pjimai - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > pjimai		Structured version Visualization version GIF version

Theorem pjimai 30534

Description: The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective", https://doi.org/10.48550/arXiv.quant-ph/0701113. (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 29572	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐴 → 𝑣 ∈ ℋ)
4		pjeq 29757	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝑣 ∈ ℋ) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
5	1, 3, 4	sylancr 587	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
6		ibar 529	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐵 → (∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤) ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
7	6	bicomd 222	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐵 → ((𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)) ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
8	5, 7	sylan9bbr 511	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
9	1	cheli 29590	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐵 → 𝑢 ∈ ℋ)
10	1	choccli 29665	. . . . . . . . . . . 12 ⊢ (⊥‘𝐵) ∈ C_ℋ
11	10	cheli 29590	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (⊥‘𝐵) → 𝑤 ∈ ℋ)
12		hvsubadd 29435	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
13	12	3comr 1124	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
14		ax-hvcom 29359	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑢 +_ℎ 𝑤) = (𝑤 +_ℎ 𝑢))
15	14	3adant2 1130	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑢 +_ℎ 𝑤) = (𝑤 +_ℎ 𝑢))
16	15	eqeq1d 2742	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑢 +_ℎ 𝑤) = 𝑣 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
17	13, 16	bitr4d 281	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑢 +_ℎ 𝑤) = 𝑣))
18	9, 3, 11, 17	syl3an 1159	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ (⊥‘𝐵)) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑢 +_ℎ 𝑤) = 𝑣))
19		eqcom 2747	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 −_ℎ 𝑤) ↔ (𝑣 −_ℎ 𝑤) = 𝑢)
20		eqcom 2747	. . . . . . . . . 10 ⊢ (𝑣 = (𝑢 +_ℎ 𝑤) ↔ (𝑢 +_ℎ 𝑤) = 𝑣)
21	18, 19, 20	3bitr4g 314	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ (⊥‘𝐵)) → (𝑢 = (𝑣 −_ℎ 𝑤) ↔ 𝑣 = (𝑢 +_ℎ 𝑤)))
22	21	3expa 1117	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) ∧ 𝑤 ∈ (⊥‘𝐵)) → (𝑢 = (𝑣 −_ℎ 𝑤) ↔ 𝑣 = (𝑢 +_ℎ 𝑤)))
23	22	rexbidva 3227	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤) ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
24	8, 23	bitr4d 281	. . . . . 6 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
25	24	rexbidva 3227	. . . . 5 ⊢ (𝑢 ∈ 𝐵 → (∃𝑣 ∈ 𝐴 ((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
26	1	pjfni 30059	. . . . . 6 ⊢ (proj_ℎ‘𝐵) Fn ℋ
27	2	shssii 29571	. . . . . 6 ⊢ 𝐴 ⊆ ℋ
28		fvelimab 6838	. . . . . 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 689	. . . . 5 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ ∃𝑣 ∈ 𝐴 ((proj_ℎ‘𝐵)‘𝑣) = 𝑢)
30	10	chshii 29585	. . . . . 6 ⊢ (⊥‘𝐵) ∈ S_ℋ
31		shsel3 29673	. . . . . 6 ⊢ ((𝐴 ∈ S_ℋ ∧ (⊥‘𝐵) ∈ S_ℋ ) → (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
32	2, 30, 31	mp2an 689	. . . . 5 ⊢ (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤))
33	25, 29, 32	3bitr4g 314	. . . 4 ⊢ (𝑢 ∈ 𝐵 → (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ 𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵))))
34	33	pm5.32ri 576	. . 3 ⊢ ((𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵) ↔ (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵))
35		imassrn 5979	. . . . . 6 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) ⊆ ran (proj_ℎ‘𝐵)
36	1	pjrni 30060	. . . . . 6 ⊢ ran (proj_ℎ‘𝐵) = 𝐵
37	35, 36	sseqtri 3962	. . . . 5 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) ⊆ 𝐵
38	37	sseli 3922	. . . 4 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) → 𝑢 ∈ 𝐵)
39	38	pm4.71i 560	. . 3 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵))
40		elin 3908	. . 3 ⊢ (𝑢 ∈ ((𝐴 +_ℋ (⊥‘𝐵)) ∩ 𝐵) ↔ (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵))
41	34, 39, 40	3bitr4i 303	. 2 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ 𝑢 ∈ ((𝐴 +_ℋ (⊥‘𝐵)) ∩ 𝐵))
42	41	eqriv 2737	1 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) = ((𝐴 +_ℋ (⊥‘𝐵)) ∩ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 ∩ cin 3891 ⊆ wss 3892 ran crn 5591 “ cima 5593 Fn wfn 6427 ‘cfv 6432 (class class class)co 7271 ℋchba 29277 +_ℎ cva 29278 −_ℎ cmv 29283 S_ℋ csh 29286 C_ℋ cch 29287 ⊥cort 29288 +_ℋ cph 29289 proj_ℎcpjh 29295

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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-inf2 9377 ax-cc 10192 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-addf 10951 ax-mulf 10952 ax-hilex 29357 ax-hfvadd 29358 ax-hvcom 29359 ax-hvass 29360 ax-hv0cl 29361 ax-hvaddid 29362 ax-hfvmul 29363 ax-hvmulid 29364 ax-hvmulass 29365 ax-hvdistr1 29366 ax-hvdistr2 29367 ax-hvmul0 29368 ax-hfi 29437 ax-his1 29440 ax-his2 29441 ax-his3 29442 ax-his4 29443 ax-hcompl 29560

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-oadd 8292 df-omul 8293 df-er 8481 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-acn 9701 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-ico 13084 df-icc 13085 df-fz 13239 df-fzo 13382 df-fl 13510 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-rlim 15196 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-cn 22376 df-cnp 22377 df-lm 22378 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cfil 24417 df-cau 24418 df-cmet 24419 df-grpo 28851 df-gid 28852 df-ginv 28853 df-gdiv 28854 df-ablo 28903 df-vc 28917 df-nv 28950 df-va 28953 df-ba 28954 df-sm 28955 df-0v 28956 df-vs 28957 df-nmcv 28958 df-ims 28959 df-dip 29059 df-ssp 29080 df-ph 29171 df-cbn 29221 df-hnorm 29326 df-hba 29327 df-hvsub 29329 df-hlim 29330 df-hcau 29331 df-sh 29565 df-ch 29579 df-oc 29610 df-ch0 29611 df-shs 29666 df-pjh 29753

This theorem is referenced by: (None)

W3C validator