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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 29532 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 36914 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | olj0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | olj0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | op0cl 36884 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝐾 ∈ OL → 0 ∈ 𝐵) |
6 | 5 | adantr 484 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
7 | eqid 2736 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | ollat 36913 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
9 | 8 | 3ad2ant1 1135 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝐾 ∈ Lat) |
10 | olj0.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
11 | 2, 10 | latjcl 17899 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
12 | 8, 11 | syl3an1 1165 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
13 | simp2 1139 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
14 | 2, 7 | latref 17901 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
15 | 8, 14 | sylan 583 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
16 | 15 | 3adant3 1134 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
17 | 2, 7, 3 | op0le 36886 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
18 | 1, 17 | sylan 583 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
19 | 18 | 3adant3 1134 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
20 | simp3 1140 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 ∈ 𝐵) | |
21 | 2, 7, 10 | latjle12 17910 | . . . . 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 1374 | . . . 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 712 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 )(le‘𝐾)𝑋) |
24 | 2, 7, 10 | latlej1 17908 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
25 | 8, 24 | syl3an1 1165 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
26 | 2, 7, 9, 12, 13, 23, 25 | latasymd 17905 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
27 | 6, 26 | mpd3an3 1464 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 lecple 16756 joincjn 17772 0.cp0 17883 Latclat 17891 OPcops 36872 OLcol 36874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-proset 17756 df-poset 17774 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-lat 17892 df-oposet 36876 df-ol 36878 |
This theorem is referenced by: olj02 36926 olm11 36927 omllaw3 36945 omlspjN 36961 2at0mat0 37225 lhp2at0nle 37735 lhple 37742 cdlemc6 37896 cdleme3c 37930 cdleme7e 37947 cdlemednpq 37999 cdlemefrs29pre00 38095 cdlemefrs29bpre0 38096 cdlemefrs29cpre1 38098 cdleme32fva 38137 cdleme42ke 38185 cdlemg12e 38347 cdlemg31d 38400 trljco 38440 cdlemkid2 38624 dihvalcqat 38939 dihmeetlem7N 39010 dihjatc1 39011 djh01 39112 |
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