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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > op0cl | Structured version Visualization version GIF version |
Description: An orthoposet has a zero element. (h0elch 29038 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
op0cl.b | ⊢ 𝐵 = (Base‘𝐾) |
op0cl.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
op0cl | ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2798 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | op0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | p0val 17643 | . 2 ⊢ (𝐾 ∈ OP → 0 = ((glb‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP) | |
6 | eqid 2798 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | op01dm 36479 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
8 | 7 | simprd 499 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (glb‘𝐾)) |
9 | 1, 2, 5, 8 | glbcl 17600 | . 2 ⊢ (𝐾 ∈ OP → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
10 | 4, 9 | eqeltrd 2890 | 1 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 dom cdm 5519 ‘cfv 6324 Basecbs 16475 lubclub 17544 glbcglb 17545 0.cp0 17639 OPcops 36468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-glb 17577 df-p0 17641 df-oposet 36472 |
This theorem is referenced by: ople0 36483 lub0N 36485 opltn0 36486 opoc1 36498 opoc0 36499 olj01 36521 olj02 36522 olm01 36532 olm02 36533 0ltat 36587 leatb 36588 hlhgt2 36685 hl0lt1N 36686 hl2at 36701 atcvr0eq 36722 lnnat 36723 atle 36732 athgt 36752 1cvratex 36769 ps-2 36774 dalemcea 36956 pmapeq0 37062 2atm2atN 37081 lhp0lt 37299 lhpn0 37300 ltrnatb 37433 cdleme3c 37526 cdleme7e 37543 dia0eldmN 38336 dia2dimlem2 38361 dia2dimlem3 38362 dib0 38460 dih0 38576 dih0bN 38577 dih0rn 38580 dihlspsnssN 38628 dihlspsnat 38629 dihatexv 38634 |
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