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Mirrors > Home > MPE Home > Th. List > latjjdir | Structured version Visualization version GIF version |
Description: Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.) |
Ref | Expression |
---|---|
latjass.b | ⊢ 𝐵 = (Base‘𝐾) |
latjass.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latjjdir | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑋 ∨ 𝑍) ∨ (𝑌 ∨ 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latjass.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latjass.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
3 | 1, 2 | latjidm 18519 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑍 ∈ 𝐵) → (𝑍 ∨ 𝑍) = 𝑍) |
4 | 3 | 3ad2antr3 1189 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 ∨ 𝑍) = 𝑍) |
5 | 4 | oveq2d 7446 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ (𝑍 ∨ 𝑍)) = ((𝑋 ∨ 𝑌) ∨ 𝑍)) |
6 | simpl 482 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
7 | simpr1 1193 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
8 | simpr2 1194 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
9 | simpr3 1195 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
10 | 1, 2 | latj4 18546 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ (𝑍 ∨ 𝑍)) = ((𝑋 ∨ 𝑍) ∨ (𝑌 ∨ 𝑍))) |
11 | 6, 7, 8, 9, 9, 10 | syl122anc 1378 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ (𝑍 ∨ 𝑍)) = ((𝑋 ∨ 𝑍) ∨ (𝑌 ∨ 𝑍))) |
12 | 5, 11 | eqtr3d 2776 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑋 ∨ 𝑍) ∨ (𝑌 ∨ 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 joincjn 18368 Latclat 18488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-proset 18351 df-poset 18370 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-lat 18489 |
This theorem is referenced by: dalem38 39692 cdleme23b 40332 |
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