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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 28991 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ) |
5 | 2, 4 | pjhthlem2 29169 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (⊥‘𝐴)𝑥 = (𝑦 +ℎ 𝑧)) |
6 | eqeq1 2825 | . . . . . . . . 9 ⊢ (𝑥 = if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) → (𝑥 = (𝑦 +ℎ 𝑧) ↔ if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) = (𝑦 +ℎ 𝑧))) | |
7 | eleq1 2900 | . . . . . . . . 9 ⊢ (𝑥 = if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) → (𝑥 ∈ 𝐴 ↔ if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐴)) | |
8 | 6, 7 | imbi12d 347 | . . . . . . . 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 7163 | . . . . . . . . . 10 ⊢ (𝑦 = if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) → (𝑦 +ℎ 𝑧) = (if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) +ℎ 𝑧)) | |
10 | 9 | eqeq2d 2832 | . . . . . . . . 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 344 | . . . . . . . 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 7164 | . . . . . . . . . 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 2832 | . . . . . . . . 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 344 | . . . . . . . 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 29004 | . . . . . . . . 9 ⊢ 𝐴 ∈ Sℋ |
16 | omlsi.4 | . . . . . . . . 9 ⊢ (𝐵 ∩ (⊥‘𝐴)) = 0ℋ | |
17 | sh0 28993 | . . . . . . . . . . 11 ⊢ (𝐵 ∈ Sℋ → 0ℎ ∈ 𝐵) | |
18 | 3, 17 | ax-mp 5 | . . . . . . . . . 10 ⊢ 0ℎ ∈ 𝐵 |
19 | 18 | elimel 4534 | . . . . . . . . 9 ⊢ if(𝑥 ∈ 𝐵, 𝑥, 0ℎ) ∈ 𝐵 |
20 | ch0 29005 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ Cℋ → 0ℎ ∈ 𝐴) | |
21 | 2, 20 | ax-mp 5 | . . . . . . . . . 10 ⊢ 0ℎ ∈ 𝐴 |
22 | 21 | elimel 4534 | . . . . . . . . 9 ⊢ if(𝑦 ∈ 𝐴, 𝑦, 0ℎ) ∈ 𝐴 |
23 | shocsh 29061 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) | |
24 | 15, 23 | ax-mp 5 | . . . . . . . . . . 11 ⊢ (⊥‘𝐴) ∈ Sℋ |
25 | sh0 28993 | . . . . . . . . . . 11 ⊢ ((⊥‘𝐴) ∈ Sℋ → 0ℎ ∈ (⊥‘𝐴)) | |
26 | 24, 25 | ax-mp 5 | . . . . . . . . . 10 ⊢ 0ℎ ∈ (⊥‘𝐴) |
27 | 26 | elimel 4534 | . . . . . . . . 9 ⊢ if(𝑧 ∈ (⊥‘𝐴), 𝑧, 0ℎ) ∈ (⊥‘𝐴) |
28 | 15, 3, 1, 16, 19, 22, 27 | omlsilem 29179 | . . . . . . . 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 4525 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ (⊥‘𝐴)) → (𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴)) |
30 | 29 | 3expia 1117 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ (⊥‘𝐴) → (𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴))) |
31 | 30 | rexlimdv 3283 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ (⊥‘𝐴)𝑥 = (𝑦 +ℎ 𝑧) → 𝑥 ∈ 𝐴)) |
32 | 31 | rexlimdva 3284 | . . . 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 3971 | . 2 ⊢ 𝐵 ⊆ 𝐴 |
35 | 1, 34 | eqssi 3983 | 1 ⊢ 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ∩ cin 3935 ⊆ wss 3936 ifcif 4467 ‘cfv 6355 (class class class)co 7156 +ℎ cva 28697 0ℎc0v 28701 Sℋ csh 28705 Cℋ cch 28706 ⊥cort 28707 0ℋc0h 28712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cc 9857 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvmulass 28784 ax-hvdistr1 28785 ax-hvdistr2 28786 ax-hvmul0 28787 ax-hfi 28856 ax-his1 28859 ax-his2 28860 ax-his3 28861 ax-his4 28862 ax-hcompl 28979 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ico 12745 df-icc 12746 df-fz 12894 df-fl 13163 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-rest 16696 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-top 21502 df-topon 21519 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lm 21837 df-haus 21923 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-cfil 23858 df-cau 23859 df-cmet 23860 df-grpo 28270 df-gid 28271 df-ginv 28272 df-gdiv 28273 df-ablo 28322 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-vs 28376 df-nmcv 28377 df-ims 28378 df-ssp 28499 df-ph 28590 df-cbn 28640 df-hnorm 28745 df-hba 28746 df-hvsub 28748 df-hlim 28749 df-hcau 28750 df-sh 28984 df-ch 28998 df-oc 29029 df-ch0 29030 |
This theorem is referenced by: omlsi 29181 ococi 29182 qlaxr3i 29413 hatomistici 30139 |
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