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Mirrors > Home > MPE Home > Th. List > nmsq | Structured version Visualization version GIF version |
Description: The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
nmsq.v | β’ π = (Baseβπ) |
nmsq.h | β’ , = (Β·πβπ) |
nmsq.n | β’ π = (normβπ) |
Ref | Expression |
---|---|
nmsq | β’ ((π β βPreHil β§ π΄ β π) β ((πβπ΄)β2) = (π΄ , π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmsq.v | . . . 4 β’ π = (Baseβπ) | |
2 | nmsq.h | . . . 4 β’ , = (Β·πβπ) | |
3 | nmsq.n | . . . 4 β’ π = (normβπ) | |
4 | 1, 2, 3 | cphnm 24634 | . . 3 β’ ((π β βPreHil β§ π΄ β π) β (πβπ΄) = (ββ(π΄ , π΄))) |
5 | 4 | oveq1d 7405 | . 2 β’ ((π β βPreHil β§ π΄ β π) β ((πβπ΄)β2) = ((ββ(π΄ , π΄))β2)) |
6 | 1, 2 | cphipcl 24632 | . . . 4 β’ ((π β βPreHil β§ π΄ β π β§ π΄ β π) β (π΄ , π΄) β β) |
7 | 6 | 3anidm23 1421 | . . 3 β’ ((π β βPreHil β§ π΄ β π) β (π΄ , π΄) β β) |
8 | 7 | sqsqrtd 15365 | . 2 β’ ((π β βPreHil β§ π΄ β π) β ((ββ(π΄ , π΄))β2) = (π΄ , π΄)) |
9 | 5, 8 | eqtrd 2771 | 1 β’ ((π β βPreHil β§ π΄ β π) β ((πβπ΄)β2) = (π΄ , π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βcfv 6529 (class class class)co 7390 βcc 11087 2c2 12246 βcexp 14006 βcsqrt 15159 Basecbs 17123 Β·πcip 17181 normcnm 24009 βPreHilccph 24607 |
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 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-pre-sup 11167 ax-addf 11168 ax-mulf 11169 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3430 df-v 3472 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 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-tp 4624 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-1st 7954 df-2nd 7955 df-tpos 8190 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-sup 9416 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-z 12538 df-dec 12657 df-uz 12802 df-rp 12954 df-fz 13464 df-seq 13946 df-exp 14007 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-struct 17059 df-sets 17076 df-slot 17094 df-ndx 17106 df-base 17124 df-ress 17153 df-plusg 17189 df-mulr 17190 df-starv 17191 df-sca 17192 df-vsca 17193 df-ip 17194 df-tset 17195 df-ple 17196 df-ds 17198 df-unif 17199 df-0g 17366 df-mgm 18540 df-sgrp 18589 df-mnd 18600 df-grp 18794 df-subg 18972 df-ghm 19053 df-cmn 19611 df-mgp 19944 df-ur 19961 df-ring 20013 df-cring 20014 df-oppr 20099 df-dvdsr 20120 df-unit 20121 df-drng 20264 df-subrg 20305 df-lmhm 20577 df-lvec 20658 df-sra 20729 df-rgmod 20730 df-cnfld 20874 df-phl 21107 df-cph 24609 |
This theorem is referenced by: cphnmf 24636 reipcl 24638 ipge0 24639 cphpyth 24657 nmparlem 24680 cphipval2 24682 cphipval 24684 pjthlem1 24878 |
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