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Mirrors > Home > MPE Home > Th. List > Mathboxes > olj01 | Structured version Visualization version GIF version |
Description: An ortholattice element joined with zero equals itself. (chj0 27740 analog.) (Contributed by NM, 19-Oct-2011.) |
Ref | Expression |
---|---|
olj0.b | ⊢ 𝐵 = (Base‘𝐾) |
olj0.j | ⊢ ∨ = (join‘𝐾) |
olj0.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
olj01 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olop 33519 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | olj0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | olj0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | op0cl 33489 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝐾 ∈ OL → 0 ∈ 𝐵) |
6 | 5 | adantr 480 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
7 | eqid 2610 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | ollat 33518 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
9 | 8 | 3ad2ant1 1075 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝐾 ∈ Lat) |
10 | olj0.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
11 | 2, 10 | latjcl 16874 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
12 | 8, 11 | syl3an1 1351 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
13 | simp2 1055 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
14 | 2, 7 | latref 16876 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
15 | 8, 14 | sylan 487 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
16 | 15 | 3adant3 1074 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
17 | 2, 7, 3 | op0le 33491 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
18 | 1, 17 | sylan 487 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
19 | 18 | 3adant3 1074 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
20 | simp3 1056 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 ∈ 𝐵) | |
21 | 2, 7, 10 | latjle12 16885 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋(le‘𝐾)𝑋 ∧ 0 (le‘𝐾)𝑋) ↔ (𝑋 ∨ 0 )(le‘𝐾)𝑋)) |
22 | 9, 13, 20, 13, 21 | syl13anc 1320 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑋 ∧ 0 (le‘𝐾)𝑋) ↔ (𝑋 ∨ 0 )(le‘𝐾)𝑋)) |
23 | 16, 19, 22 | mpbi2and 958 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 )(le‘𝐾)𝑋) |
24 | 2, 7, 10 | latlej1 16883 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
25 | 8, 24 | syl3an1 1351 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
26 | 2, 7, 9, 12, 13, 23, 25 | latasymd 16880 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
27 | 6, 26 | mpd3an3 1417 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 0.cp0 16860 Latclat 16868 OPcops 33477 OLcol 33479 |
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 ax-un 6847 |
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-oprab 6553 df-preset 16751 df-poset 16769 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-lat 16869 df-oposet 33481 df-ol 33483 |
This theorem is referenced by: olj02 33531 olm11 33532 omllaw3 33550 omlspjN 33566 2at0mat0 33829 lhp2at0nle 34339 lhple 34346 cdlemc6 34501 cdleme3c 34535 cdleme7e 34552 cdlemednpq 34604 cdlemefrs29pre00 34701 cdlemefrs29bpre0 34702 cdlemefrs29cpre1 34704 cdleme32fva 34743 cdleme42ke 34791 cdlemg12e 34953 cdlemg31d 35006 trljco 35046 cdlemkid2 35230 dihvalcqat 35546 dihmeetlem7N 35617 dihjatc1 35618 djh01 35719 |
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