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Mirrors > Home > MPE Home > Th. List > latabs2 | Structured version Visualization version GIF version |
Description: Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 29221 analog.) (Contributed by NM, 8-Nov-2011.) |
Ref | Expression |
---|---|
latabs1.b | ⊢ 𝐵 = (Base‘𝐾) |
latabs1.j | ⊢ ∨ = (join‘𝐾) |
latabs1.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latabs2 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latabs1.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2818 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | latabs1.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | 1, 2, 3 | latlej1 17658 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 𝑌)) |
5 | 1, 3 | latjcl 17649 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
6 | latabs1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
7 | 1, 2, 6 | latleeqm1 17677 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵) → (𝑋(le‘𝐾)(𝑋 ∨ 𝑌) ↔ (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)) |
8 | 5, 7 | syld3an3 1401 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)(𝑋 ∨ 𝑌) ↔ (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)) |
9 | 4, 8 | mpbid 233 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 joincjn 17542 meetcmee 17543 Latclat 17643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-proset 17526 df-poset 17544 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-lat 17644 |
This theorem is referenced by: latdisdlem 17787 cmtbr3N 36270 cdlemc6 37212 cdlemkid1 37938 cdlemkid2 37940 |
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