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Mirrors > Home > MPE Home > Th. List > latjlej1 | Structured version Visualization version GIF version |
Description: Add join to both sides of a lattice ordering. (chlej1i 31355 analog.) (Contributed by NM, 8-Nov-2011.) |
Ref | Expression |
---|---|
latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
latlej.l | ⊢ ≤ = (le‘𝐾) |
latlej.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latjlej1 | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∨ 𝑍) ≤ (𝑌 ∨ 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latlej.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latlej.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
3 | latlej.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
4 | 1, 2, 3 | latlej1 18443 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑌 ≤ (𝑌 ∨ 𝑍)) |
5 | 4 | 3adant3r1 1179 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ≤ (𝑌 ∨ 𝑍)) |
6 | simpl 481 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
7 | simpr1 1191 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
8 | simpr2 1192 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
9 | 1, 3 | latjcl 18434 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∨ 𝑍) ∈ 𝐵) |
10 | 9 | 3adant3r1 1179 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∨ 𝑍) ∈ 𝐵) |
11 | 1, 2 | lattr 18439 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑌 ∨ 𝑍) ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ (𝑌 ∨ 𝑍)) → 𝑋 ≤ (𝑌 ∨ 𝑍))) |
12 | 6, 7, 8, 10, 11 | syl13anc 1369 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ (𝑌 ∨ 𝑍)) → 𝑋 ≤ (𝑌 ∨ 𝑍))) |
13 | 5, 12 | mpan2d 692 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → 𝑋 ≤ (𝑌 ∨ 𝑍))) |
14 | 1, 2, 3 | latlej2 18444 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ≤ (𝑌 ∨ 𝑍)) |
15 | 14 | 3adant3r1 1179 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ≤ (𝑌 ∨ 𝑍)) |
16 | 13, 15 | jctird 525 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ≤ (𝑌 ∨ 𝑍) ∧ 𝑍 ≤ (𝑌 ∨ 𝑍)))) |
17 | simpr3 1193 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
18 | 7, 17, 10 | 3jca 1125 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ (𝑌 ∨ 𝑍) ∈ 𝐵)) |
19 | 1, 2, 3 | latjle12 18445 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ (𝑌 ∨ 𝑍) ∈ 𝐵)) → ((𝑋 ≤ (𝑌 ∨ 𝑍) ∧ 𝑍 ≤ (𝑌 ∨ 𝑍)) ↔ (𝑋 ∨ 𝑍) ≤ (𝑌 ∨ 𝑍))) |
20 | 18, 19 | syldan 589 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ (𝑌 ∨ 𝑍) ∧ 𝑍 ≤ (𝑌 ∨ 𝑍)) ↔ (𝑋 ∨ 𝑍) ≤ (𝑌 ∨ 𝑍))) |
21 | 16, 20 | sylibd 238 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∨ 𝑍) ≤ (𝑌 ∨ 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 lecple 17243 joincjn 18306 Latclat 18426 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-poset 18308 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-lat 18427 |
This theorem is referenced by: latjlej2 18449 latjlej12 18450 ps-2 39081 dalem5 39270 cdlema1N 39394 dalawlem3 39476 dalawlem6 39479 dalawlem7 39480 dalawlem11 39484 dalawlem12 39485 cdleme20d 39915 trlcolem 40329 cdlemh1 40418 |
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