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Mirrors > Home > MPE Home > Th. List > Mathboxes > op0le | Structured version Visualization version GIF version |
Description: Orthoposet zero is less than or equal to any element. (ch0le 29553 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
op0le.b | ⊢ 𝐵 = (Base‘𝐾) |
op0le.l | ⊢ ≤ = (le‘𝐾) |
op0le.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
op0le | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op0le.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2739 | . 2 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | op0le.l | . 2 ⊢ ≤ = (le‘𝐾) | |
4 | op0le.z | . 2 ⊢ 0 = (0.‘𝐾) | |
5 | simpl 486 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) | |
6 | simpr 488 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2739 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
8 | 1, 7, 2 | op01dm 36970 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
9 | 8 | simprd 499 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (glb‘𝐾)) |
10 | 9 | adantr 484 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ dom (glb‘𝐾)) |
11 | 1, 2, 3, 4, 5, 6, 10 | p0le 17967 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5069 dom cdm 5568 ‘cfv 6400 Basecbs 16792 lecple 16841 lubclub 17848 glbcglb 17849 0.cp0 17961 OPcops 36959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-id 5471 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-glb 17885 df-p0 17963 df-oposet 36963 |
This theorem is referenced by: ople0 36974 opnlen0 36975 lub0N 36976 opltn0 36977 olj01 37012 olm01 37023 leatb 37079 1cvratex 37260 llnn0 37303 lplnn0N 37334 lvoln0N 37378 dalemcea 37447 ltrnatb 37924 tendo0tp 38576 cdlemk39s-id 38727 dia0eldmN 38827 dib0 38951 dih0 39067 dihmeetlem18N 39111 |
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