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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > op1cl | Structured version Visualization version GIF version |
Description: An orthoposet has a unity element. (helch 31272 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 2735 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | op1cl.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
4 | 1, 2, 3 | p1val 18486 | . 2 ⊢ (𝐾 ∈ OP → 1 = ((lub‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP) | |
6 | eqid 2735 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
7 | 1, 2, 6 | op01dm 39165 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
8 | 7 | simpld 494 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (lub‘𝐾)) |
9 | 1, 2, 5, 8 | lubcl 18415 | . 2 ⊢ (𝐾 ∈ OP → ((lub‘𝐾)‘𝐵) ∈ 𝐵) |
10 | 4, 9 | eqeltrd 2839 | 1 ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 dom cdm 5689 ‘cfv 6563 Basecbs 17245 lubclub 18367 glbcglb 18368 1.cp1 18482 OPcops 39154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-lub 18404 df-p1 18484 df-oposet 39158 |
This theorem is referenced by: op1le 39174 glb0N 39175 opoc1 39184 opoc0 39185 olm11 39209 olm12 39210 ncvr1 39254 hlhgt2 39372 hl0lt1N 39373 hl2at 39388 athgt 39439 1cvrco 39455 1cvrjat 39458 pmap1N 39750 pol1N 39893 lhp2lt 39984 lhpexnle 39989 dih1 41269 dih1rn 41270 dih1cnv 41271 dihglb2 41325 dochocss 41349 dihjatc 41400 |
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