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Mirrors > Home > MPE Home > Th. List > latjlej2 | Structured version Visualization version GIF version |
Description: Add join to both sides of a lattice ordering. (chlej2i 29257 analog.) (Contributed by NM, 8-Nov-2011.) |
Ref | Expression |
---|---|
latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
latlej.l | ⊢ ≤ = (le‘𝐾) |
latlej.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latjlej2 | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑍 ∨ 𝑋) ≤ (𝑍 ∨ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latlej.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latlej.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | latlej.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | 1, 2, 3 | latjlej1 17667 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∨ 𝑍) ≤ (𝑌 ∨ 𝑍))) |
5 | 1, 3 | latjcom 17661 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∨ 𝑍) = (𝑍 ∨ 𝑋)) |
6 | 5 | 3adant3r2 1180 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ 𝑍) = (𝑍 ∨ 𝑋)) |
7 | 1, 3 | latjcom 17661 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∨ 𝑍) = (𝑍 ∨ 𝑌)) |
8 | 7 | 3adant3r1 1179 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∨ 𝑍) = (𝑍 ∨ 𝑌)) |
9 | 6, 8 | breq12d 5043 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑍) ≤ (𝑌 ∨ 𝑍) ↔ (𝑍 ∨ 𝑋) ≤ (𝑍 ∨ 𝑌))) |
10 | 4, 9 | sylibd 242 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑍 ∨ 𝑋) ≤ (𝑍 ∨ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 joincjn 17546 Latclat 17647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-poset 17548 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-lat 17648 |
This theorem is referenced by: latjlej12 17669 cvrat3 36738 2llnjaN 36862 2lplnja 36915 dalawlem3 37169 dalawlem6 37172 dalawlem11 37177 lhpj1 37318 cdleme1 37523 cdleme9 37549 cdleme11g 37561 cdleme28a 37666 cdleme30a 37674 cdleme32c 37739 cdlemi1 38114 cdlemk11 38145 cdlemk11u 38167 cdlemk51 38249 cdlemm10N 38414 cdlemn10 38502 |
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