Theorem dlatmjdi 18579

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 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (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.

Step	Hyp	Ref	Expression
1		isdlat.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		isdlat.j	. . . 4 ⊢ ∨ = (join‘𝐾)
3		isdlat.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
4	1, 2, 3	isdlat 18578	. . 3 ⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
5	4	simprbi 502	. 2 ⊢ (𝐾 ∈ DLat → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))
6		oveq1 7418	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ (𝑦 ∨ 𝑧)) = (𝑋 ∧ (𝑦 ∨ 𝑧)))
7		oveq1 7418	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑦) = (𝑋 ∧ 𝑦))
8		oveq1 7418	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑧) = (𝑋 ∧ 𝑧))
9	7, 8	oveq12d 7429	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)))
10	6, 9	eqeq12d 2785	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑦 ∨ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧))))
11		oveq1 7418	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ∨ 𝑧) = (𝑌 ∨ 𝑧))
12	11	oveq2d 7427	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ∧ (𝑦 ∨ 𝑧)) = (𝑋 ∧ (𝑌 ∨ 𝑧)))
13		oveq2 7419	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 ∧ 𝑦) = (𝑋 ∧ 𝑌))
14	13	oveq1d 7426	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)))
15	12, 14	eqeq12d 2785	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ∧ (𝑦 ∨ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑌 ∨ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧))))
16		oveq2 7419	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑌 ∨ 𝑧) = (𝑌 ∨ 𝑍))
17	16	oveq2d 7427	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑋 ∧ (𝑌 ∨ 𝑧)) = (𝑋 ∧ (𝑌 ∨ 𝑍)))
18		oveq2 7419	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ∧ 𝑧) = (𝑋 ∧ 𝑍))
19	18	oveq2d 7427	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)))
20	17, 19	eqeq12d 2785	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋 ∧ (𝑌 ∨ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍))))
21	10, 15, 20	rspc3v 3606	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍))))
22	5, 21	mpan9 515	1 ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  joincjn 18367  meetcmee 18368  Latclat 18487  DLatcdlat 18576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-dlat 18577

This theorem is referenced by:  dlatjmdi  18582