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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 2724 | . . 3 β’ (toβPreHilβπ) = (toβPreHilβπ) | |
| 2 | ipcau.v | . . 3 β’ π = (Baseβπ) | |
| 3 | eqid 2724 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
| 4 | simp1 1133 | . . . 4 β’ ((π β βPreHil β§ π β π β§ π β π) β π β βPreHil) | |
| 5 | cphphl 25043 | . . . 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 2724 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
| 8 | 3, 7 | cphsca 25051 | . . . 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 25056 | . . . 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 25070 | . . . 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 2724 | . . 3 β’ (normβ(toβPreHilβπ)) = (normβ(toβPreHilβπ)) | |
| 16 | eqid 2724 | . . 3 β’ ((π , π) / (π , π)) = ((π , π) / (π , π)) | |
| 17 | simp2 1134 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β π β π) | |
| 18 | simp3 1135 | . . 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 25106 | . 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 25102 | . . . . 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 6884 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β (πβπ) = ((normβ(toβPreHilβπ))βπ)) |
| 24 | 22 | fveq1d 6884 | . . 3 β’ ((π β βPreHil β§ π β π β§ π β π) β (πβπ) = ((normβ(toβPreHilβπ))βπ)) |
| 25 | 23, 24 | oveq12d 7420 | . 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 5167 | 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 1084 = wceq 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 (class class class)co 7402 βcr 11106 0cc0 11107 Β· cmul 11112 β€ cle 11248 / cdiv 11870 βcsqrt 15182 abscabs 15183 Basecbs 17149 βΎs cress 17178 Scalarcsca 17205 Β·πcip 17207 βfldccnfld 21234 PreHilcphl 21506 normcnm 24429 βPreHilccph 25038 toβPreHilctcph 25039 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ico 13331 df-fz 13486 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-topgen 17394 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-ghm 19135 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-drng 20585 df-staf 20684 df-srng 20685 df-lmod 20704 df-lmhm 20866 df-lvec 20947 df-sra 21017 df-rgmod 21018 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-cnfld 21235 df-phl 21508 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-xms 24170 df-ms 24171 df-nm 24435 df-ngp 24436 df-tng 24437 df-nlm 24439 df-clm 24934 df-cph 25040 df-tcph 25041 |
| This theorem is referenced by: ipcnlem2 25116 |
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