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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 unit. (chss 29008 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 2823 | . 2 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | ople1.l | . 2 ⊢ ≤ = (le‘𝐾) | |
4 | ople1.u | . 2 ⊢ 1 = (1.‘𝐾) | |
5 | simpl 485 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) | |
6 | simpr 487 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2823 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
8 | 1, 2, 7 | op01dm 36321 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
9 | 8 | simpld 497 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (lub‘𝐾)) |
10 | 9 | adantr 483 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ dom (lub‘𝐾)) |
11 | 1, 2, 3, 4, 5, 6, 10 | ple1 17656 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 Basecbs 16485 lecple 16574 lubclub 17554 glbcglb 17555 1.cp1 17650 OPcops 36310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-lub 17586 df-p1 17652 df-oposet 36314 |
This theorem is referenced by: op1le 36330 glb0N 36331 opoc1 36340 ncvr1 36410 1cvrat 36614 pmap1N 36905 pol1N 37048 dih1 38424 dihjatc 38555 |
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