| 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 31274 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 2737 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
| 3 | op0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
| 4 | 1, 2, 3 | p0val 18472 | . 2 ⊢ (𝐾 ∈ OP → 0 = ((glb‘𝐾)‘𝐵)) |
| 5 | id 22 | . . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP) | |
| 6 | eqid 2737 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
| 7 | 1, 6, 2 | op01dm 39184 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
| 8 | 7 | simprd 495 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (glb‘𝐾)) |
| 9 | 1, 2, 5, 8 | glbcl 18415 | . 2 ⊢ (𝐾 ∈ OP → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
| 10 | 4, 9 | eqeltrd 2841 | 1 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 dom cdm 5685 ‘cfv 6561 Basecbs 17247 lubclub 18355 glbcglb 18356 0.cp0 18468 OPcops 39173 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-glb 18392 df-p0 18470 df-oposet 39177 |
| This theorem is referenced by: ople0 39188 lub0N 39190 opltn0 39191 opoc1 39203 opoc0 39204 olj01 39226 olj02 39227 olm01 39237 olm02 39238 0ltat 39292 leatb 39293 hlhgt2 39391 hl0lt1N 39392 hl2at 39407 atcvr0eq 39428 lnnat 39429 atle 39438 athgt 39458 1cvratex 39475 ps-2 39480 dalemcea 39662 pmapeq0 39768 2atm2atN 39787 lhp0lt 40005 lhpn0 40006 ltrnatb 40139 cdleme3c 40232 cdleme7e 40249 dia0eldmN 41042 dia2dimlem2 41067 dia2dimlem3 41068 dib0 41166 dih0 41282 dih0bN 41283 dih0rn 41286 dihlspsnssN 41334 dihlspsnat 41335 dihatexv 41340 |
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