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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 2728 | . . 3 β’ (toβPreHilβπ) = (toβPreHilβπ) | |
| 2 | ipcau.v | . . 3 β’ π = (Baseβπ) | |
| 3 | eqid 2728 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
| 4 | simp1 1134 | . . . 4 β’ ((π β βPreHil β§ π β π β§ π β π) β π β βPreHil) | |
| 5 | cphphl 25112 | . . . 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 2728 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
| 8 | 3, 7 | cphsca 25120 | . . . 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 25125 | . . . 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 579 | . . 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 25139 | . . . 4 β’ ((π β βPreHil β§ π₯ β π) β 0 β€ (π₯ , π₯)) |
| 14 | 4, 13 | sylan 579 | . . 3 β’ (((π β βPreHil β§ π β π β§ π β π) β§ π₯ β π) β 0 β€ (π₯ , π₯)) |
| 15 | eqid 2728 | . . 3 β’ (normβ(toβPreHilβπ)) = (normβ(toβPreHilβπ)) | |
| 16 | eqid 2728 | . . 3 β’ ((π , π) / (π , π)) = ((π , π) / (π , π)) | |
| 17 | simp2 1135 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β π β π) | |
| 18 | simp3 1136 | . . 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 25175 | . 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 25171 | . . . . 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 6899 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β (πβπ) = ((normβ(toβPreHilβπ))βπ)) |
| 24 | 22 | fveq1d 6899 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β (πβπ) = ((normβ(toβPreHilβπ))βπ)) |
| 25 | 23, 24 | oveq12d 7438 | . 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 5176 | 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 1085 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 (class class class)co 7420 βcr 11138 0cc0 11139 Β· cmul 11144 β€ cle 11280 / cdiv 11902 βcsqrt 15213 abscabs 15214 Basecbs 17180 βΎs cress 17209 Scalarcsca 17236 Β·πcip 17238 βfldccnfld 21279 PreHilcphl 21556 normcnm 24498 βPreHilccph 25107 toβPreHilctcph 25108 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ico 13363 df-fz 13518 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-topgen 17425 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-ghm 19168 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-drng 20626 df-staf 20725 df-srng 20726 df-lmod 20745 df-lmhm 20907 df-lvec 20988 df-sra 21058 df-rgmod 21059 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-phl 21558 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-xms 24239 df-ms 24240 df-nm 24504 df-ngp 24505 df-tng 24506 df-nlm 24508 df-clm 25003 df-cph 25109 df-tcph 25110 |
| This theorem is referenced by: ipcnlem2 25185 |
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