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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 30761 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 2730 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | op1cl.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
4 | 1, 2, 3 | p1val 18387 | . 2 ⊢ (𝐾 ∈ OP → 1 = ((lub‘𝐾)‘𝐵)) |
5 | id 22 | . . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP) | |
6 | eqid 2730 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
7 | 1, 2, 6 | op01dm 38358 | . . . 4 ⊢ (𝐾 ∈ OP → (𝐵 ∈ dom (lub‘𝐾) ∧ 𝐵 ∈ dom (glb‘𝐾))) |
8 | 7 | simpld 493 | . . 3 ⊢ (𝐾 ∈ OP → 𝐵 ∈ dom (lub‘𝐾)) |
9 | 1, 2, 5, 8 | lubcl 18316 | . 2 ⊢ (𝐾 ∈ OP → ((lub‘𝐾)‘𝐵) ∈ 𝐵) |
10 | 4, 9 | eqeltrd 2831 | 1 ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 dom cdm 5677 ‘cfv 6544 Basecbs 17150 lubclub 18268 glbcglb 18269 1.cp1 18383 OPcops 38347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-lub 18305 df-p1 18385 df-oposet 38351 |
This theorem is referenced by: op1le 38367 glb0N 38368 opoc1 38377 opoc0 38378 olm11 38402 olm12 38403 ncvr1 38447 hlhgt2 38565 hl0lt1N 38566 hl2at 38581 athgt 38632 1cvrco 38648 1cvrjat 38651 pmap1N 38943 pol1N 39086 lhp2lt 39177 lhpexnle 39182 dih1 40462 dih1rn 40463 dih1cnv 40464 dihglb2 40518 dochocss 40542 dihjatc 40593 |
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