![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > pjimai | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
pjima.1 | ⊢ 𝐴 ∈ Sℋ |
pjima.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
pjimai | ⊢ ((projℎ‘𝐵) “ 𝐴) = ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjima.2 | . . . . . . . . 9 ⊢ 𝐵 ∈ Cℋ | |
2 | pjima.1 | . . . . . . . . . 10 ⊢ 𝐴 ∈ Sℋ | |
3 | 2 | sheli 31101 | . . . . . . . . 9 ⊢ (𝑣 ∈ 𝐴 → 𝑣 ∈ ℋ) |
4 | pjeq 31286 | . . . . . . . . 9 ⊢ ((𝐵 ∈ Cℋ ∧ 𝑣 ∈ ℋ) → (((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +ℎ 𝑤)))) | |
5 | 1, 3, 4 | sylancr 585 | . . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → (((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +ℎ 𝑤)))) |
6 | ibar 527 | . . . . . . . . 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 509 | . . . . . . 7 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) → (((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +ℎ 𝑤))) |
9 | 1 | cheli 31119 | . . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐵 → 𝑢 ∈ ℋ) |
10 | 1 | choccli 31194 | . . . . . . . . . . . 12 ⊢ (⊥‘𝐵) ∈ Cℋ |
11 | 10 | cheli 31119 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ (⊥‘𝐵) → 𝑤 ∈ ℋ) |
12 | hvsubadd 30964 | . . . . . . . . . . . . 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 30888 | . . . . . . . . . . . . . 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 2727 | . . . . . . . . . . . 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 1157 | . . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ (⊥‘𝐵)) → ((𝑣 −ℎ 𝑤) = 𝑢 ↔ (𝑢 +ℎ 𝑤) = 𝑣)) |
19 | eqcom 2732 | . . . . . . . . . 10 ⊢ (𝑢 = (𝑣 −ℎ 𝑤) ↔ (𝑣 −ℎ 𝑤) = 𝑢) | |
20 | eqcom 2732 | . . . . . . . . . 10 ⊢ (𝑣 = (𝑢 +ℎ 𝑤) ↔ (𝑢 +ℎ 𝑤) = 𝑣) | |
21 | 18, 19, 20 | 3bitr4g 313 | . . . . . . . . 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 3166 | . . . . . . 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) → (((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −ℎ 𝑤))) |
25 | 24 | rexbidva 3166 | . . . . 5 ⊢ (𝑢 ∈ 𝐵 → (∃𝑣 ∈ 𝐴 ((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −ℎ 𝑤))) |
26 | 1 | pjfni 31588 | . . . . . 6 ⊢ (projℎ‘𝐵) Fn ℋ |
27 | 2 | shssii 31100 | . . . . . 6 ⊢ 𝐴 ⊆ ℋ |
28 | fvelimab 6970 | . . . . . 6 ⊢ (((projℎ‘𝐵) Fn ℋ ∧ 𝐴 ⊆ ℋ) → (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ ∃𝑣 ∈ 𝐴 ((projℎ‘𝐵)‘𝑣) = 𝑢)) | |
29 | 26, 27, 28 | mp2an 690 | . . . . 5 ⊢ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ ∃𝑣 ∈ 𝐴 ((projℎ‘𝐵)‘𝑣) = 𝑢) |
30 | 10 | chshii 31114 | . . . . . 6 ⊢ (⊥‘𝐵) ∈ Sℋ |
31 | shsel3 31202 | . . . . . 6 ⊢ ((𝐴 ∈ Sℋ ∧ (⊥‘𝐵) ∈ Sℋ ) → (𝑢 ∈ (𝐴 +ℋ (⊥‘𝐵)) ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −ℎ 𝑤))) | |
32 | 2, 30, 31 | mp2an 690 | . . . . 5 ⊢ (𝑢 ∈ (𝐴 +ℋ (⊥‘𝐵)) ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −ℎ 𝑤)) |
33 | 25, 29, 32 | 3bitr4g 313 | . . . 4 ⊢ (𝑢 ∈ 𝐵 → (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ 𝑢 ∈ (𝐴 +ℋ (⊥‘𝐵)))) |
34 | 33 | pm5.32ri 574 | . . 3 ⊢ ((𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵) ↔ (𝑢 ∈ (𝐴 +ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵)) |
