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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 28928 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 35352 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
2 | olj0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | olj0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | op0cl 35322 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝐾 ∈ OL → 0 ∈ 𝐵) |
6 | 5 | adantr 474 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
7 | eqid 2777 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | ollat 35351 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
9 | 8 | 3ad2ant1 1124 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝐾 ∈ Lat) |
10 | olj0.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
11 | 2, 10 | latjcl 17437 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
12 | 8, 11 | syl3an1 1163 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) ∈ 𝐵) |
13 | simp2 1128 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
14 | 2, 7 | latref 17439 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
15 | 8, 14 | sylan 575 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
16 | 15 | 3adant3 1123 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
17 | 2, 7, 3 | op0le 35324 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
18 | 1, 17 | sylan 575 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
19 | 18 | 3adant3 1123 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
20 | simp3 1129 | . . . . 5 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 0 ∈ 𝐵) | |
21 | 2, 7, 10 | latjle12 17448 | . . . . 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 1440 | . . . 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 702 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 )(le‘𝐾)𝑋) |
24 | 2, 7, 10 | latlej1 17446 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
25 | 8, 24 | syl3an1 1163 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 0 )) |
26 | 2, 7, 9, 12, 13, 23, 25 | latasymd 17443 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
27 | 6, 26 | mpd3an3 1535 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 lecple 16345 joincjn 17330 0.cp0 17423 Latclat 17431 OPcops 35310 OLcol 35312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-proset 17314 df-poset 17332 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-lat 17432 df-oposet 35314 df-ol 35316 |
This theorem is referenced by: olj02 35364 olm11 35365 omllaw3 35383 omlspjN 35399 2at0mat0 35663 lhp2at0nle 36173 lhple 36180 cdlemc6 36334 cdleme3c 36368 cdleme7e 36385 cdlemednpq 36437 cdlemefrs29pre00 36533 cdlemefrs29bpre0 36534 cdlemefrs29cpre1 36536 cdleme32fva 36575 cdleme42ke 36623 cdlemg12e 36785 cdlemg31d 36838 trljco 36878 cdlemkid2 37062 dihvalcqat 37377 dihmeetlem7N 37448 dihjatc1 37449 djh01 37550 |
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