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Mirrors > Home > HSE Home > Th. List > pjinormii | Structured version Visualization version GIF version |
Description: The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjidm.1 | ⊢ 𝐻 ∈ Cℋ |
pjidm.2 | ⊢ 𝐴 ∈ ℋ |
Ref | Expression |
---|---|
pjinormii | ⊢ (((projℎ‘𝐻)‘𝐴) ·ih 𝐴) = ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjidm.1 | . . . 4 ⊢ 𝐻 ∈ Cℋ | |
2 | pjidm.2 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
3 | 1, 2 | pjhclii 29771 | . . 3 ⊢ ((projℎ‘𝐻)‘𝐴) ∈ ℋ |
4 | 3 | normsqi 29481 | . 2 ⊢ ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) = (((projℎ‘𝐻)‘𝐴) ·ih ((projℎ‘𝐻)‘𝐴)) |
5 | 1, 3, 2 | pjadjii 30023 | . 2 ⊢ (((projℎ‘𝐻)‘((projℎ‘𝐻)‘𝐴)) ·ih 𝐴) = (((projℎ‘𝐻)‘𝐴) ·ih ((projℎ‘𝐻)‘𝐴)) |
6 | 1, 2 | pjidmi 30022 | . . 3 ⊢ ((projℎ‘𝐻)‘((projℎ‘𝐻)‘𝐴)) = ((projℎ‘𝐻)‘𝐴) |
7 | 6 | oveq1i 7279 | . 2 ⊢ (((projℎ‘𝐻)‘((projℎ‘𝐻)‘𝐴)) ·ih 𝐴) = (((projℎ‘𝐻)‘𝐴) ·ih 𝐴) |
8 | 4, 5, 7 | 3eqtr2ri 2773 | 1 ⊢ (((projℎ‘𝐻)‘𝐴) ·ih 𝐴) = ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 ‘cfv 6428 (class class class)co 7269 2c2 12017 ↑cexp 13771 ℋchba 29268 ·ih csp 29271 normℎcno 29272 Cℋ cch 29278 projℎcpjh 29286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-inf2 9388 ax-cc 10180 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 ax-pre-sup 10938 ax-addf 10939 ax-mulf 10940 ax-hilex 29348 ax-hfvadd 29349 ax-hvcom 29350 ax-hvass 29351 ax-hv0cl 29352 ax-hvaddid 29353 ax-hfvmul 29354 ax-hvmulid 29355 ax-hvmulass 29356 ax-hvdistr1 29357 ax-hvdistr2 29358 ax-hvmul0 29359 ax-hfi 29428 ax-his1 29431 ax-his2 29432 ax-his3 29433 ax-his4 29434 ax-hcompl 29551 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-se 5542 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-isom 6437 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-oadd 8290 df-omul 8291 df-er 8487 df-map 8606 df-pm 8607 df-ixp 8675 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-fsupp 9118 df-fi 9159 df-sup 9190 df-inf 9191 df-oi 9258 df-card 9686 df-acn 9689 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-nn 11963 df-2 12025 df-3 12026 df-4 12027 df-5 12028 df-6 12029 df-7 12030 df-8 12031 df-9 12032 df-n0 12223 df-z 12309 df-dec 12427 df-uz 12572 df-q 12678 df-rp 12720 df-xneg 12837 df-xadd 12838 df-xmul 12839 df-ioo 13072 df-ico 13074 df-icc 13075 df-fz 13229 df-fzo 13372 df-fl 13501 df-seq 13711 df-exp 13772 df-hash 14034 df-cj 14799 df-re 14800 df-im 14801 df-sqrt 14935 df-abs 14936 df-clim 15186 df-rlim 15187 df-sum 15387 df-struct 16837 df-sets 16854 df-slot 16872 df-ndx 16884 df-base 16902 df-ress 16931 df-plusg 16964 df-mulr 16965 df-starv 16966 df-sca 16967 df-vsca 16968 df-ip 16969 df-tset 16970 df-ple 16971 df-ds 16973 df-unif 16974 df-hom 16975 df-cco 16976 df-rest 17122 df-topn 17123 df-0g 17141 df-gsum 17142 df-topgen 17143 df-pt 17144 df-prds 17147 df-xrs 17202 df-qtop 17207 df-imas 17208 df-xps 17210 df-mre 17284 df-mrc 17285 df-acs 17287 df-mgm 18315 df-sgrp 18364 df-mnd 18375 df-submnd 18420 df-mulg 18690 df-cntz 18912 df-cmn 19377 df-psmet 20578 df-xmet 20579 df-met 20580 df-bl 20581 df-mopn 20582 df-fbas 20583 df-fg 20584 df-cnfld 20587 df-top 22032 df-topon 22049 df-topsp 22071 df-bases 22085 df-cld 22159 df-ntr 22160 df-cls 22161 df-nei 22238 df-cn 22367 df-cnp 22368 df-lm 22369 df-haus 22455 df-tx 22702 df-hmeo 22895 df-fil 22986 df-fm 23078 df-flim 23079 df-flf 23080 df-xms 23462 df-ms 23463 df-tms 23464 df-cfil 24408 df-cau 24409 df-cmet 24410 df-grpo 28842 df-gid 28843 df-ginv 28844 df-gdiv 28845 df-ablo 28894 df-vc 28908 df-nv 28941 df-va 28944 df-ba 28945 df-sm 28946 df-0v 28947 df-vs 28948 df-nmcv 28949 df-ims 28950 df-dip 29050 df-ssp 29071 df-ph 29162 df-cbn 29212 df-hnorm 29317 df-hba 29318 df-hvsub 29320 df-hlim 29321 df-hcau 29322 df-sh 29556 df-ch 29570 df-oc 29601 df-ch0 29602 df-shs 29657 df-pjh 29744 |
This theorem is referenced by: pjssge0ii 30031 pjdifnormii 30032 pjinormi 30036 |
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