35 | imassrn 6075 | . . . . . 6 ⊢ ((projℎ‘𝐵) “ 𝐴) ⊆ ran (projℎ‘𝐵) | |
36 | 1 | pjrni 31589 | . . . . . 6 ⊢ ran (projℎ‘𝐵) = 𝐵 |
37 | 35, 36 | sseqtri 4013 | . . . . 5 ⊢ ((projℎ‘𝐵) “ 𝐴) ⊆ 𝐵 |
38 | 37 | sseli 3972 | . . . 4 ⊢ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) → 𝑢 ∈ 𝐵) |
39 | 38 | pm4.71i 558 | . . 3 ⊢ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵)) |
40 | elin 3960 | . . 3 ⊢ (𝑢 ∈ ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵) ↔ (𝑢 ∈ (𝐴 +ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵)) | |
41 | 34, 39, 40 | 3bitr4i 302 | . 2 ⊢ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ 𝑢 ∈ ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵)) |
42 | 41 | eqriv 2722 | 1 ⊢ ((projℎ‘𝐵) “ 𝐴) = ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 ∩ cin 3943 ⊆ wss 3944 ran crn 5679 “ cima 5681 Fn wfn 6544 ‘cfv 6549 (class class class)co 7419 ℋchba 30806 +ℎ cva 30807 −ℎ cmv 30812 Sℋ csh 30815 Cℋ cch 30816 ⊥cort 30817 +ℋ cph 30818 projℎcpjh 30824 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cc 10465 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225 ax-hilex 30886 ax-hfvadd 30887 ax-hvcom 30888 ax-hvass 30889 ax-hv0cl 30890 ax-hvaddid 30891 ax-hfvmul 30892 ax-hvmulid 30893 ax-hvmulass 30894 ax-hvdistr1 30895 ax-hvdistr2 30896 ax-hvmul0 30897 ax-hfi 30966 ax-his1 30969 ax-his2 30970 ax-his3 30971 ax-his4 30972 ax-hcompl 31089 |
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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-card 9969 df-acn 9972 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-seq 14008 df-exp 14068 df-hash 14331 df-cj 15087 df-re 15088 df-im 15089 df-sqrt 15223 df-abs 15224 df-clim 15473 df-rlim 15474 df-sum 15674 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-starv 17256 df-sca 17257 df-vsca 17258 df-ip 17259 df-tset 17260 df-ple 17261 df-ds 17263 df-unif 17264 df-hom 17265 df-cco 17266 df-rest 17412 df-topn 17413 df-0g 17431 df-gsum 17432 df-topgen 17433 df-pt 17434 df-prds 17437 df-xrs 17492 df-qtop 17497 df-imas 17498 df-xps 17500 df-mre 17574 df-mrc 17575 df-acs 17577 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18749 df-mulg 19037 df-cntz 19285 df-cmn 19754 df-psmet 21293 df-xmet 21294 df-met 21295 df-bl 21296 df-mopn 21297 df-fbas 21298 df-fg 21299 df-cnfld 21302 df-top 22845 df-topon 22862 df-topsp 22884 df-bases 22898 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-cn 23180 df-cnp 23181 df-lm 23182 df-haus 23268 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24275 df-ms 24276 df-tms 24277 df-cfil 25232 df-cau 25233 df-cmet 25234 df-grpo 30380 df-gid 30381 df-ginv 30382 df-gdiv 30383 df-ablo 30432 df-vc 30446 df-nv 30479 df-va 30482 df-ba 30483 df-sm 30484 df-0v 30485 df-vs 30486 df-nmcv 30487 df-ims 30488 df-dip 30588 df-ssp 30609 df-ph 30700 df-cbn 30750 df-hnorm 30855 df-hba 30856 df-hvsub 30858 df-hlim 30859 df-hcau 30860 df-sh 31094 df-ch 31108 df-oc 31139 df-ch0 31140 df-shs 31195 df-pjh 31282 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |