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Theorem spansnpji 31445

Description: A subset of Hilbert space is orthogonal to the span of the singleton of a projection onto its orthocomplement. (Contributed by NM, 4-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
spansnpj.1	⊢ 𝐴 ⊆ ℋ
spansnpj.2	⊢ 𝐵 ∈ ℋ

**Assertion**
Ref	Expression
spansnpji	⊢ 𝐴 ⊆ (⊥‘(span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))

**Proof of Theorem spansnpji**
Step	Hyp	Ref	Expression
1		spansnpj.1	. . 3 ⊢ 𝐴 ⊆ ℋ
2		ococss 31160	. . 3 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
3	1, 2	ax-mp 5	. 2 ⊢ 𝐴 ⊆ (⊥‘(⊥‘𝐴))
4		occl 31171	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ C_ℋ )
5	1, 4	ax-mp 5	. . . . . 6 ⊢ (⊥‘𝐴) ∈ C_ℋ
6	5	chssii 31098	. . . . 5 ⊢ (⊥‘𝐴) ⊆ ℋ
7		spansnpj.2	. . . . . . 7 ⊢ 𝐵 ∈ ℋ
8	5, 7	pjclii 31288	. . . . . 6 ⊢ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ (⊥‘𝐴)
9		snssi 4812	. . . . . 6 ⊢ (((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ (⊥‘𝐴) → {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ (⊥‘𝐴))
10	8, 9	ax-mp 5	. . . . 5 ⊢ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ (⊥‘𝐴)
11		spanss 31215	. . . . 5 ⊢ (((⊥‘𝐴) ⊆ ℋ ∧ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ (⊥‘𝐴)) → (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ⊆ (span‘(⊥‘𝐴)))
12	6, 10, 11	mp2an 690	. . . 4 ⊢ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ⊆ (span‘(⊥‘𝐴))
13	5	chshii 31094	. . . . 5 ⊢ (⊥‘𝐴) ∈ S_ℋ
14		spanid 31214	. . . . 5 ⊢ ((⊥‘𝐴) ∈ S_ℋ → (span‘(⊥‘𝐴)) = (⊥‘𝐴))
15	13, 14	ax-mp 5	. . . 4 ⊢ (span‘(⊥‘𝐴)) = (⊥‘𝐴)
16	12, 15	sseqtri 4014	. . 3 ⊢ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ⊆ (⊥‘𝐴)
17	5, 7	pjhclii 31289	. . . . 5 ⊢ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ
18	17	spansnchi 31429	. . . 4 ⊢ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ∈ C_ℋ
19	18, 5	chsscon3i 31328	. . 3 ⊢ ((span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ⊆ (⊥‘𝐴) ↔ (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
20	16, 19	mpbi 229	. 2 ⊢ (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
21	3, 20	sstri 3987	1 ⊢ 𝐴 ⊆ (⊥‘(span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 ⊆ wss 3945 {csn 4629 ‘cfv 6547 ℋchba 30786 S_ℋ csh 30795 C_ℋ cch 30796 ⊥cort 30797 spancspn 30799 proj_ℎcpjh 30804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cc 10458 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 ax-hilex 30866 ax-hfvadd 30867 ax-hvcom 30868 ax-hvass 30869 ax-hv0cl 30870 ax-hvaddid 30871 ax-hfvmul 30872 ax-hvmulid 30873 ax-hvmulass 30874 ax-hvdistr1 30875 ax-hvdistr2 30876 ax-hvmul0 30877 ax-hfi 30946 ax-his1 30949 ax-his2 30950 ax-his3 30951 ax-his4 30952 ax-hcompl 31069

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-rlim 15466 df-sum 15666 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22827 df-topon 22844 df-topsp 22866 df-bases 22880 df-cld 22954 df-ntr 22955 df-cls 22956 df-nei 23033 df-cn 23162 df-cnp 23163 df-lm 23164 df-haus 23250 df-tx 23497 df-hmeo 23690 df-fil 23781 df-fm 23873 df-flim 23874 df-flf 23875 df-xms 24257 df-ms 24258 df-tms 24259 df-cfil 25214 df-cau 25215 df-cmet 25216 df-grpo 30360 df-gid 30361 df-ginv 30362 df-gdiv 30363 df-ablo 30412 df-vc 30426 df-nv 30459 df-va 30462 df-ba 30463 df-sm 30464 df-0v 30465 df-vs 30466 df-nmcv 30467 df-ims 30468 df-dip 30568 df-ssp 30589 df-ph 30680 df-cbn 30730 df-hnorm 30835 df-hba 30836 df-hvsub 30838 df-hlim 30839 df-hcau 30840 df-sh 31074 df-ch 31088 df-oc 31119 df-ch0 31120 df-shs 31175 df-span 31176 df-pjh 31262

This theorem is referenced by: spansnji 31513

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