Theorem isdlat 18428

Description: Property of being a distributive lattice. (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
isdlat	⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐾   𝑥,𝐵,𝑦,𝑧   𝑥, ∨ ,𝑦,𝑧   𝑥, ∧ ,𝑦,𝑧

Proof of Theorem isdlat

Dummy variables 𝑘 𝑏 𝑗 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isdlat.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2784	. . . 4 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6822	. . . . . 6 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
5		isdlat.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
6	4, 5	eqtr4di 2784	. . . . 5 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
7		fveq2 6822	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
8		isdlat.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
9	7, 8	eqtr4di 2784	. . . . . 6 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
10	9	sbceq1d 3741	. . . . 5 ⊢ (𝑘 = 𝐾 → ([(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ [ ∧ / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
11	6, 10	sbceqbid 3743	. . . 4 ⊢ (𝑘 = 𝐾 → ([(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ [ ∨ / 𝑗][ ∧ / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
12	3, 11	sbceqbid 3743	. . 3 ⊢ (𝑘 = 𝐾 → ([(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ [𝐵 / 𝑏][ ∨ / 𝑗][ ∧ / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
13	2	fvexi 6836	. . . 4 ⊢ 𝐵 ∈ V
14	5	fvexi 6836	. . . 4 ⊢ ∨ ∈ V
15	8	fvexi 6836	. . . 4 ⊢ ∧ ∈ V
16		raleq 3289	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
17	16	raleqbi1dv 3304	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
18	17	raleqbi1dv 3304	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))))
19		simpr 484	. . . . . . . . . 10 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → 𝑚 = ∧ )
20		eqidd 2732	. . . . . . . . . 10 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → 𝑥 = 𝑥)
21		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → 𝑗 = ∨ )
22	21	oveqd 7363	. . . . . . . . . 10 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → (𝑦𝑗𝑧) = (𝑦 ∨ 𝑧))
23	19, 20, 22	oveq123d 7367	. . . . . . . . 9 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → (𝑥𝑚(𝑦𝑗𝑧)) = (𝑥 ∧ (𝑦 ∨ 𝑧)))
24	19	oveqd 7363	. . . . . . . . . 10 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → (𝑥𝑚𝑦) = (𝑥 ∧ 𝑦))
25	19	oveqd 7363	. . . . . . . . . 10 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → (𝑥𝑚𝑧) = (𝑥 ∧ 𝑧))
26	21, 24, 25	oveq123d 7367	. . . . . . . . 9 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))
27	23, 26	eqeq12d 2747	. . . . . . . 8 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → ((𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
28	27	ralbidv 3155	. . . . . . 7 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → (∀𝑧 ∈ 𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
29	28	2ralbidv 3196	. . . . . 6 ⊢ ((𝑗 = ∨ ∧ 𝑚 = ∧ ) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
30	18, 29	sylan9bb 509	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ (𝑗 = ∨ ∧ 𝑚 = ∧ )) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
31	30	3impb 1114	. . . 4 ⊢ ((𝑏 = 𝐵 ∧ 𝑗 = ∨ ∧ 𝑚 = ∧ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
32	13, 14, 15, 31	sbc3ie 3814	. . 3 ⊢ ([𝐵 / 𝑏][ ∨ / 𝑗][ ∧ / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))
33	12, 32	bitrdi 287	. 2 ⊢ (𝑘 = 𝐾 → ([(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
34		df-dlat 18427	. 2 ⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))}
35	33, 34	elrab2 3645	1 ⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  [wsbc 3736  ‘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:  dlatmjdi  18429  dlatl  18430  odudlatb  18431  topdlat  49114