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Mirrors > Home > MPE Home > Th. List > sii | Structured version Visualization version GIF version |
Description: Schwarz inequality. Part of Lemma 3-2.1(a) of [Kreyszig] p. 137. This is also called the Cauchy-Schwarz inequality by some authors and Bunjakovaskij-Cauchy-Schwarz inequality by others. See also theorems bcseqi 27361, bcsiALT 27420, bcsiHIL 27421, csbren 22990. This is Metamath 100 proof #78. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sii.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
sii.6 | ⊢ 𝑁 = (normCV‘𝑈) |
sii.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
sii.9 | ⊢ 𝑈 ∈ CPreHilOLD |
Ref | Expression |
---|---|
sii | ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6556 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝐴𝑃𝐵) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐵)) | |
2 | 1 | fveq2d 6107 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (abs‘(𝐴𝑃𝐵)) = (abs‘(if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐵))) |
3 | fveq2 6103 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → (𝑁‘𝐴) = (𝑁‘if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)))) | |
4 | 3 | oveq1d 6564 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((𝑁‘𝐴) · (𝑁‘𝐵)) = ((𝑁‘if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))) · (𝑁‘𝐵))) |
5 | 2, 4 | breq12d 4596 | . 2 ⊢ (𝐴 = if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) → ((abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)) ↔ (abs‘(if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐵)) ≤ ((𝑁‘if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))) · (𝑁‘𝐵)))) |
6 | oveq2 6557 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐵) = (if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))) | |
7 | 6 | fveq2d 6107 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (abs‘(if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐵)) = (abs‘(if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))))) |
8 | fveq2 6103 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → (𝑁‘𝐵) = (𝑁‘if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))) | |
9 | 8 | oveq2d 6565 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → ((𝑁‘if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))) · (𝑁‘𝐵)) = ((𝑁‘if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))) · (𝑁‘if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈))))) |
10 | 7, 9 | breq12d 4596 | . 2 ⊢ (𝐵 = if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) → ((abs‘(if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃𝐵)) ≤ ((𝑁‘if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))) · (𝑁‘𝐵)) ↔ (abs‘(if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))) ≤ ((𝑁‘if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))) · (𝑁‘if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))))) |
11 | sii.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
12 | sii.6 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
13 | sii.7 | . . 3 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
14 | sii.9 | . . 3 ⊢ 𝑈 ∈ CPreHilOLD | |
15 | eqid 2610 | . . . 4 ⊢ (0vec‘𝑈) = (0vec‘𝑈) | |
16 | 11, 15, 14 | elimph 27059 | . . 3 ⊢ if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈)) ∈ 𝑋 |
17 | 11, 15, 14 | elimph 27059 | . . 3 ⊢ if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)) ∈ 𝑋 |
18 | 11, 12, 13, 14, 16, 17 | siii 27092 | . 2 ⊢ (abs‘(if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))𝑃if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))) ≤ ((𝑁‘if(𝐴 ∈ 𝑋, 𝐴, (0vec‘𝑈))) · (𝑁‘if(𝐵 ∈ 𝑋, 𝐵, (0vec‘𝑈)))) |
19 | 5, 10, 18 | dedth2h 4090 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘(𝐴𝑃𝐵)) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ifcif 4036 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 · cmul 9820 ≤ cle 9954 abscabs 13822 BaseSetcba 26825 0veccn0v 26827 normCVcnmcv 26829 ·𝑖OLDcdip 26939 CPreHilOLDccphlo 27051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-cn 20841 df-cnp 20842 df-t1 20928 df-haus 20929 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-dip 26940 df-ph 27052 |
This theorem is referenced by: ipblnfi 27095 htthlem 27158 |
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