) · (1 / (abs‘(𝐴 ·ih 𝐵)))) = 1)
4629, 44, 453eqtrd 2776 . . . . . . . 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 234 . . . . . 6 ((𝐴 ·ih 𝐵) ≠ 0 → (((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵))) ∈ ℂ ∧ (abs‘((𝐴 ·ih 𝐵) / (abs‘(𝐴 ·ih 𝐵)))) = 1))
493, 6normlem7tALT 30627 . . . . . 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 5170 . . . 4 ((𝐴 ·ih 𝐵) ≠ 0 → (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
5215, 51sylbir 234 . . 3 (¬ (𝐴 ·ih 𝐵) = 0 → (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
5310recni 11232 . . . . . 6 (norm𝐵) ∈ ℂ
549recni 11232 . . . . . 6 (norm𝐴) ∈ ℂ
55 normval 30632 . . . . . . . 8 (𝐵 ∈ ℋ → (norm𝐵) = (√‘(𝐵 ·ih 𝐵)))
566, 55ax-mp 5 . . . . . . 7 (norm𝐵) = (√‘(𝐵 ·ih 𝐵))
57 normval 30632 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) = (√‘(𝐴 ·ih 𝐴)))
583, 57ax-mp 5 . . . . . . 7 (norm𝐴) = (√‘(𝐴 ·ih 𝐴))
5956, 58oveq12i 7423 . . . . . 6 ((norm𝐵) · (norm𝐴)) = ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))
6053, 54, 59mulcomli 11227 . . . . 5 ((norm𝐴) · (norm𝐵)) = ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))
6160breq2i 5156 . . . 4 ((abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)) ↔ (abs‘(𝐴 ·ih 𝐵)) ≤ ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))))
62 2pos 12319 . . . . 5 0 < 2
63 hiidge0 30606 . . . . . . . 8 (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·ih 𝐵))
64 hiidrcl 30603 . . . . . . . . . 10 (𝐵 ∈ ℋ → (𝐵 ·ih 𝐵) ∈ ℝ)
656, 64ax-mp 5 . . . . . . . . 9 (𝐵 ·ih 𝐵) ∈ ℝ
6665sqrtcli 15322 . . . . . . . 8 (0 ≤ (𝐵 ·ih 𝐵) → (√‘(𝐵 ·ih 𝐵)) ∈ ℝ)
676, 63, 66mp2b 10 . . . . . . 7 (√‘(𝐵 ·ih 𝐵)) ∈ ℝ
68 hiidge0 30606 . . . . . . . 8 (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
69 hiidrcl 30603 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
703, 69ax-mp 5 . . . . . . . . 9 (𝐴 ·ih 𝐴) ∈ ℝ
7170sqrtcli 15322 . . . . . . . 8 (0 ≤ (𝐴 ·ih 𝐴) → (√‘(𝐴 ·ih 𝐴)) ∈ ℝ)
723, 68, 71mp2b 10 . . . . . . 7 (√‘(𝐴 ·ih 𝐴)) ∈ ℝ
7367, 72remulcli 11234 . . . . . 6 ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴))) ∈ ℝ
74 2re 12290 . . . . . 6 2 ∈ ℝ
7525, 73, 74lemul2i 12141 . . . . 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 274 . . 3 ((abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)) ↔ (2 · (abs‘(𝐴 ·ih 𝐵))) ≤ (2 · ((√‘(𝐵 ·ih 𝐵)) · (√‘(𝐴 ·ih 𝐴)))))
7852, 77sylibr 233 . 2 (¬ (𝐴 ·ih 𝐵) = 0 → (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵)))
7914, 78pm2.61i 182 1 (abs‘(𝐴 ·ih 𝐵)) ≤ ((norm𝐴) · (norm𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940   class class class wbr 5148  cfv 6543  (class class class)co 7411  cc 11110  cr 11111  0cc0 11112  1c1 11113   + caddc 11115   · cmul 11117   < clt 11252  cle 11253   / cdiv 11875  2c2 12271  ccj 15047  csqrt 15184  abscabs 15185  chba 30427   ·ih csp 30430  normcno 30431
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-hfvadd 30508  ax-hv0cl 30511  ax-hfvmul 30513  ax-hvmulass 30515  ax-hvmul0 30518  ax-hfi 30587  ax-his1 30590  ax-his2 30591  ax-his3 30592  ax-his4 30593
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-hnorm 30476  df-hvsub 30479
This theorem is referenced by: (None)
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