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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 31052 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 2727 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | op0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | p0val 18410 | . 2 ⊢ (𝐾 ∈ OP → 0 = ((glb‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP) | |
6 | eqid 2727 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | op01dm 38592 | . . . 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 18353 | . 2 ⊢ (𝐾 ∈ OP → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
10 | 4, 9 | eqeltrd 2828 | 1 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 dom cdm 5672 ‘cfv 6542 Basecbs 17171 lubclub 18292 glbcglb 18293 0.cp0 18406 OPcops 38581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-glb 18330 df-p0 18408 df-oposet 38585 |
This theorem is referenced by: ople0 38596 lub0N 38598 opltn0 38599 opoc1 38611 opoc0 38612 olj01 38634 olj02 38635 olm01 38645 olm02 38646 0ltat 38700 leatb 38701 hlhgt2 38799 hl0lt1N 38800 hl2at 38815 atcvr0eq 38836 lnnat 38837 atle 38846 athgt 38866 1cvratex 38883 ps-2 38888 dalemcea 39070 pmapeq0 39176 2atm2atN 39195 lhp0lt 39413 lhpn0 39414 ltrnatb 39547 cdleme3c 39640 cdleme7e 39657 dia0eldmN 40450 dia2dimlem2 40475 dia2dimlem3 40476 dib0 40574 dih0 40690 dih0bN 40691 dih0rn 40694 dihlspsnssN 40742 dihlspsnat 40743 dihatexv 40748 |
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