Theorem dlatmjdi 18429

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 18428	. . 3 ⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
5	4	simprbi 496	. 2 ⊢ (𝐾 ∈ DLat → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))
6		oveq1 7353	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ (𝑦 ∨ 𝑧)) = (𝑋 ∧ (𝑦 ∨ 𝑧)))
7		oveq1 7353	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑦) = (𝑋 ∧ 𝑦))
8		oveq1 7353	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∧ 𝑧) = (𝑋 ∧ 𝑧))
9	7, 8	oveq12d 7364	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)))
10	6, 9	eqeq12d 2747	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑦 ∨ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧))))
11		oveq1 7353	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ∨ 𝑧) = (𝑌 ∨ 𝑧))
12	11	oveq2d 7362	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ∧ (𝑦 ∨ 𝑧)) = (𝑋 ∧ (𝑌 ∨ 𝑧)))
13		oveq2 7354	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 ∧ 𝑦) = (𝑋 ∧ 𝑌))
14	13	oveq1d 7361	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)))
15	12, 14	eqeq12d 2747	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ∧ (𝑦 ∨ 𝑧)) = ((𝑋 ∧ 𝑦) ∨ (𝑋 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑌 ∨ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧))))
16		oveq2 7354	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑌 ∨ 𝑧) = (𝑌 ∨ 𝑍))
17	16	oveq2d 7362	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑋 ∧ (𝑌 ∨ 𝑧)) = (𝑋 ∧ (𝑌 ∨ 𝑍)))
18		oveq2 7354	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ∧ 𝑧) = (𝑋 ∧ 𝑍))
19	18	oveq2d 7362	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)))
20	17, 19	eqeq12d 2747	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋 ∧ (𝑌 ∨ 𝑧)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑧)) ↔ (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍))))
21	10, 15, 20	rspc3v 3588	. 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 506	1 ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  joincjn 18217  meetcmee 18218  Latclat 18337  DLatcdlat 18426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-dlat 18427

This theorem is referenced by:  dlatjmdi  18432