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Mirrors > Home > HSE Home > Th. List > pjinormi | 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, 2-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjadjt.1 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjinormi | ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐻)‘𝐴) ·ih 𝐴) = ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6904 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((projℎ‘𝐻)‘𝐴) = ((projℎ‘𝐻)‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) | |
2 | id 22 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → 𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) | |
3 | 1, 2 | oveq12d 7447 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (((projℎ‘𝐻)‘𝐴) ·ih 𝐴) = (((projℎ‘𝐻)‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) |
4 | 2fveq3 6909 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (normℎ‘((projℎ‘𝐻)‘𝐴)) = (normℎ‘((projℎ‘𝐻)‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))) | |
5 | 4 | oveq1d 7444 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) = ((normℎ‘((projℎ‘𝐻)‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))↑2)) |
6 | 3, 5 | eqeq12d 2752 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((((projℎ‘𝐻)‘𝐴) ·ih 𝐴) = ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2) ↔ (((projℎ‘𝐻)‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = ((normℎ‘((projℎ‘𝐻)‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))↑2))) |
7 | pjadjt.1 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
8 | ifhvhv0 31031 | . . 3 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ | |
9 | 7, 8 | pjinormii 31685 | . 2 ⊢ (((projℎ‘𝐻)‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ·ih if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) = ((normℎ‘((projℎ‘𝐻)‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))↑2) |
10 | 6, 9 | dedth 4582 | 1 ⊢ (𝐴 ∈ ℋ → (((projℎ‘𝐻)‘𝐴) ·ih 𝐴) = ((normℎ‘((projℎ‘𝐻)‘𝐴))↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ifcif 4524 ‘cfv 6559 (class class class)co 7429 2c2 12317 ↑cexp 14098 ℋchba 30928 ·ih csp 30931 normℎcno 30932 0ℎc0v 30933 Cℋ cch 30938 projℎcpjh 30946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-inf2 9677 ax-cc 10471 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 ax-addf 11230 ax-mulf 11231 ax-hilex 31008 ax-hfvadd 31009 ax-hvcom 31010 ax-hvass 31011 ax-hv0cl 31012 ax-hvaddid 31013 ax-hfvmul 31014 ax-hvmulid 31015 ax-hvmulass 31016 ax-hvdistr1 31017 ax-hvdistr2 31018 ax-hvmul0 31019 ax-hfi 31088 ax-his1 31091 ax-his2 31092 ax-his3 31093 ax-his4 31094 ax-hcompl 31211 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-oadd 8506 df-omul 8507 df-er 8741 df-map 8864 df-pm 8865 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9398 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-acn 9978 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-q 12987 df-rp 13031 df-xneg 13150 df-xadd 13151 df-xmul 13152 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-starv 17308 df-sca 17309 df-vsca 17310 df-ip 17311 df-tset 17312 df-ple 17313 df-ds 17315 df-unif 17316 df-hom 17317 df-cco 17318 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-pt 17485 df-prds 17488 df-xrs 17543 df-qtop 17548 df-imas 17549 df-xps 17551 df-mre 17625 df-mrc 17626 df-acs 17628 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-submnd 18793 df-mulg 19082 df-cntz 19331 df-cmn 19796 df-psmet 21348 df-xmet 21349 df-met 21350 df-bl 21351 df-mopn 21352 df-fbas 21353 df-fg 21354 df-cnfld 21357 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-cld 23017 df-ntr 23018 df-cls 23019 df-nei 23096 df-cn 23225 df-cnp 23226 df-lm 23227 df-haus 23313 df-tx 23560 df-hmeo 23753 df-fil 23844 df-fm 23936 df-flim 23937 df-flf 23938 df-xms 24320 df-ms 24321 df-tms 24322 df-cfil 25279 df-cau 25280 df-cmet 25281 df-grpo 30502 df-gid 30503 df-ginv 30504 df-gdiv 30505 df-ablo 30554 df-vc 30568 df-nv 30601 df-va 30604 df-ba 30605 df-sm 30606 df-0v 30607 df-vs 30608 df-nmcv 30609 df-ims 30610 df-dip 30710 df-ssp 30731 df-ph 30822 df-cbn 30872 df-hnorm 30977 df-hba 30978 df-hvsub 30980 df-hlim 30981 df-hcau 30982 df-sh 31216 df-ch 31230 df-oc 31261 df-ch0 31262 df-shs 31317 df-pjh 31404 |
This theorem is referenced by: pjige0i 31699 pjssposi 32181 |
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