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Mirrors > Home > MPE Home > Th. List > ipcau | Structured version Visualization version GIF version |
Description: The Cauchy-Schwarz inequality for a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.) |
Ref | Expression |
---|---|
ipcau.v | ⊢ 𝑉 = (Base‘𝑊) |
ipcau.h | ⊢ , = (·𝑖‘𝑊) |
ipcau.n | ⊢ 𝑁 = (norm‘𝑊) |
Ref | Expression |
---|---|
ipcau | ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁‘𝑋) · (𝑁‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . 3 ⊢ (toℂPreHil‘𝑊) = (toℂPreHil‘𝑊) | |
2 | ipcau.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | eqid 2825 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | simp1 1170 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑊 ∈ ℂPreHil) | |
5 | cphphl 23347 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑊 ∈ PreHil) |
7 | eqid 2825 | . . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | 3, 7 | cphsca 23355 | . . . 4 ⊢ (𝑊 ∈ ℂPreHil → (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) |
9 | 4, 8 | syl 17 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) |
10 | ipcau.h | . . 3 ⊢ , = (·𝑖‘𝑊) | |
11 | 3, 7 | cphsqrtcl 23360 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘(Scalar‘𝑊))) |
12 | 4, 11 | sylan 575 | . . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ (Base‘(Scalar‘𝑊))) |
13 | 2, 10 | ipge0 23374 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) |
14 | 4, 13 | sylan 575 | . . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) |
15 | eqid 2825 | . . 3 ⊢ (norm‘(toℂPreHil‘𝑊)) = (norm‘(toℂPreHil‘𝑊)) | |
16 | eqid 2825 | . . 3 ⊢ ((𝑌 , 𝑋) / (𝑌 , 𝑌)) = ((𝑌 , 𝑋) / (𝑌 , 𝑌)) | |
17 | simp2 1171 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
18 | simp3 1172 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉) | |
19 | 1, 2, 3, 6, 9, 10, 12, 14, 7, 15, 16, 17, 18 | ipcau2 23409 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (abs‘(𝑋 , 𝑌)) ≤ (((norm‘(toℂPreHil‘𝑊))‘𝑋) · ((norm‘(toℂPreHil‘𝑊))‘𝑌))) |
20 | ipcau.n | . . . . . 6 ⊢ 𝑁 = (norm‘𝑊) | |
21 | 1, 20 | cphtcphnm 23405 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (norm‘(toℂPreHil‘𝑊))) |
22 | 4, 21 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 𝑁 = (norm‘(toℂPreHil‘𝑊))) |
23 | 22 | fveq1d 6439 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘𝑋) = ((norm‘(toℂPreHil‘𝑊))‘𝑋)) |
24 | 22 | fveq1d 6439 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘𝑌) = ((norm‘(toℂPreHil‘𝑊))‘𝑌)) |
25 | 23, 24 | oveq12d 6928 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (((norm‘(toℂPreHil‘𝑊))‘𝑋) · ((norm‘(toℂPreHil‘𝑊))‘𝑌))) |
26 | 19, 25 | breqtrrd 4903 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁‘𝑋) · (𝑁‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 0cc0 10259 · cmul 10264 ≤ cle 10399 / cdiv 11016 √csqrt 14357 abscabs 14358 Basecbs 16229 ↾s cress 16230 Scalarcsca 16315 ·𝑖cip 16317 ℂfldccnfld 20113 PreHilcphl 20338 normcnm 22758 ℂPreHilccph 23342 toℂPreHilctcph 23343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ico 12476 df-fz 12627 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-topgen 16464 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-ghm 18016 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-rnghom 19078 df-drng 19112 df-subrg 19141 df-staf 19208 df-srng 19209 df-lmod 19228 df-lmhm 19388 df-lvec 19469 df-sra 19540 df-rgmod 19541 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-cnfld 20114 df-phl 20340 df-top 21076 df-topon 21093 df-topsp 21115 df-bases 21128 df-xms 22502 df-ms 22503 df-nm 22764 df-ngp 22765 df-tng 22766 df-nlm 22768 df-clm 23239 df-cph 23344 df-tcph 23345 |
This theorem is referenced by: ipcnlem2 23419 |
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