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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 2778 | . . 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 2778 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | op01dm 35337 | . . . 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 2859 | 1 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 dom cdm 5355 ‘cfv 6135 Basecbs 16255 lubclub 17328 glbcglb 17329 0.cp0 17423 OPcops 35326 |
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 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 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 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 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 35330 |
This theorem is referenced by: ople0 35341 lub0N 35343 opltn0 35344 opoc1 35356 opoc0 35357 olj01 35379 olj02 35380 olm01 35390 olm02 35391 0ltat 35445 leatb 35446 hlhgt2 35543 hl0lt1N 35544 hl2at 35559 atcvr0eq 35580 lnnat 35581 atle 35590 athgt 35610 1cvratex 35627 ps-2 35632 dalemcea 35814 pmapeq0 35920 2atm2atN 35939 lhp0lt 36157 lhpn0 36158 ltrnatb 36291 cdleme3c 36384 cdleme7e 36401 dia0eldmN 37194 dia2dimlem2 37219 dia2dimlem3 37220 dib0 37318 dih0 37434 dih0bN 37435 dih0rn 37438 dihlspsnssN 37486 dihlspsnat 37487 dihatexv 37492 |
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