MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dlatmjdi Structured version   Visualization version   GIF version

Theorem dlatmjdi 18581
Description: In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
isdlat.b 𝐵 = (Base‘𝐾)
isdlat.j = (join‘𝐾)
isdlat.m = (meet‘𝐾)
Assertion
Ref Expression
dlatmjdi ((𝐾 ∈ DLat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Proof of Theorem dlatmjdi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdlat.b . . . 4 𝐵 = (Base‘𝐾)
2 isdlat.j . . . 4 = (join‘𝐾)
3 isdlat.m . . . 4 = (meet‘𝐾)
41, 2, 3isdlat 18580 . . 3 (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
54simprbi 496 . 2 (𝐾 ∈ DLat → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)))
6 oveq1 7438 . . . 4 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
7 oveq1 7438 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
8 oveq1 7438 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑧) = (𝑋 𝑧))
97, 8oveq12d 7449 . . . 4 (𝑥 = 𝑋 → ((𝑥 𝑦) (𝑥 𝑧)) = ((𝑋 𝑦) (𝑋 𝑧)))
106, 9eqeq12d 2751 . . 3 (𝑥 = 𝑋 → ((𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)) ↔ (𝑋 (𝑦 𝑧)) = ((𝑋 𝑦) (𝑋 𝑧))))
11 oveq1 7438 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1211oveq2d 7447 . . . 4 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
13 oveq2 7439 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
1413oveq1d 7446 . . . 4 (𝑦 = 𝑌 → ((𝑋 𝑦) (𝑋 𝑧)) = ((𝑋 𝑌) (𝑋 𝑧)))
1512, 14eqeq12d 2751 . . 3 (𝑦 = 𝑌 → ((𝑋 (𝑦 𝑧)) = ((𝑋 𝑦) (𝑋 𝑧)) ↔ (𝑋 (𝑌 𝑧)) = ((𝑋 𝑌) (𝑋 𝑧))))
16 oveq2 7439 . . . . 5 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
1716oveq2d 7447 . . . 4 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
18 oveq2 7439 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧) = (𝑋 𝑍))
1918oveq2d 7447 . . . 4 (𝑧 = 𝑍 → ((𝑋 𝑌) (𝑋 𝑧)) = ((𝑋 𝑌) (𝑋 𝑍)))
2017, 19eqeq12d 2751 . . 3 (𝑧 = 𝑍 → ((𝑋 (𝑌 𝑧)) = ((𝑋 𝑌) (𝑋 𝑧)) ↔ (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍))))
2110, 15, 20rspc3v 3638 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍))))
225, 21mpan9 506 1 ((𝐾 ∈ DLat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  joincjn 18369  meetcmee 18370  Latclat 18489  DLatcdlat 18578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-dlat 18579
This theorem is referenced by:  dlatjmdi  18584
  Copyright terms: Public domain W3C validator