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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 27496 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 2610 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | op0cl.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | p0val 16864 | . 2 ⊢ (𝐾 ∈ OP → 0 = ((glb‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP) | |
6 | eqid 2610 | . . . . 5 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
7 | 1, 6, 2 | op01dm 33488 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
8 | 7 | simprd 478 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (glb‘𝐾)) |
9 | 1, 2, 5, 8 | glbcl 16821 | . 2 ⊢ (𝐾 ∈ OP → ((glb‘𝐾)‘𝐵) ∈ 𝐵) |
10 | 4, 9 | eqeltrd 2688 | 1 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 dom cdm 5038 ‘cfv 5804 Basecbs 15695 lubclub 16765 glbcglb 16766 0.cp0 16860 OPcops 33477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-glb 16798 df-p0 16862 df-oposet 33481 |
This theorem is referenced by: ople0 33492 lub0N 33494 opltn0 33495 opoc1 33507 opoc0 33508 olj01 33530 olj02 33531 olm01 33541 olm02 33542 0ltat 33596 leatb 33597 hlhgt2 33693 hl0lt1N 33694 hl2at 33709 atcvr0eq 33730 lnnat 33731 atle 33740 athgt 33760 1cvratex 33777 ps-2 33782 dalemcea 33964 pmapeq0 34070 2atm2atN 34089 lhp0lt 34307 lhpn0 34308 ltrnatb 34441 ltrnmwOLD 34456 cdleme3c 34535 cdleme7e 34552 dia0eldmN 35347 dia2dimlem2 35372 dia2dimlem3 35373 dib0 35471 dih0 35587 dih0bN 35588 dih0rn 35591 dihlspsnssN 35639 dihlspsnat 35640 dihatexv 35645 |
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