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Mirrors > Home > HSE Home > Th. List > osumi | Structured version Visualization version GIF version |
Description: If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. Note that the (countable) Axiom of Choice is used for this proof via pjhth 28803, although "the hard part" of this proof, chscl 29051, requires no choice. (Contributed by NM, 28-Oct-1999.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
osum.1 | ⊢ 𝐴 ∈ Cℋ |
osum.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
osumi | ⊢ (𝐴 ⊆ (⊥‘𝐵) → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | osum.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝐴 ⊆ (⊥‘𝐵) → 𝐴 ∈ Cℋ ) |
3 | osum.2 | . . . 4 ⊢ 𝐵 ∈ Cℋ | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝐴 ⊆ (⊥‘𝐵) → 𝐵 ∈ Cℋ ) |
5 | 1, 3 | chsscon2i 28873 | . . . 4 ⊢ (𝐴 ⊆ (⊥‘𝐵) ↔ 𝐵 ⊆ (⊥‘𝐴)) |
6 | 5 | biimpi 208 | . . 3 ⊢ (𝐴 ⊆ (⊥‘𝐵) → 𝐵 ⊆ (⊥‘𝐴)) |
7 | 2, 4, 6 | chscl 29051 | . 2 ⊢ (𝐴 ⊆ (⊥‘𝐵) → (𝐴 +ℋ 𝐵) ∈ Cℋ ) |
8 | 1 | chshii 28635 | . . 3 ⊢ 𝐴 ∈ Sℋ |
9 | 3 | chshii 28635 | . . 3 ⊢ 𝐵 ∈ Sℋ |
10 | 8, 9 | shjshseli 28903 | . 2 ⊢ ((𝐴 +ℋ 𝐵) ∈ Cℋ ↔ (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵)) |
11 | 7, 10 | sylib 210 | 1 ⊢ (𝐴 ⊆ (⊥‘𝐵) → (𝐴 +ℋ 𝐵) = (𝐴 ∨ℋ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 ‘cfv 6127 (class class class)co 6910 Cℋ cch 28337 ⊥cort 28338 +ℋ cph 28339 ∨ℋ chj 28341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cc 9579 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 ax-hilex 28407 ax-hfvadd 28408 ax-hvcom 28409 ax-hvass 28410 ax-hv0cl 28411 ax-hvaddid 28412 ax-hfvmul 28413 ax-hvmulid 28414 ax-hvmulass 28415 ax-hvdistr1 28416 ax-hvdistr2 28417 ax-hvmul0 28418 ax-hfi 28487 ax-his1 28490 ax-his2 28491 ax-his3 28492 ax-his4 28493 ax-hcompl 28610 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-omul 7836 df-er 8014 df-map 8129 df-pm 8130 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-fi 8592 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-acn 9088 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ioo 12474 df-ico 12476 df-icc 12477 df-fz 12627 df-fzo 12768 df-fl 12895 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-clim 14603 df-rlim 14604 df-sum 14801 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-rest 16443 df-topn 16444 df-0g 16462 df-gsum 16463 df-topgen 16464 df-pt 16465 df-prds 16468 df-xrs 16522 df-qtop 16527 df-imas 16528 df-xps 16530 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-mulg 17902 df-cntz 18107 df-cmn 18555 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-fbas 20110 df-fg 20111 df-cnfld 20114 df-top 21076 df-topon 21093 df-topsp 21115 df-bases 21128 df-cld 21201 df-ntr 21202 df-cls 21203 df-nei 21280 df-cn 21409 df-cnp 21410 df-lm 21411 df-haus 21497 df-tx 21743 df-hmeo 21936 df-fil 22027 df-fm 22119 df-flim 22120 df-flf 22121 df-xms 22502 df-ms 22503 df-tms 22504 df-cfil 23430 df-cau 23431 df-cmet 23432 df-grpo 27899 df-gid 27900 df-ginv 27901 df-gdiv 27902 df-ablo 27951 df-vc 27965 df-nv 27998 df-va 28001 df-ba 28002 df-sm 28003 df-0v 28004 df-vs 28005 df-nmcv 28006 df-ims 28007 df-dip 28107 df-ssp 28128 df-ph 28219 df-cbn 28270 df-hnorm 28376 df-hba 28377 df-hvsub 28379 df-hlim 28380 df-hcau 28381 df-sh 28615 df-ch 28629 df-oc 28660 df-ch0 28661 df-shs 28718 df-chj 28720 df-pjh 28805 |
This theorem is referenced by: osumcori 29053 osumcor2i 29054 osum 29055 spansnji 29056 5oai 29071 3oalem5 29076 pjsumi 29120 pjdsi 29122 pjds3i 29123 pjssumi 29581 |
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