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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ople1 | Structured version Visualization version GIF version |
Description: Any element is less than the orthoposet unity. (chss 31162 analog.) (Contributed by NM, 23-Oct-2011.) |
Ref | Expression |
---|---|
ople1.b | ⊢ 𝐵 = (Base‘𝐾) |
ople1.l | ⊢ ≤ = (le‘𝐾) |
ople1.u | ⊢ 1 = (1.‘𝐾) |
Ref | Expression |
---|---|
ople1 | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ople1.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2726 | . 2 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | ople1.l | . 2 ⊢ ≤ = (le‘𝐾) | |
4 | ople1.u | . 2 ⊢ 1 = (1.‘𝐾) | |
5 | simpl 481 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) | |
6 | simpr 483 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2726 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
8 | 1, 2, 7 | op01dm 38881 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
9 | 8 | simpld 493 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (lub‘𝐾)) |
10 | 9 | adantr 479 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ dom (lub‘𝐾)) |
11 | 1, 2, 3, 4, 5, 6, 10 | ple1 18455 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 dom cdm 5682 ‘cfv 6554 Basecbs 17213 lecple 17273 lubclub 18334 glbcglb 18335 1.cp1 18449 OPcops 38870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-lub 18371 df-p1 18451 df-oposet 38874 |
This theorem is referenced by: op1le 38890 glb0N 38891 opoc1 38900 ncvr1 38970 1cvrat 39175 pmap1N 39466 pol1N 39609 dih1 40985 dihjatc 41116 |
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