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Mirrors > Home > MPE Home > Th. List > latjidm | Structured version Visualization version GIF version |
Description: Lattice join is idempotent. (Contributed by NM, 8-Oct-2011.) |
Ref | Expression |
---|---|
latidm.b | ⊢ 𝐵 = (Base‘𝐾) |
latidm.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latjidm | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latidm.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2760 | . 2 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | simpl 474 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) | |
4 | latidm.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | 1, 4 | latjcl 17252 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) ∈ 𝐵) |
6 | 5 | 3anidm23 1532 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) ∈ 𝐵) |
7 | simpr 479 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
8 | 1, 2 | latref 17254 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
9 | 1, 2, 4 | latjle12 17263 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋(le‘𝐾)𝑋 ∧ 𝑋(le‘𝐾)𝑋) ↔ (𝑋 ∨ 𝑋)(le‘𝐾)𝑋)) |
10 | 3, 7, 7, 7, 9 | syl13anc 1479 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑋 ∧ 𝑋(le‘𝐾)𝑋) ↔ (𝑋 ∨ 𝑋)(le‘𝐾)𝑋)) |
11 | 8, 8, 10 | mpbi2and 994 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋)(le‘𝐾)𝑋) |
12 | 1, 2, 4 | latlej1 17261 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 𝑋)) |
13 | 12 | 3anidm23 1532 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 𝑋)) |
14 | 1, 2, 3, 6, 7, 11, 13 | latasymd 17258 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 lecple 16150 joincjn 17145 Latclat 17246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-preset 17129 df-poset 17147 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-lat 17247 |
This theorem is referenced by: lubsn 17295 latjjdi 17304 latjjdir 17305 cvlsupr2 35133 hlatjidm 35158 cvrat3 35231 snatpsubN 35539 dalawlem7 35666 cdleme11 36060 cdleme23b 36140 cdlemg33a 36496 trljco 36530 doca2N 36917 djajN 36928 |
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