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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 28684 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 2777 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | op0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | p0val 17427 | . 2 ⊢ (𝐾 ∈ OP → 0 = ((glb‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP) | |
6 | eqid 2777 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | op01dm 35321 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
8 | 7 | simprd 491 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (glb‘𝐾)) |
9 | 1, 2, 5, 8 | glbcl 17384 | . 2 ⊢ (𝐾 ∈ OP → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
10 | 4, 9 | eqeltrd 2858 | 1 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 dom cdm 5355 ‘cfv 6135 Basecbs 16255 lubclub 17328 glbcglb 17329 0.cp0 17423 OPcops 35310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-glb 17361 df-p0 17425 df-oposet 35314 |
This theorem is referenced by: ople0 35325 lub0N 35327 opltn0 35328 opoc1 35340 opoc0 35341 olj01 35363 olj02 35364 olm01 35374 olm02 35375 0ltat 35429 leatb 35430 hlhgt2 35527 hl0lt1N 35528 hl2at 35543 atcvr0eq 35564 lnnat 35565 atle 35574 athgt 35594 1cvratex 35611 ps-2 35616 dalemcea 35798 pmapeq0 35904 2atm2atN 35923 lhp0lt 36141 lhpn0 36142 ltrnatb 36275 cdleme3c 36368 cdleme7e 36385 dia0eldmN 37178 dia2dimlem2 37203 dia2dimlem3 37204 dib0 37302 dih0 37418 dih0bN 37419 dih0rn 37422 dihlspsnssN 37470 dihlspsnat 37471 dihatexv 37476 |
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