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Mirrors > Home > HSE Home > Th. List > ococ | Structured version Visualization version GIF version |
Description: Complement of complement of a closed subspace of Hilbert space. Theorem 3.7(ii) of [Beran] p. 102. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ococ | ⊢ (𝐴 ∈ Cℋ → (⊥‘(⊥‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fveq3 6848 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, ℋ) → (⊥‘(⊥‘𝐴)) = (⊥‘(⊥‘if(𝐴 ∈ Cℋ , 𝐴, ℋ)))) | |
2 | id 22 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, ℋ) → 𝐴 = if(𝐴 ∈ Cℋ , 𝐴, ℋ)) | |
3 | 1, 2 | eqeq12d 2753 | . 2 ⊢ (𝐴 = if(𝐴 ∈ Cℋ , 𝐴, ℋ) → ((⊥‘(⊥‘𝐴)) = 𝐴 ↔ (⊥‘(⊥‘if(𝐴 ∈ Cℋ , 𝐴, ℋ))) = if(𝐴 ∈ Cℋ , 𝐴, ℋ))) |
4 | ifchhv 30189 | . . 3 ⊢ if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∈ Cℋ | |
5 | 4 | ococi 30350 | . 2 ⊢ (⊥‘(⊥‘if(𝐴 ∈ Cℋ , 𝐴, ℋ))) = if(𝐴 ∈ Cℋ , 𝐴, ℋ) |
6 | 3, 5 | dedth 4545 | 1 ⊢ (𝐴 ∈ Cℋ → (⊥‘(⊥‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ifcif 4487 ‘cfv 6497 ℋchba 29864 Cℋ cch 29874 ⊥cort 29875 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cc 10372 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 ax-addf 11131 ax-mulf 11132 ax-hilex 29944 ax-hfvadd 29945 ax-hvcom 29946 ax-hvass 29947 ax-hv0cl 29948 ax-hvaddid 29949 ax-hfvmul 29950 ax-hvmulid 29951 ax-hvmulass 29952 ax-hvdistr1 29953 ax-hvdistr2 29954 ax-hvmul0 29955 ax-hfi 30024 ax-his1 30027 ax-his2 30028 ax-his3 30029 ax-his4 30030 ax-hcompl 30147 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-oadd 8417 df-omul 8418 df-er 8649 df-map 8768 df-pm 8769 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fi 9348 df-sup 9379 df-inf 9380 df-oi 9447 df-card 9876 df-acn 9879 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-n0 12415 df-z 12501 df-uz 12765 df-q 12875 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-ico 13271 df-icc 13272 df-fz 13426 df-fl 13698 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 df-rlim 15372 df-rest 17305 df-topgen 17326 df-psmet 20791 df-xmet 20792 df-met 20793 df-bl 20794 df-mopn 20795 df-fbas 20796 df-fg 20797 df-top 22246 df-topon 22263 df-bases 22299 df-cld 22373 df-ntr 22374 df-cls 22375 df-nei 22452 df-lm 22583 df-haus 22669 df-fil 23200 df-fm 23292 df-flim 23293 df-flf 23294 df-cfil 24622 df-cau 24623 df-cmet 24624 df-grpo 29438 df-gid 29439 df-ginv 29440 df-gdiv 29441 df-ablo 29490 df-vc 29504 df-nv 29537 df-va 29540 df-ba 29541 df-sm 29542 df-0v 29543 df-vs 29544 df-nmcv 29545 df-ims 29546 df-ssp 29667 df-ph 29758 df-cbn 29808 df-hnorm 29913 df-hba 29914 df-hvsub 29916 df-hlim 29917 df-hcau 29918 df-sh 30152 df-ch 30166 df-oc 30197 df-ch0 30198 |
This theorem is referenced by: dfch2 30352 ococin 30353 shlub 30359 pjhtheu2 30361 shjshseli 30438 chsscon1 30446 chpsscon1 30449 chpsscon2 30450 chdmm2 30471 chdmm3 30472 chdmm4 30473 chdmj1 30474 chdmj2 30475 chdmj3 30476 chdmj4 30477 fh2 30564 hstle 31175 hstoh 31177 mddmd 31246 |
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