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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 28640 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 2824 | . 2 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | ople1.l | . 2 ⊢ ≤ = (le‘𝐾) | |
4 | ople1.u | . 2 ⊢ 1 = (1.‘𝐾) | |
5 | simpl 476 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) | |
6 | simpr 479 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2824 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
8 | 1, 2, 7 | op01dm 35257 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
9 | 8 | simpld 490 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (lub‘𝐾)) |
10 | 9 | adantr 474 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐵 ∈ dom (lub‘𝐾)) |
11 | 1, 2, 3, 4, 5, 6, 10 | ple1 17396 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4872 dom cdm 5341 ‘cfv 6122 Basecbs 16221 lecple 16311 lubclub 17294 glbcglb 17295 1.cp1 17390 OPcops 35246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-lub 17326 df-p1 17392 df-oposet 35250 |
This theorem is referenced by: op1le 35266 glb0N 35267 opoc1 35276 ncvr1 35346 1cvrat 35550 pmap1N 35841 pol1N 35984 dih1 37360 dihjatc 37491 |
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