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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 29145 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 2818 | . 2 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | op0le.l | . 2 ⊢ ≤ = (le‘𝐾) | |
4 | op0le.z | . 2 ⊢ 0 = (0.‘𝐾) | |
5 | simpl 483 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) | |
6 | simpr 485 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2818 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
8 | 1, 7, 2 | op01dm 36199 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
9 | 8 | simprd 496 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (glb‘𝐾)) |
10 | 9 | adantr 481 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ dom (glb‘𝐾)) |
11 | 1, 2, 3, 4, 5, 6, 10 | p0le 17641 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 Basecbs 16471 lecple 16560 lubclub 17540 glbcglb 17541 0.cp0 17635 OPcops 36188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-glb 17573 df-p0 17637 df-oposet 36192 |
This theorem is referenced by: ople0 36203 opnlen0 36204 lub0N 36205 opltn0 36206 olj01 36241 olm01 36252 leatb 36308 1cvratex 36489 llnn0 36532 lplnn0N 36563 lvoln0N 36607 dalemcea 36676 ltrnatb 37153 tendo0tp 37805 cdlemk39s-id 37956 dia0eldmN 38056 dib0 38180 dih0 38296 dihmeetlem18N 38340 |
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