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Theorem dlatmjdi 18480
Description: In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
isdlat.b 𝐡 = (Baseβ€˜πΎ)
isdlat.j ∨ = (joinβ€˜πΎ)
isdlat.m ∧ = (meetβ€˜πΎ)
Assertion
Ref Expression
dlatmjdi ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑍)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑍)))

Proof of Theorem dlatmjdi
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdlat.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 isdlat.j . . . 4 ∨ = (joinβ€˜πΎ)
3 isdlat.m . . . 4 ∧ = (meetβ€˜πΎ)
41, 2, 3isdlat 18479 . . 3 (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ∧ (𝑦 ∨ 𝑧)) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ 𝑧))))
54simprbi 495 . 2 (𝐾 ∈ DLat β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ∧ (𝑦 ∨ 𝑧)) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ 𝑧)))
6 oveq1 7418 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ ∧ (𝑦 ∨ 𝑧)) = (𝑋 ∧ (𝑦 ∨ 𝑧)))
7 oveq1 7418 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯ ∧ 𝑦) = (𝑋 ∧ 𝑦))
8 oveq1 7418 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘₯ ∧ 𝑧) = (𝑋 ∧ 𝑧))
97, 8oveq12d 7429 . . . 4 (π‘₯ = 𝑋 β†’ ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)))
106, 9eqeq12d 2746 . . 3 (π‘₯ = 𝑋 β†’ ((π‘₯ ∧ (𝑦 ∨ 𝑧)) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ 𝑧)) ↔ (𝑋 ∧ (𝑦 ∨ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧))))
11 oveq1 7418 . . . . 5 (𝑦 = π‘Œ β†’ (𝑦 ∨ 𝑧) = (π‘Œ ∨ 𝑧))
1211oveq2d 7427 . . . 4 (𝑦 = π‘Œ β†’ (𝑋 ∧ (𝑦 ∨ 𝑧)) = (𝑋 ∧ (π‘Œ ∨ 𝑧)))
13 oveq2 7419 . . . . 5 (𝑦 = π‘Œ β†’ (𝑋 ∧ 𝑦) = (𝑋 ∧ π‘Œ))
1413oveq1d 7426 . . . 4 (𝑦 = π‘Œ β†’ ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑧)))
1512, 14eqeq12d 2746 . . 3 (𝑦 = π‘Œ β†’ ((𝑋 ∧ (𝑦 ∨ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)) ↔ (𝑋 ∧ (π‘Œ ∨ 𝑧)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑧))))
16 oveq2 7419 . . . . 5 (𝑧 = 𝑍 β†’ (π‘Œ ∨ 𝑧) = (π‘Œ ∨ 𝑍))
1716oveq2d 7427 . . . 4 (𝑧 = 𝑍 β†’ (𝑋 ∧ (π‘Œ ∨ 𝑧)) = (𝑋 ∧ (π‘Œ ∨ 𝑍)))
18 oveq2 7419 . . . . 5 (𝑧 = 𝑍 β†’ (𝑋 ∧ 𝑧) = (𝑋 ∧ 𝑍))
1918oveq2d 7427 . . . 4 (𝑧 = 𝑍 β†’ ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑧)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑍)))
2017, 19eqeq12d 2746 . . 3 (𝑧 = 𝑍 β†’ ((𝑋 ∧ (π‘Œ ∨ 𝑧)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑧)) ↔ (𝑋 ∧ (π‘Œ ∨ 𝑍)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑍))))
2110, 15, 20rspc3v 3626 . 2 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ∧ (𝑦 ∨ 𝑧)) = ((π‘₯ ∧ 𝑦) ∨ (π‘₯ ∧ 𝑧)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑍)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑍))))
225, 21mpan9 505 1 ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 ∧ (π‘Œ ∨ 𝑍)) = ((𝑋 ∧ π‘Œ) ∨ (𝑋 ∧ 𝑍)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  joincjn 18268  meetcmee 18269  Latclat 18388  DLatcdlat 18477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-dlat 18478
This theorem is referenced by:  dlatjmdi  18483
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