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Theorem dlatmjdi 17807
 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 17806 . . 3 (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧))))
54simprbi 499 . 2 (𝐾 ∈ DLat → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)))
6 oveq1 7166 . . . 4 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
7 oveq1 7166 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
8 oveq1 7166 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑧) = (𝑋 𝑧))
97, 8oveq12d 7177 . . . 4 (𝑥 = 𝑋 → ((𝑥 𝑦) (𝑥 𝑧)) = ((𝑋 𝑦) (𝑋 𝑧)))
106, 9eqeq12d 2840 . . 3 (𝑥 = 𝑋 → ((𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)) ↔ (𝑋 (𝑦 𝑧)) = ((𝑋 𝑦) (𝑋 𝑧))))
11 oveq1 7166 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1211oveq2d 7175 . . . 4 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
13 oveq2 7167 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
1413oveq1d 7174 . . . 4 (𝑦 = 𝑌 → ((𝑋 𝑦) (𝑋 𝑧)) = ((𝑋 𝑌) (𝑋 𝑧)))
1512, 14eqeq12d 2840 . . 3 (𝑦 = 𝑌 → ((𝑋 (𝑦 𝑧)) = ((𝑋 𝑦) (𝑋 𝑧)) ↔ (𝑋 (𝑌 𝑧)) = ((𝑋 𝑌) (𝑋 𝑧))))
16 oveq2 7167 . . . . 5 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
1716oveq2d 7175 . . . 4 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
18 oveq2 7167 . . . . 5 (𝑧 = 𝑍 → (𝑋 𝑧) = (𝑋 𝑍))
1918oveq2d 7175 . . . 4 (𝑧 = 𝑍 → ((𝑋 𝑌) (𝑋 𝑧)) = ((𝑋 𝑌) (𝑋 𝑍)))
2017, 19eqeq12d 2840 . . 3 (𝑧 = 𝑍 → ((𝑋 (𝑌 𝑧)) = ((𝑋 𝑌) (𝑋 𝑧)) ↔ (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍))))
2110, 15, 20rspc3v 3639 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 (𝑦 𝑧)) = ((𝑥 𝑦) (𝑥 𝑧)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍))))
225, 21mpan9 509 1 ((𝐾 ∈ DLat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ‘cfv 6358  (class class class)co 7159  Basecbs 16486  joincjn 17557  meetcmee 17558  Latclat 17658  DLatcdlat 17804 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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-nul 5213 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-iota 6317  df-fv 6366  df-ov 7162  df-dlat 17805 This theorem is referenced by:  dlatjmdi  17810
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