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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atl0le | Structured version Visualization version GIF version |
Description: Orthoposet zero is less than or equal to any element. (ch0le 28889 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
atl0le.b | ⊢ 𝐵 = (Base‘𝐾) |
atl0le.l | ⊢ ≤ = (le‘𝐾) |
atl0le.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
atl0le | ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atl0le.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2778 | . 2 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | atl0le.l | . 2 ⊢ ≤ = (le‘𝐾) | |
4 | atl0le.z | . 2 ⊢ 0 = (0.‘𝐾) | |
5 | simpl 476 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ AtLat) | |
6 | simpr 479 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2778 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
8 | 1, 7, 2 | atl0dm 35465 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐵 ∈ dom (glb‘𝐾)) |
9 | 8 | adantr 474 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ dom (glb‘𝐾)) |
10 | 1, 2, 3, 4, 5, 6, 9 | p0le 17440 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 dom cdm 5357 ‘cfv 6137 Basecbs 16266 lecple 16356 lubclub 17339 glbcglb 17340 0.cp0 17434 AtLatcal 35427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-glb 17372 df-p0 17436 df-atl 35461 |
This theorem is referenced by: atlle0 35468 atlltn0 35469 atcvreq0 35477 trlval4 36351 dian0 37202 dia0 37215 dihmeetlem4preN 37469 |
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