pjimai - Hilbert Space Explorer

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Theorem pjimai 32263

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 31301	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐴 → 𝑣 ∈ ℋ)
4		pjeq 31486	. . . . . . . . 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 31319	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐵 → 𝑢 ∈ ℋ)
10	1	choccli 31394	. . . . . . . . . . . 12 ⊢ (⊥‘𝐵) ∈ C_ℋ
11	10	cheli 31319	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (⊥‘𝐵) → 𝑤 ∈ ℋ)
12		hvsubadd 31164	. . . . . . . . . . . . 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 31088	. . . . . . . . . . . . . 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 3160	. . . . . . 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 3160	. . . . 5 ⊢ (𝑢 ∈ 𝐵 → (∃𝑣 ∈ 𝐴 ((proj_ℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −_ℎ 𝑤)))
26	1	pjfni 31788	. . . . . 6 ⊢ (proj_ℎ‘𝐵) Fn ℋ
27	2	shssii 31300	. . . . . 6 ⊢ 𝐴 ⊆ ℋ
28		fvelimab 6914	. . . . . 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 31314	. . . . . 6 ⊢ (⊥‘𝐵) ∈ S_ℋ
31		shsel3 31402	. . . . . 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 6038	. . . . . 6 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) ⊆ ran (proj_ℎ‘𝐵)
36	1	pjrni 31789	. . . . . 6 ⊢ ran (proj_ℎ‘𝐵) = 𝐵
37	35, 36	sseqtri 3984	. . . . 5 ⊢ ((proj_ℎ‘𝐵) “ 𝐴) ⊆ 𝐵
38	37	sseli 3931	. . . 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 3919	. . 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 3062 ∩ cin 3902 ⊆ wss 3903 ran crn 5633 “ cima 5635 Fn wfn 6495 ‘cfv 6500 (class class class)co 7368 ℋchba 31006 +_ℎ cva 31007 −_ℎ cmv 31012 S_ℋ csh 31015 C_ℋ cch 31016 ⊥cort 31017 +_ℋ cph 31018 proj_ℎcpjh 31024

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31086 ax-hfvadd 31087 ax-hvcom 31088 ax-hvass 31089 ax-hv0cl 31090 ax-hvaddid 31091 ax-hfvmul 31092 ax-hvmulid 31093 ax-hvmulass 31094 ax-hvdistr1 31095 ax-hvdistr2 31096 ax-hvmul0 31097 ax-hfi 31166 ax-his1 31169 ax-his2 31170 ax-his3 31171 ax-his4 31172 ax-hcompl 31289

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 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-cn 23183 df-cnp 23184 df-lm 23185 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cfil 25223 df-cau 25224 df-cmet 25225 df-grpo 30580 df-gid 30581 df-ginv 30582 df-gdiv 30583 df-ablo 30632 df-vc 30646 df-nv 30679 df-va 30682 df-ba 30683 df-sm 30684 df-0v 30685 df-vs 30686 df-nmcv 30687 df-ims 30688 df-dip 30788 df-ssp 30809 df-ph 30900 df-cbn 30950 df-hnorm 31055 df-hba 31056 df-hvsub 31058 df-hlim 31059 df-hcau 31060 df-sh 31294 df-ch 31308 df-oc 31339 df-ch0 31340 df-shs 31395 df-pjh 31482

This theorem is referenced by: (None)

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