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| Mirrors > Home > MPE Home > Th. List > Mathboxes > opoc0 | Structured version Visualization version GIF version | ||
| Description: Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.) | 
| Ref | Expression | 
|---|---|
| opoc1.z | ⊢ 0 = (0.‘𝐾) | 
| opoc1.u | ⊢ 1 = (1.‘𝐾) | 
| opoc1.o | ⊢ ⊥ = (oc‘𝐾) | 
| Ref | Expression | 
|---|---|
| opoc0 | ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opoc1.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
| 2 | opoc1.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
| 3 | opoc1.o | . . 3 ⊢ ⊥ = (oc‘𝐾) | |
| 4 | 1, 2, 3 | opoc1 39203 | . 2 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 1 ) = 0 ) | 
| 5 | eqid 2737 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 6 | 5, 2 | op1cl 39186 | . . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) | 
| 7 | 5, 1 | op0cl 39185 | . . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) | 
| 8 | 5, 3 | opcon1b 39199 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾) ∧ 0 ∈ (Base‘𝐾)) → (( ⊥ ‘ 1 ) = 0 ↔ ( ⊥ ‘ 0 ) = 1 )) | 
| 9 | 6, 7, 8 | mpd3an23 1465 | . 2 ⊢ (𝐾 ∈ OP → (( ⊥ ‘ 1 ) = 0 ↔ ( ⊥ ‘ 0 ) = 1 )) | 
| 10 | 4, 9 | mpbid 232 | 1 ⊢ (𝐾 ∈ OP → ( ⊥ ‘ 0 ) = 1 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Basecbs 17247 occoc 17305 0.cp0 18468 1.cp1 18469 OPcops 39173 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-proset 18340 df-poset 18359 df-lub 18391 df-glb 18392 df-p0 18470 df-p1 18471 df-oposet 39177 | 
| This theorem is referenced by: 1cvrjat 39477 doch0 41360 | 
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