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Mirrors > Home > HSE Home > Th. List > omlsii | Structured version Visualization version GIF version |
Description: Subspace inference form of orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
omlsi.1 | ⊢ 𝐴 ∈ Cℋ |
omlsi.2 | ⊢ 𝐵 ∈ Sℋ |
omlsi.3 | ⊢ 𝐴 ⊆ 𝐵 |
omlsi.4 | ⊢ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ |
Ref | Expression |
---|---|
omlsii | ⊢ 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omlsi.3 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | omlsi.1 | . . . . 5 ⊢ 𝐴 ∈ Cℋ | |
3 | omlsi.2 | . . . . . 6 ⊢ 𝐵 ∈ Sℋ | |
4 | 3 | sheli 29162 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ) |
5 | 2, 4 | pjhthlem2 29340 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (⊥‘𝐴)𝑥 = (𝑦 +ℎ 𝑧)) |
6 | eqeq1 2743 | . . . . . . . . 9 ⊢ (𝑥 = if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) → (𝑥 = (𝑦 +ℎ 𝑧) ↔ if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (𝑦 +ℎ 𝑧))) | |
7 | eleq1 2821 | . . . . . . . . 9 ⊢ (𝑥 = if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) → (𝑥 ∈ 𝐴 ↔ if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐴)) | |
8 | 6, 7 | imbi12d 348 | . . . . . . . 8 ⊢ (𝑥 = if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) → ((𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴) ↔ (if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (𝑦 +ℎ 𝑧) → if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐴))) |
9 | oveq1 7190 | . . . . . . . . . 10 ⊢ (𝑦 = if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) → (𝑦 +ℎ 𝑧) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ 𝑧)) | |
10 | 9 | eqeq2d 2750 | . . . . . . . . 9 ⊢ (𝑦 = if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) → (if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (𝑦 +ℎ 𝑧) ↔ if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ 𝑧))) |
11 | 10 | imbi1d 345 | . . . . . . . 8 ⊢ (𝑦 = if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) → ((if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (𝑦 +ℎ 𝑧) → if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐴) ↔ (if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ 𝑧) → if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐴))) |
12 | oveq2 7191 | . . . . . . . . . 10 ⊢ (𝑧 = if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ) → (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ 𝑧) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ))) | |
13 | 12 | eqeq2d 2750 | . . . . . . . . 9 ⊢ (𝑧 = if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ) → (if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ 𝑧) ↔ if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ)))) |
14 | 13 | imbi1d 345 | . . . . . . . 8 ⊢ (𝑧 = if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ) → ((if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ 𝑧) → if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐴) ↔ (if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ)) → if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐴))) |
15 | 2 | chshii 29175 | . . . . . . . . 9 ⊢ 𝐴 ∈ Sℋ |
16 | omlsi.4 | . . . . . . . . 9 ⊢ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ | |
17 | sh0 29164 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ Sℋ → 0ℎ ∈ 𝐵) | |
18 | 3, 17 | ax-mp 5 | . . . . . . . . . 10 ⊢ 0ℎ ∈ 𝐵 |
19 | 18 | elimel 4493 | . . . . . . . . 9 ⊢ if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐵 |
20 | ch0 29176 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ Cℋ → 0ℎ ∈ 𝐴) | |
21 | 2, 20 | ax-mp 5 | . . . . . . . . . 10 ⊢ 0ℎ ∈ 𝐴 |
22 | 21 | elimel 4493 | . . . . . . . . 9 ⊢ if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) ∈ 𝐴 |
23 | shocsh 29232 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) | |
24 | 15, 23 | ax-mp 5 | . . . . . . . . . . 11 ⊢ (⊥‘𝐴) ∈ Sℋ |
25 | sh0 29164 | . . . . . . . . . . 11 ⊢ ((⊥‘𝐴) ∈ Sℋ → 0ℎ ∈ (⊥‘𝐴)) | |
26 | 24, 25 | ax-mp 5 | . . . . . . . . . 10 ⊢ 0ℎ ∈ (⊥‘𝐴) |
27 | 26 | elimel 4493 | . . . . . . . . 9 ⊢ if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ) ∈ (⊥‘𝐴) |
28 | 15, 3, 1, 16, 19, 22, 27 | omlsilem 29350 | . . . . . . . 8 ⊢ (if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ)) → if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐴) |
29 | 8, 11, 14, 28 | dedth3h 4484 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ (⊥‘𝐴)) → (𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴)) |
30 | 29 | 3expia 1122 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ (⊥‘𝐴) → (𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴))) |
31 | 30 | rexlimdv 3194 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ (⊥‘𝐴)𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴)) |
32 | 31 | rexlimdva 3195 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (⊥‘𝐴)𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴)) |
33 | 5, 32 | mpd 15 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴) |
34 | 33 | ssriv 3891 | . 2 ⊢ 𝐵 ⊆ 𝐴 |
35 | 1, 34 | eqssi 3903 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃wrex 3055 ∩ cin 3852 ⊆ wss 3853 ifcif 4424 ‘cfv 6350 (class class class)co 7183 +ℎ cva 28868 0ℎc0v 28872 Sℋ csh 28876 Cℋ cch 28877 ⊥cort 28878 0ℋc0h 28883 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-cc 9948 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 ax-addf 10707 ax-mulf 10708 ax-hilex 28947 ax-hfvadd 28948 ax-hvcom 28949 ax-hvass 28950 ax-hv0cl 28951 ax-hvaddid 28952 ax-hfvmul 28953 ax-hvmulid 28954 ax-hvmulass 28955 ax-hvdistr1 28956 ax-hvdistr2 28957 ax-hvmul0 28958 ax-hfi 29027 ax-his1 29030 ax-his2 29031 ax-his3 29032 ax-his4 29033 ax-hcompl 29150 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-oadd 8148 df-omul 8149 df-er 8333 df-map 8452 df-pm 8453 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fi 8961 df-sup 8992 df-inf 8993 df-oi 9060 df-card 9454 df-acn 9457 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-n0 11990 df-z 12076 df-uz 12338 df-q 12444 df-rp 12486 df-xneg 12603 df-xadd 12604 df-xmul 12605 df-ico 12840 df-icc 12841 df-fz 12995 df-fl 13266 df-seq 13474 df-exp 13535 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-clim 14948 df-rlim 14949 df-rest 16812 df-topgen 16833 df-psmet 20222 df-xmet 20223 df-met 20224 df-bl 20225 df-mopn 20226 df-fbas 20227 df-fg 20228 df-top 21658 df-topon 21675 df-bases 21710 df-cld 21783 df-ntr 21784 df-cls 21785 df-nei 21862 df-lm 21993 df-haus 22079 df-fil 22610 df-fm 22702 df-flim 22703 df-flf 22704 df-cfil 24020 df-cau 24021 df-cmet 24022 df-grpo 28441 df-gid 28442 df-ginv 28443 df-gdiv 28444 df-ablo 28493 df-vc 28507 df-nv 28540 df-va 28543 df-ba 28544 df-sm 28545 df-0v 28546 df-vs 28547 df-nmcv 28548 df-ims 28549 df-ssp 28670 df-ph 28761 df-cbn 28811 df-hnorm 28916 df-hba 28917 df-hvsub 28919 df-hlim 28920 df-hcau 28921 df-sh 29155 df-ch 29169 df-oc 29200 df-ch0 29201 |
This theorem is referenced by: omlsi 29352 ococi 29353 qlaxr3i 29584 hatomistici 30310 |
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