pjimai - Hilbert Space Explorer

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Theorem pjimai 29959

Description: The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective" http://www.arxiv.org/pdf/quant-ph/0701113 p. 20. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
pjima.1	⊢ 𝐴 ∈ S_ℋ
pjima.2	⊢ 𝐵 ∈ C_ℋ

**Assertion**
Ref	Expression
pjimai	⊢ ((proj_ℎ‘𝐵) “ 𝐴) = ((𝐴 +_ℋ (⊥‘𝐵)) ∩ 𝐵)

Proof of Theorem pjimai Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjima.2	. . . . . . . . 9 ⊢ 𝐵 ∈ C_ℋ
2		pjima.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ S_ℋ
3	2	sheli 28997	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐴 → 𝑣 ∈ ℋ)
4		pjeq 29182	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝑣 ∈ ℋ) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
5	1, 3, 4	sylancr 590	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
6		ibar 532	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐵 → (∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤) ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
7	6	bicomd 226	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐵 → ((𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)) ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
8	5, 7	sylan9bbr 514	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
9	1	cheli 29015	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐵 → 𝑢 ∈ ℋ)
10	1	choccli 29090	. . . . . . . . . . . 12 ⊢ (⊥‘𝐵) ∈ C_ℋ
11	10	cheli 29015	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (⊥‘𝐵) → 𝑤 ∈ ℋ)
12		hvsubadd 28860	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
13	12	3comr 1122	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
14		ax-hvcom 28784	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑢 +_ℎ 𝑤) = (𝑤 +_ℎ 𝑢))
15	14	3adant2 1128	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑢 +_ℎ 𝑤) = (𝑤 +_ℎ 𝑢))
16	15	eqeq1d 2800	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑢 +_ℎ 𝑤) = 𝑣 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
17	13, 16	bitr4d 285	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑢 +_ℎ 𝑤) = 𝑣))
18	9, 3, 11, 17	syl3an 1157	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ (⊥‘𝐵)) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑢 +_ℎ 𝑤) = 𝑣))
19		eqcom 2805	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 −_ℎ 𝑤) ↔ (𝑣 −_ℎ 𝑤) = 𝑢)
20		eqcom 2805	. . . . . . . . . 10 ⊢ (𝑣 = (𝑢 +_ℎ 𝑤) ↔ (𝑢 +_ℎ 𝑤) = 𝑣)
21	18, 19, 20	3bitr4g 317	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ (⊥‘𝐵)) → (𝑢 = (𝑣 −_ℎ 𝑤) ↔ 𝑣 = (𝑢 +_ℎ 𝑤)))
22	21	3expa 1115	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) ∧ 𝑤 ∈ (⊥‘𝐵)) → (𝑢 = (𝑣 −_ℎ 𝑤) ↔ 𝑣 = (𝑢 +_ℎ 𝑤)))
23	22	rexbidva 3255	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤) ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
24	8, 23	bitr4d 285	. . . . . 6 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
25	24	rexbidva 3255	. . . . 5 ⊢ (𝑢 ∈ 𝐵 → (∃𝑣 ∈ 𝐴 ((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
26	1	pjfni 29484	. . . . . 6 ⊢ (proj_ℎ‘𝐵) Fn ℋ
27	2	shssii 28996	. . . . . 6 ⊢ 𝐴 ⊆ ℋ
28		fvelimab 6712	. . . . . 6 ⊢ (((proj_ℎ‘𝐵) Fn ℋ ∧ 𝐴 ⊆ ℋ) → (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ ∃𝑣 ∈ 𝐴 ((proj_ℎ‘𝐵)‘𝑣) = 𝑢))
29	26, 27, 28	mp2an 691	. . . . 5 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ ∃𝑣 ∈ 𝐴 ((proj_ℎ‘𝐵)‘𝑣) = 𝑢)
30	10	chshii 29010	. . . . . 6 ⊢ (⊥‘𝐵) ∈ S_ℋ
31		shsel3 29098	. . . . . 6 ⊢ ((𝐴 ∈ S_ℋ ∧ (⊥‘𝐵) ∈ S_ℋ ) → (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
32	2, 30, 31	mp2an 691	. . . . 5 ⊢ (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤))
33	25, 29, 32	3bitr4g 317	. . . 4 ⊢ (𝑢 ∈ 𝐵 → (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ 𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵))))
34	33	pm5.32ri 579	. . 3 ⊢ ((𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵) ↔ (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵))
35		imassrn 5907	. . . . . 6 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) ⊆ ran (proj_ℎ‘𝐵)
36	1	pjrni 29485	. . . . . 6 ⊢ ran (proj_ℎ‘𝐵) = 𝐵
37	35, 36	sseqtri 3951	. . . . 5 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) ⊆ 𝐵
38	37	sseli 3911	. . . 4 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) → 𝑢 ∈ 𝐵)
39	38	pm4.71i 563	. . 3 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵))
40		elin 3897	. . 3 ⊢ (𝑢 ∈ ((𝐴 +_ℋ (⊥‘𝐵)) ∩ 𝐵) ↔ (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵))
41	34, 39, 40	3bitr4i 306	. 2 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ 𝑢 ∈ ((𝐴 +_ℋ (⊥‘𝐵)) ∩ 𝐵))
42	41	eqriv 2795	1 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) = ((𝐴 +_ℋ (⊥‘𝐵)) ∩ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 ran crn 5520 “ cima 5522 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ℋchba 28702 +_ℎ cva 28703 −_ℎ cmv 28708 S_ℋ csh 28711 C_ℋ cch 28712 ⊥cort 28713 +_ℋ cph 28714 proj_ℎcpjh 28720

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 ax-hcompl 28985

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-lm 21834 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cfil 23859 df-cau 23860 df-cmet 23861 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-dip 28484 df-ssp 28505 df-ph 28596 df-cbn 28646 df-hnorm 28751 df-hba 28752 df-hvsub 28754 df-hlim 28755 df-hcau 28756 df-sh 28990 df-ch 29004 df-oc 29035 df-ch0 29036 df-shs 29091 df-pjh 29178

This theorem is referenced by: (None)

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