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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 31327 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 38718 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | olj0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | olj0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | op0cl 38688 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝐾 ∈ OL → 0 ∈ 𝐵) |
6 | 5 | adantr 479 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
7 | eqid 2728 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | ollat 38717 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
9 | 8 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝐾 ∈ Lat) |
10 | olj0.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
11 | 2, 10 | latjcl 18438 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
12 | 8, 11 | syl3an1 1160 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
13 | simp2 1134 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
14 | 2, 7 | latref 18440 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
15 | 8, 14 | sylan 578 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
16 | 15 | 3adant3 1129 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
17 | 2, 7, 3 | op0le 38690 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
18 | 1, 17 | sylan 578 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
19 | 18 | 3adant3 1129 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
20 | simp3 1135 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 ∈ 𝐵) | |
21 | 2, 7, 10 | latjle12 18449 | . . . . 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 1369 | . . . 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 710 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 )(le‘𝐾)𝑋) |
24 | 2, 7, 10 | latlej1 18447 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
25 | 8, 24 | syl3an1 1160 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
26 | 2, 7, 9, 12, 13, 23, 25 | latasymd 18444 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
27 | 6, 26 | mpd3an3 1458 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 lecple 17247 joincjn 18310 0.cp0 18422 Latclat 18430 OPcops 38676 OLcol 38678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18294 df-poset 18312 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-lat 18431 df-oposet 38680 df-ol 38682 |
This theorem is referenced by: olj02 38730 olm11 38731 omllaw3 38749 omlspjN 38765 2at0mat0 39030 lhp2at0nle 39540 lhple 39547 cdlemc6 39701 cdleme3c 39735 cdleme7e 39752 cdlemednpq 39804 cdlemefrs29pre00 39900 cdlemefrs29bpre0 39901 cdlemefrs29cpre1 39903 cdleme32fva 39942 cdleme42ke 39990 cdlemg12e 40152 cdlemg31d 40205 trljco 40245 cdlemkid2 40429 dihvalcqat 40744 dihmeetlem7N 40815 dihjatc1 40816 djh01 40917 |
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