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Mirrors > Home > MPE Home > Th. List > Mathboxes > opococ | Structured version Visualization version GIF version |
Description: Double negative law for orthoposets. (ococ 30522 analog.) (Contributed by NM, 13-Sep-2011.) |
Ref | Expression |
---|---|
opoccl.b | ⊢ 𝐵 = (Base‘𝐾) |
opoccl.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opococ | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
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 | simp2d 1143 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
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: opcon3b 37871 opcon2b 37872 oplecon3b 37875 oplecon1b 37876 opltcon1b 37880 opltcon2b 37881 oldmm2 37893 oldmm3N 37894 oldmm4 37895 oldmj1 37896 oldmj2 37897 oldmj3 37898 oldmj4 37899 olm11 37902 omllaw4 37921 cmt2N 37925 glbconN 38052 glbconNOLD 38053 1cvratex 38149 1cvrjat 38151 polval2N 38582 2polpmapN 38589 2polvalN 38590 2polatN 38608 lhpoc2N 38691 doch2val2 40040 dochocss 40042 dochoc 40043 |
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