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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 30452 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 37757 | . . . 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 5132 ‘cfv 6523 (class class class)co 7384 Basecbs 17116 lecple 17176 occoc 17177 joincjn 18236 meetcmee 18237 0.cp0 18348 1.cp1 18349 OPcops 37747 |
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 5290 |
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 3426 df-v 3468 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-br 5133 df-dm 5670 df-iota 6475 df-fv 6531 df-ov 7387 df-oposet 37751 |
This theorem is referenced by: opcon3b 37771 opcon2b 37772 oplecon3b 37775 oplecon1b 37776 opltcon1b 37780 opltcon2b 37781 oldmm2 37793 oldmm3N 37794 oldmm4 37795 oldmj1 37796 oldmj2 37797 oldmj3 37798 oldmj4 37799 olm11 37802 omllaw4 37821 cmt2N 37825 glbconN 37952 glbconNOLD 37953 1cvratex 38049 1cvrjat 38051 polval2N 38482 2polpmapN 38489 2polvalN 38490 2polatN 38508 lhpoc2N 38591 doch2val2 39940 dochocss 39942 dochoc 39943 |
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