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Mirrors > Home > MPE Home > Th. List > latabs1 | Structured version Visualization version GIF version |
Description: Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 29299 analog.) (Contributed by NM, 8-Nov-2011.) |
Ref | Expression |
---|---|
latabs1.b | ⊢ 𝐵 = (Base‘𝐾) |
latabs1.j | ⊢ ∨ = (join‘𝐾) |
latabs1.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latabs1 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latabs1.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2798 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | latabs1.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | 1, 2, 3 | latmle1 17678 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌)(le‘𝐾)𝑋) |
5 | 1, 3 | latmcl 17654 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
6 | latabs1.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
7 | 1, 2, 6 | latleeqj2 17666 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑋 ∧ 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋)) |
8 | 7 | 3com23 1123 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → ((𝑋 ∧ 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋)) |
9 | 5, 8 | syld3an3 1406 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋)) |
10 | 4, 9 | mpbid 235 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 joincjn 17546 meetcmee 17547 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-proset 17530 df-poset 17548 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-lat 17648 |
This theorem is referenced by: latdisdlem 17791 cvrexchlem 36715 |
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