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 32234

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 31272	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐴 → 𝑣 ∈ ℋ)
4		pjeq 31457	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝑣 ∈ ℋ) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
5	1, 3, 4	sylancr 588	. . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
6		ibar 528	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝐵 → (∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤) ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤))))
7	6	bicomd 223	. . . . . . . 8 ⊢ (𝑢 ∈ 𝐵 → ((𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)) ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
8	5, 7	sylan9bbr 510	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
9	1	cheli 31290	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐵 → 𝑢 ∈ ℋ)
10	1	choccli 31365	. . . . . . . . . . . 12 ⊢ (⊥‘𝐵) ∈ C_ℋ
11	10	cheli 31290	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (⊥‘𝐵) → 𝑤 ∈ ℋ)
12		hvsubadd 31135	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
13	12	3comr 1126	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
14		ax-hvcom 31059	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑢 +_ℎ 𝑤) = (𝑤 +_ℎ 𝑢))
15	14	3adant2 1132	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑢 +_ℎ 𝑤) = (𝑤 +_ℎ 𝑢))
16	15	eqeq1d 2739	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑢 +_ℎ 𝑤) = 𝑣 ↔ (𝑤 +_ℎ 𝑢) = 𝑣))
17	13, 16	bitr4d 282	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑢 +_ℎ 𝑤) = 𝑣))
18	9, 3, 11, 17	syl3an 1161	. . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ (⊥‘𝐵)) → ((𝑣 −_ℎ 𝑤) = 𝑢 ↔ (𝑢 +_ℎ 𝑤) = 𝑣))
19		eqcom 2744	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 −_ℎ 𝑤) ↔ (𝑣 −_ℎ 𝑤) = 𝑢)
20		eqcom 2744	. . . . . . . . . 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 1119	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) ∧ 𝑤 ∈ (⊥‘𝐵)) → (𝑢 = (𝑣 −_ℎ 𝑤) ↔ 𝑣 = (𝑢 +_ℎ 𝑤)))
23	22	rexbidva 3159	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤) ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +_ℎ 𝑤)))
24	8, 23	bitr4d 282	. . . . . 6 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
25	24	rexbidva 3159	. . . . 5 ⊢ (𝑢 ∈ 𝐵 → (∃𝑣 ∈ 𝐴 ((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
26	1	pjfni 31759	. . . . . 6 ⊢ (proj_ℎ‘𝐵) Fn ℋ
27	2	shssii 31271	. . . . . 6 ⊢ 𝐴 ⊆ ℋ
28		fvelimab 6907	. . . . . 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 693	. . . . 5 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ ∃𝑣 ∈ 𝐴 ((proj_ℎ‘𝐵)‘𝑣) = 𝑢)
30	10	chshii 31285	. . . . . 6 ⊢ (⊥‘𝐵) ∈ S_ℋ
31		shsel3 31373	. . . . . 6 ⊢ ((𝐴 ∈ S_ℋ ∧ (⊥‘𝐵) ∈ S_ℋ ) → (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
32	2, 30, 31	mp2an 693	. . . . 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 575	. . 3 ⊢ ((𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵) ↔ (𝑢 ∈ (𝐴 +_ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵))
35		imassrn 6031	. . . . . 6 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) ⊆ ran (proj_ℎ‘𝐵)
36	1	pjrni 31760	. . . . . 6 ⊢ ran (proj_ℎ‘𝐵) = 𝐵
37	35, 36	sseqtri 3983	. . . . 5 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) ⊆ 𝐵
38	37	sseli 3930	. . . 4 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) → 𝑢 ∈ 𝐵)
39	38	pm4.71i 559	. . 3 ⊢ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ↔ (𝑢 ∈ ((proj_ℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵))
40		elin 3918	. . 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 2734	1 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) = ((𝐴 +_ℋ (⊥‘𝐵)) ∩ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 ∩ cin 3901 ⊆ wss 3902 ran crn 5626 “ cima 5628 Fn wfn 6488 ‘cfv 6493 (class class class)co 7360 ℋchba 30977 +_ℎ cva 30978 −_ℎ cmv 30983 S_ℋ csh 30986 C_ℋ cch 30987 ⊥cort 30988 +_ℋ cph 30989 proj_ℎcpjh 30995

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110 ax-hilex 31057 ax-hfvadd 31058 ax-hvcom 31059 ax-hvass 31060 ax-hv0cl 31061 ax-hvaddid 31062 ax-hfvmul 31063 ax-hvmulid 31064 ax-hvmulass 31065 ax-hvdistr1 31066 ax-hvdistr2 31067 ax-hvmul0 31068 ax-hfi 31137 ax-his1 31140 ax-his2 31141 ax-his3 31142 ax-his4 31143 ax-hcompl 31260

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-acn 9858 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-rlim 15416 df-sum 15614 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-cn 23175 df-cnp 23176 df-lm 23177 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-tms 24270 df-cfil 25215 df-cau 25216 df-cmet 25217 df-grpo 30551 df-gid 30552 df-ginv 30553 df-gdiv 30554 df-ablo 30603 df-vc 30617 df-nv 30650 df-va 30653 df-ba 30654 df-sm 30655 df-0v 30656 df-vs 30657 df-nmcv 30658 df-ims 30659 df-dip 30759 df-ssp 30780 df-ph 30871 df-cbn 30921 df-hnorm 31026 df-hba 31027 df-hvsub 31029 df-hlim 31030 df-hcau 31031 df-sh 31265 df-ch 31279 df-oc 31310 df-ch0 31311 df-shs 31366 df-pjh 31453

This theorem is referenced by: (None)

W3C validator