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Mirrors > Home > MPE Home > Th. List > Mathboxes > opoccl | Structured version Visualization version GIF version |
Description: Closure of orthocomplement operation. (choccl 30422 analog.) (Contributed by NM, 20-Oct-2011.) |
Ref | Expression |
---|---|
opoccl.b | ⊢ 𝐵 = (Base‘𝐾) |
opoccl.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opoccl | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opoccl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2731 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opoccl.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
4 | eqid 2731 | . . . . 5 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | eqid 2731 | . . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | eqid 2731 | . . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | eqid 2731 | . . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 37857 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾))) |
9 | 8 | 3anidm23 1421 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋))) ∧ (𝑋(join‘𝐾)( ⊥ ‘𝑋)) = (1.‘𝐾) ∧ (𝑋(meet‘𝐾)( ⊥ ‘𝑋)) = (0.‘𝐾))) |
10 | 9 | simp1d 1142 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋)))) |
11 | 10 | simp1d 1142 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 lecple 17186 occoc 17187 joincjn 18246 meetcmee 18247 0.cp0 18358 1.cp1 18359 OPcops 37847 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-dm 5679 df-iota 6484 df-fv 6540 df-ov 7396 df-oposet 37851 |
This theorem is referenced by: opcon2b 37872 oplecon3b 37875 oplecon1b 37876 opoc1 37877 opltcon3b 37879 opltcon1b 37880 opltcon2b 37881 riotaocN 37884 oldmm1 37892 oldmm2 37893 oldmm3N 37894 oldmm4 37895 oldmj1 37896 oldmj2 37897 oldmj3 37898 oldmj4 37899 olm11 37902 latmassOLD 37904 omllaw2N 37919 omllaw4 37921 cmtcomlemN 37923 cmt2N 37925 cmt3N 37926 cmt4N 37927 cmtbr2N 37928 cmtbr3N 37929 cmtbr4N 37930 lecmtN 37931 omlfh1N 37933 omlfh3N 37934 omlspjN 37936 cvrcon3b 37952 cvrcmp2 37959 atlatmstc 37994 glbconN 38052 glbconNOLD 38053 glbconxN 38054 cvrexch 38096 1cvrco 38148 1cvratex 38149 1cvrjat 38151 polval2N 38582 polsubN 38583 2polpmapN 38589 2polvalN 38590 poldmj1N 38604 pmapj2N 38605 polatN 38607 2polatN 38608 pnonsingN 38609 ispsubcl2N 38623 polsubclN 38628 poml4N 38629 pmapojoinN 38644 pl42lem1N 38655 lhpoc2N 38691 lhpocnle 38692 lhpmod2i2 38714 lhpmod6i1 38715 lhprelat3N 38716 trlcl 38840 trlle 38860 docaclN 39800 doca2N 39802 djajN 39813 dih1 39962 dih1dimatlem 40005 dochcl 40029 dochvalr3 40039 doch2val2 40040 dochss 40041 dochocss 40042 dochoc 40043 dochnoncon 40067 djhlj 40077 |
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