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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 31037 | . . . . . . . . 9 ⊢ (𝑣 ∈ 𝐴 → 𝑣 ∈ ℋ) |
4 | pjeq 31222 | . . . . . . . . 9 ⊢ ((𝐵 ∈ Cℋ ∧ 𝑣 ∈ ℋ) → (((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ (𝑢 ∈ 𝐵 ∧ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +ℎ 𝑤)))) | |
5 | 1, 3, 4 | sylancr 586 | . . . . . . . 8 ⊢ (𝑣 ∈ 𝐴 → (((projℎ‘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 222 | . . . . . . . 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) → (((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑣 = (𝑢 +ℎ 𝑤))) |
9 | 1 | cheli 31055 | . . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐵 → 𝑢 ∈ ℋ) |
10 | 1 | choccli 31130 | . . . . . . . . . . . 12 ⊢ (⊥‘𝐵) ∈ Cℋ |
11 | 10 | cheli 31055 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ (⊥‘𝐵) → 𝑤 ∈ ℋ) |
12 | hvsubadd 30900 | . . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ ∧ 𝑢 ∈ ℋ) → ((𝑣 −ℎ 𝑤) = 𝑢 ↔ (𝑤 +ℎ 𝑢) = 𝑣)) | |
13 | 12 | 3comr 1123 | . . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → ((𝑣 −ℎ 𝑤) = 𝑢 ↔ (𝑤 +ℎ 𝑢) = 𝑣)) |
14 | ax-hvcom 30824 | . . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑢 +ℎ 𝑤) = (𝑤 +ℎ 𝑢)) | |
15 | 14 | 3adant2 1129 | . . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ℋ ∧ 𝑣 ∈ ℋ ∧ 𝑤 ∈ ℋ) → (𝑢 +ℎ 𝑤) = (𝑤 +ℎ 𝑢)) |
16 | 15 | eqeq1d 2730 | . . . . . . . . . . . 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 1158 | . . . . . . . . . 10 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ (⊥‘𝐵)) → ((𝑣 −ℎ 𝑤) = 𝑢 ↔ (𝑢 +ℎ 𝑤) = 𝑣)) |
19 | eqcom 2735 | . . . . . . . . . 10 ⊢ (𝑢 = (𝑣 −ℎ 𝑤) ↔ (𝑣 −ℎ 𝑤) = 𝑢) | |
20 | eqcom 2735 | . . . . . . . . . 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 1116 | . . . . . . . 8 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐴) ∧ 𝑤 ∈ (⊥‘𝐵)) → (𝑢 = (𝑣 −ℎ 𝑤) ↔ 𝑣 = (𝑢 +ℎ 𝑤))) |
23 | 22 | rexbidva 3173 | . . . . . . 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) → (((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −ℎ 𝑤))) |
25 | 24 | rexbidva 3173 | . . . . 5 ⊢ (𝑢 ∈ 𝐵 → (∃𝑣 ∈ 𝐴 ((projℎ‘𝐵)‘𝑣) = 𝑢 ↔ ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ (⊥‘𝐵)𝑢 = (𝑣 −ℎ 𝑤))) |
26 | 1 | pjfni 31524 | . . . . . 6 ⊢ (projℎ‘𝐵) Fn ℋ |
27 | 2 | shssii 31036 | . . . . . 6 ⊢ 𝐴 ⊆ ℋ |
28 | fvelimab 6971 | . . . . . 6 ⊢ (((projℎ‘𝐵) Fn ℋ ∧ 𝐴 ⊆ ℋ) → (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ ∃𝑣 ∈ 𝐴 ((projℎ‘𝐵)‘𝑣) = 𝑢)) | |
29 | 26, 27, 28 | mp2an 691 | . . . . 5 ⊢ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ ∃𝑣 ∈ 𝐴 ((projℎ‘𝐵)‘𝑣) = 𝑢) |
30 | 10 | chshii 31050 | . . . . . 6 ⊢ (⊥‘𝐵) ∈ Sℋ |
31 | shsel3 31138 | . . . . . 6 ⊢ ((𝐴 ∈ Sℋ ∧ (⊥‘𝐵) ∈ Sℋ ) → (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 314 | . . . 4 ⊢ (𝑢 ∈ 𝐵 → (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ 𝑢 ∈ (𝐴 +ℋ (⊥‘𝐵)))) |
34 | 33 | pm5.32ri 575 | . . 3 ⊢ ((𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵) ↔ (𝑢 ∈ (𝐴 +ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵)) |
35 | imassrn 6074 | . . . . . 6 ⊢ ((projℎ‘𝐵) “ 𝐴) ⊆ ran (projℎ‘𝐵) | |
36 | 1 | pjrni 31525 | . . . . . 6 ⊢ ran (projℎ‘𝐵) = 𝐵 |
37 | 35, 36 | sseqtri 4016 | . . . . 5 ⊢ ((projℎ‘𝐵) “ 𝐴) ⊆ 𝐵 |
38 | 37 | sseli 3976 | . . . 4 ⊢ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) → 𝑢 ∈ 𝐵) |
39 | 38 | pm4.71i 559 | . . 3 ⊢ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ∧ 𝑢 ∈ 𝐵)) |
40 | elin 3963 | . . 3 ⊢ (𝑢 ∈ ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵) ↔ (𝑢 ∈ (𝐴 +ℋ (⊥‘𝐵)) ∧ 𝑢 ∈ 𝐵)) | |
41 | 34, 39, 40 | 3bitr4i 303 | . 2 ⊢ (𝑢 ∈ ((projℎ‘𝐵) “ 𝐴) ↔ 𝑢 ∈ ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵)) |
42 | 41 | eqriv 2725 | 1 ⊢ ((projℎ‘𝐵) “ 𝐴) = ((𝐴 +ℋ (⊥‘𝐵)) ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 ∩ cin 3946 ⊆ wss 3947 ran crn 5679 “ cima 5681 Fn wfn 6543 ‘cfv 6548 (class class class)co 7420 ℋchba 30742 +ℎ cva 30743 −ℎ cmv 30748 Sℋ csh 30751 Cℋ cch 30752 ⊥cort 30753 +ℋ cph 30754 projℎcpjh 30760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cc 10459 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 ax-hilex 30822 ax-hfvadd 30823 ax-hvcom 30824 ax-hvass 30825 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 ax-hvmulass 30830 ax-hvdistr1 30831 ax-hvdistr2 30832 ax-hvmul0 30833 ax-hfi 30902 ax-his1 30905 ax-his2 30906 ax-his3 30907 ax-his4 30908 ax-hcompl 31025 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 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 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-acn 9966 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-rlim 15466 df-sum 15666 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-cn 23144 df-cnp 23145 df-lm 23146 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cfil 25196 df-cau 25197 df-cmet 25198 df-grpo 30316 df-gid 30317 df-ginv 30318 df-gdiv 30319 df-ablo 30368 df-vc 30382 df-nv 30415 df-va 30418 df-ba 30419 df-sm 30420 df-0v 30421 df-vs 30422 df-nmcv 30423 df-ims 30424 df-dip 30524 df-ssp 30545 df-ph 30636 df-cbn 30686 df-hnorm 30791 df-hba 30792 df-hvsub 30794 df-hlim 30795 df-hcau 30796 df-sh 31030 df-ch 31044 df-oc 31075 df-ch0 31076 df-shs 31131 df-pjh 31218 |
This theorem is referenced by: (None) |
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