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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 29186 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 2824 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | opoccl.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
4 | eqid 2824 | . . . . 5 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | eqid 2824 | . . . . 5 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
6 | eqid 2824 | . . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | eqid 2824 | . . . . 5 ⊢ (1.‘𝐾) = (1.‘𝐾) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oposlem 36322 | . . . 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 1417 | . . 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 1138 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (( ⊥ ‘𝑋) ∈ 𝐵 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ (𝑋(le‘𝐾)𝑋 → ( ⊥ ‘𝑋)(le‘𝐾)( ⊥ ‘𝑋)))) |
11 | 10 | simp2d 1139 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 lecple 16575 occoc 16576 joincjn 17557 meetcmee 17558 0.cp0 17650 1.cp1 17651 OPcops 36312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-dm 5568 df-iota 6317 df-fv 6366 df-ov 7162 df-oposet 36316 |
This theorem is referenced by: opcon3b 36336 opcon2b 36337 oplecon3b 36340 oplecon1b 36341 opltcon1b 36345 opltcon2b 36346 oldmm2 36358 oldmm3N 36359 oldmm4 36360 oldmj1 36361 oldmj2 36362 oldmj3 36363 oldmj4 36364 olm11 36367 omllaw4 36386 cmt2N 36390 glbconN 36517 1cvratex 36613 1cvrjat 36615 polval2N 37046 2polpmapN 37053 2polvalN 37054 2polatN 37072 lhpoc2N 37155 doch2val2 38504 dochocss 38506 dochoc 38507 |
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