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Mathbox for Norm Megill |
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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 31469 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 2734 | . 2 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | op0le.l | . 2 ⊢ ≤ = (le‘𝐾) | |
4 | op0le.z | . 2 ⊢ 0 = (0.‘𝐾) | |
5 | simpl 482 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) | |
6 | simpr 484 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2734 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
8 | 1, 7, 2 | op01dm 39164 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
9 | 8 | simprd 495 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (glb‘𝐾)) |
10 | 9 | adantr 480 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ dom (glb‘𝐾)) |
11 | 1, 2, 3, 4, 5, 6, 10 | p0le 18486 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 dom cdm 5688 ‘cfv 6562 Basecbs 17244 lecple 17304 lubclub 18366 glbcglb 18367 0.cp0 18480 OPcops 39153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-glb 18404 df-p0 18482 df-oposet 39157 |
This theorem is referenced by: ople0 39168 opnlen0 39169 lub0N 39170 opltn0 39171 olj01 39206 olm01 39217 leatb 39273 1cvratex 39455 llnn0 39498 lplnn0N 39529 lvoln0N 39573 dalemcea 39642 ltrnatb 40119 tendo0tp 40771 cdlemk39s-id 40922 dia0eldmN 41022 dib0 41146 dih0 41262 dihmeetlem18N 41306 |
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