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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 29875 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 2738 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | latabs1.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | 1, 2, 3 | latmle1 18180 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌)(le‘𝐾)𝑋) |
5 | 1, 3 | latmcl 18156 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
6 | latabs1.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
7 | 1, 2, 6 | latleeqj2 18168 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑋 ∧ 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋)) |
8 | 7 | 3com23 1125 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → ((𝑋 ∧ 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋)) |
9 | 5, 8 | syld3an3 1408 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∧ 𝑌)(le‘𝐾)𝑋 ↔ (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋)) |
10 | 4, 9 | mpbid 231 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5076 ‘cfv 6435 (class class class)co 7277 Basecbs 16910 lecple 16967 joincjn 18027 meetcmee 18028 Latclat 18147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-proset 18011 df-poset 18029 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-lat 18148 |
This theorem is referenced by: latdisdlem 18212 cvrexchlem 37430 |
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