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Mirrors > Home > MPE Home > Th. List > Mathboxes > op1cl | Structured version Visualization version GIF version |
Description: An orthoposet has a unit element. (helch 27484 analog.) (Contributed by NM, 22-Oct-2011.) |
Ref | Expression |
---|---|
op1cl.b | ⊢ 𝐵 = (Base‘𝐾) |
op1cl.u | ⊢ 1 = (1.‘𝐾) |
Ref | Expression |
---|---|
op1cl | ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op1cl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2610 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | op1cl.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
4 | 1, 2, 3 | p1val 16865 | . 2 ⊢ (𝐾 ∈ OP → 1 = ((lub‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP) | |
6 | eqid 2610 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
7 | 1, 2, 6 | op01dm 33488 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
8 | 7 | simpld 474 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (lub‘𝐾)) |
9 | 1, 2, 5, 8 | lubcl 16808 | . 2 ⊢ (𝐾 ∈ OP → ((lub‘𝐾)‘𝐵) ∈ 𝐵) |
10 | 4, 9 | eqeltrd 2688 | 1 ⊢ (𝐾 ∈ OP → 1 ∈ 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 1.cp1 16861 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-lub 16797 df-p1 16863 df-oposet 33481 |
This theorem is referenced by: op1le 33497 glb0N 33498 opoc1 33507 opoc0 33508 olm11 33532 olm12 33533 ncvr1 33577 hlhgt2 33693 hl0lt1N 33694 hl2at 33709 athgt 33760 1cvrco 33776 1cvrjat 33779 pmap1N 34071 pol1N 34214 lhp2lt 34305 lhpexnle 34310 dih1 35593 dih1rn 35594 dih1cnv 35595 dihglb2 35649 dochocss 35673 dihjatc 35724 |
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