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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. Part of Lemma 3.2-1(a) of [Kreyszig] p. 137. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 11-Oct-2015.) |
| Ref | Expression |
|---|---|
| ipcau.v | β’ π = (Baseβπ) |
| ipcau.h | β’ , = (Β·πβπ) |
| ipcau.n | β’ π = (normβπ) |
| Ref | Expression |
|---|---|
| ipcau | β’ ((π β βPreHil β§ π β π β§ π β π) β (absβ(π , π)) β€ ((πβπ) Β· (πβπ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2732 | . . 3 β’ (toβPreHilβπ) = (toβPreHilβπ) | |
| 2 | ipcau.v | . . 3 β’ π = (Baseβπ) | |
| 3 | eqid 2732 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
| 4 | simp1 1136 | . . . 4 β’ ((π β βPreHil β§ π β π β§ π β π) β π β βPreHil) | |
| 5 | cphphl 24679 | . . . 4 β’ (π β βPreHil β π β PreHil) | |
| 6 | 4, 5 | syl 17 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β π β PreHil) |
| 7 | eqid 2732 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
| 8 | 3, 7 | cphsca 24687 | . . . 4 β’ (π β βPreHil β (Scalarβπ) = (βfld βΎs (Baseβ(Scalarβπ)))) |
| 9 | 4, 8 | syl 17 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β (Scalarβπ) = (βfld βΎs (Baseβ(Scalarβπ)))) |
| 10 | ipcau.h | . . 3 β’ , = (Β·πβπ) | |
| 11 | 3, 7 | cphsqrtcl 24692 | . . . 4 β’ ((π β βPreHil β§ (π₯ β (Baseβ(Scalarβπ)) β§ π₯ β β β§ 0 β€ π₯)) β (ββπ₯) β (Baseβ(Scalarβπ))) |
| 12 | 4, 11 | sylan 580 | . . 3 β’ (((π β βPreHil β§ π β π β§ π β π) β§ (π₯ β (Baseβ(Scalarβπ)) β§ π₯ β β β§ 0 β€ π₯)) β (ββπ₯) β (Baseβ(Scalarβπ))) |
| 13 | 2, 10 | ipge0 24706 | . . . 4 β’ ((π β βPreHil β§ π₯ β π) β 0 β€ (π₯ , π₯)) |
| 14 | 4, 13 | sylan 580 | . . 3 β’ (((π β βPreHil β§ π β π β§ π β π) β§ π₯ β π) β 0 β€ (π₯ , π₯)) |
| 15 | eqid 2732 | . . 3 β’ (normβ(toβPreHilβπ)) = (normβ(toβPreHilβπ)) | |
| 16 | eqid 2732 | . . 3 β’ ((π , π) / (π , π)) = ((π , π) / (π , π)) | |
| 17 | simp2 1137 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β π β π) | |
| 18 | simp3 1138 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β π β π) | |
| 19 | 1, 2, 3, 6, 9, 10, 12, 14, 7, 15, 16, 17, 18 | ipcau2 24742 | . 2 β’ ((π β βPreHil β§ π β π β§ π β π) β (absβ(π , π)) β€ (((normβ(toβPreHilβπ))βπ) Β· ((normβ(toβPreHilβπ))βπ))) |
| 20 | ipcau.n | . . . . . 6 β’ π = (normβπ) | |
| 21 | 1, 20 | cphtcphnm 24738 | . . . . 5 β’ (π β βPreHil β π = (normβ(toβPreHilβπ))) |
| 22 | 4, 21 | syl 17 | . . . 4 β’ ((π β βPreHil β§ π β π β§ π β π) β π = (normβ(toβPreHilβπ))) |
| 23 | 22 | fveq1d 6890 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β (πβπ) = ((normβ(toβPreHilβπ))βπ)) |
| 24 | 22 | fveq1d 6890 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β (πβπ) = ((normβ(toβPreHilβπ))βπ)) |
| 25 | 23, 24 | oveq12d 7423 | . 2 β’ ((π β βPreHil β§ π β π β§ π β π) β ((πβπ) Β· (πβπ)) = (((normβ(toβPreHilβπ))βπ) Β· ((normβ(toβPreHilβπ))βπ))) |
| 26 | 19, 25 | breqtrrd 5175 | 1 β’ ((π β βPreHil β§ π β π β§ π β π) β (absβ(π , π)) β€ ((πβπ) Β· (πβπ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 class class class wbr 5147 βcfv 6540 (class class class)co 7405 βcr 11105 0cc0 11106 Β· cmul 11111 β€ cle 11245 / cdiv 11867 βcsqrt 15176 abscabs 15177 Basecbs 17140 βΎs cress 17169 Scalarcsca 17196 Β·πcip 17198 βfldccnfld 20936 PreHilcphl 21168 normcnm 24076 βPreHilccph 24674 toβPreHilctcph 24675 |
| 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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
| 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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-topgen 17385 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-ghm 19084 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-rnghom 20243 df-drng 20309 df-subrg 20353 df-staf 20445 df-srng 20446 df-lmod 20465 df-lmhm 20625 df-lvec 20706 df-sra 20777 df-rgmod 20778 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-cnfld 20937 df-phl 21170 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-xms 23817 df-ms 23818 df-nm 24082 df-ngp 24083 df-tng 24084 df-nlm 24086 df-clm 24570 df-cph 24676 df-tcph 24677 |
| This theorem is referenced by: ipcnlem2 24752 |
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