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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 28656 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 2748 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | latabs1.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | 1, 2, 3 | latlej1 17232 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 ∨ 𝑌)) |
5 | 1, 3 | latjcl 17223 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
6 | latabs1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
7 | 1, 2, 6 | latleeqm1 17251 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 ∨ 𝑌) ∈ 𝐵) → (𝑋(le‘𝐾)(𝑋 ∨ 𝑌) ↔ (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)) |
8 | 5, 7 | syld3an3 1508 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(le‘𝐾)(𝑋 ∨ 𝑌) ↔ (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)) |
9 | 4, 8 | mpbid 222 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1072 = wceq 1620 ∈ wcel 2127 class class class wbr 4792 ‘cfv 6037 (class class class)co 6801 Basecbs 16030 lecple 16121 joincjn 17116 meetcmee 17117 Latclat 17217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-preset 17100 df-poset 17118 df-lub 17146 df-glb 17147 df-join 17148 df-meet 17149 df-lat 17218 |
This theorem is referenced by: latdisdlem 17361 cmtbr3N 35013 cdlemc6 35955 cdlemkid1 36681 cdlemkid2 36683 |
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