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Mirrors > Home > MPE Home > Th. List > cldcss2 | Structured version Visualization version GIF version |
Description: Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
Ref | Expression |
---|---|
cldcss.v | ⊢ 𝑉 = (Base‘𝑊) |
cldcss.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
cldcss.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
cldcss.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
cldcss2 | ⊢ (𝑊 ∈ ℂHil → 𝐶 = (𝐿 ∩ (Clsd‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cldcss.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | cldcss.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝑊) | |
3 | cldcss.l | . . . 4 ⊢ 𝐿 = (LSubSp‘𝑊) | |
4 | cldcss.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
5 | 1, 2, 3, 4 | cldcss 24193 | . . 3 ⊢ (𝑊 ∈ ℂHil → (𝑥 ∈ 𝐶 ↔ (𝑥 ∈ 𝐿 ∧ 𝑥 ∈ (Clsd‘𝐽)))) |
6 | elin 3859 | . . 3 ⊢ (𝑥 ∈ (𝐿 ∩ (Clsd‘𝐽)) ↔ (𝑥 ∈ 𝐿 ∧ 𝑥 ∈ (Clsd‘𝐽))) | |
7 | 5, 6 | bitr4di 292 | . 2 ⊢ (𝑊 ∈ ℂHil → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ (𝐿 ∩ (Clsd‘𝐽)))) |
8 | 7 | eqrdv 2736 | 1 ⊢ (𝑊 ∈ ℂHil → 𝐶 = (𝐿 ∩ (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∩ cin 3842 ‘cfv 6339 Basecbs 16586 TopOpenctopn 16798 LSubSpclss 19822 ClSubSpccss 20477 Clsdccld 21767 ℂHilchl 24086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-rest 16799 df-topn 16800 df-0g 16818 df-gsum 16819 df-topgen 16820 df-pt 16821 df-prds 16824 df-xrs 16878 df-qtop 16883 df-imas 16884 df-xps 16886 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-ghm 18474 df-cntz 18565 df-lsm 18879 df-pj1 18880 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-cring 19419 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-rnghom 19589 df-drng 19623 df-subrg 19652 df-staf 19735 df-srng 19736 df-lmod 19755 df-lss 19823 df-lmhm 19913 df-lvec 19994 df-sra 20063 df-rgmod 20064 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-fbas 20214 df-fg 20215 df-cnfld 20218 df-phl 20442 df-ipf 20443 df-ocv 20479 df-css 20480 df-pj 20519 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-cn 21978 df-cnp 21979 df-t1 22065 df-haus 22066 df-cmp 22138 df-tx 22313 df-hmeo 22506 df-fil 22597 df-flim 22690 df-fcls 22692 df-xms 23073 df-ms 23074 df-tms 23075 df-nm 23335 df-ngp 23336 df-tng 23337 df-nlm 23339 df-cncf 23630 df-clm 23815 df-cph 23920 df-tcph 23921 df-cfil 24007 df-cmet 24009 df-cms 24087 df-bn 24088 df-hl 24089 |
This theorem is referenced by: hlhil 24195 |
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