Theorem llnexchb2lem 40567

Description: Lemma for llnexchb2 40568. (Contributed by NM, 17-Nov-2012.)

Hypotheses

Ref	Expression
llnexch.l	⊢ ≤ = (le‘𝐾)
llnexch.j	⊢ ∨ = (join‘𝐾)
llnexch.m	⊢ ∧ = (meet‘𝐾)
llnexch.a	⊢ 𝐴 = (Atoms‘𝐾)
llnexch.n	⊢ 𝑁 = (LLines‘𝐾)

Assertion

Ref	Expression
llnexchb2lem	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) ↔ (𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄))))

Proof of Theorem llnexchb2lem

Step	Hyp	Ref	Expression
1		simpl11 1265	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ HL)
2		simpl21 1268	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ 𝐴)
3		simpl12 1266	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑋 ∈ 𝑁)
4		eqid 2769	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		llnexch.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
6	4, 5	llnbase 40208	. . . . . . 7 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾))
7	3, 6	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑋 ∈ (Base‘𝐾))
8	1	hllatd 40063	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ Lat)
9		simpl13 1267	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑌 ∈ 𝑁)
10	4, 5	llnbase 40208	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾))
11	9, 10	syl 18	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑌 ∈ (Base‘𝐾))
12		llnexch.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
13	4, 12	latmcl 18496	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
14	8, 7, 11, 13	syl3anc 1396	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
15		llnexch.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
16	4, 15, 12	latmle1 18520	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
17	8, 7, 11, 16	syl3anc 1396	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
18		llnexch.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
19		llnexch.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
20	4, 15, 18, 12, 19	atmod2i2 40561	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) ∧ (𝑋 ∧ 𝑌) ≤ 𝑋) → ((𝑋 ∧ 𝑃) ∨ (𝑋 ∧ 𝑌)) = (𝑋 ∧ (𝑃 ∨ (𝑋 ∧ 𝑌))))
21	1, 2, 7, 14, 17, 20	syl131anc 1408	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → ((𝑋 ∧ 𝑃) ∨ (𝑋 ∧ 𝑌)) = (𝑋 ∧ (𝑃 ∨ (𝑋 ∧ 𝑌))))
22	4, 19	atbase 39988	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
23	2, 22	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ (Base‘𝐾))
24	4, 12	latmcom 18519	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋))
25	8, 7, 23, 24	syl3anc 1396	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋))
26		simpl23 1270	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑃 ≤ 𝑋)
27		hlatl 40059	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28	1, 27	syl 18	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ AtLat)
29		eqid 2769	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
30	4, 15, 12, 29, 19	atnle 40016	. . . . . . . . 9 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = (0.‘𝐾)))
31	28, 2, 7, 30	syl3anc 1396	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = (0.‘𝐾)))
32	26, 31	mpbid 235	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∧ 𝑋) = (0.‘𝐾))
33	25, 32	eqtrd 2804	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑃) = (0.‘𝐾))
34	33	oveq1d 7426	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → ((𝑋 ∧ 𝑃) ∨ (𝑋 ∧ 𝑌)) = ((0.‘𝐾) ∨ (𝑋 ∧ 𝑌)))
35		simpr 489	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄))
36		hlcvl 40058	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
37	1, 36	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ CvLat)
38		simpl3 1210	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑌) ∈ 𝐴)
39		simpl22 1269	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
40		breq1 5114	. . . . . . . . . . . 12 ⊢ (𝑃 = (𝑋 ∧ 𝑌) → (𝑃 ≤ 𝑋 ↔ (𝑋 ∧ 𝑌) ≤ 𝑋))
41	17, 40	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑃 = (𝑋 ∧ 𝑌) → 𝑃 ≤ 𝑋))
42	41	necon3bd 2978	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (¬ 𝑃 ≤ 𝑋 → 𝑃 ≠ (𝑋 ∧ 𝑌)))
43	26, 42	mpd 16	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ (𝑋 ∧ 𝑌))
44	43	necomd 3019	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑌) ≠ 𝑃)
45	15, 18, 19	cvlatexchb1 40033	. . . . . . . 8 ⊢ ((𝐾 ∈ CvLat ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≠ 𝑃) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ (𝑋 ∧ 𝑌)) = (𝑃 ∨ 𝑄)))
46	37, 38, 39, 2, 44, 45	syl131anc 1408	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ (𝑋 ∧ 𝑌)) = (𝑃 ∨ 𝑄)))
47	35, 46	mpbid 235	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ (𝑋 ∧ 𝑌)) = (𝑃 ∨ 𝑄))
48	47	oveq2d 7427	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ (𝑋 ∧ 𝑌))) = (𝑋 ∧ (𝑃 ∨ 𝑄)))
49	21, 34, 48	3eqtr3rd 2813	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) = ((0.‘𝐾) ∨ (𝑋 ∧ 𝑌)))
50		hlol 40060	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
51	1, 50	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ OL)
52	4, 18, 29	olj02 39925	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ (𝑋 ∧ 𝑌)) = (𝑋 ∧ 𝑌))
53	51, 14, 52	syl2anc 595	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → ((0.‘𝐾) ∨ (𝑋 ∧ 𝑌)) = (𝑋 ∧ 𝑌))
54	49, 53	eqtr2d 2805	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄)))
55	54	ex 417	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) → (𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄))))
56		simp11 1220	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
57	56	hllatd 40063	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58		simp12 1221	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑋 ∈ 𝑁)
59	58, 6	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑋 ∈ (Base‘𝐾))
60		simp21 1223	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑃 ∈ 𝐴)
61		simp22 1224	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑄 ∈ 𝐴)
62	4, 18, 19	hlatjcl 40066	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
63	56, 60, 61, 62	syl3anc 1396	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
64	4, 15, 12	latmle2 18521	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
65	57, 59, 63, 64	syl3anc 1396	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
66		breq1 5114	. . 3 ⊢ ((𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄)) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) ↔ (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄)))
67	65, 66	syl5ibrcom 250	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → ((𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄)) → (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)))
68	55, 67	impbid 215	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) ↔ (𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   class class class wbr 5111  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  lecple 17317  joincjn 18367  meetcmee 18368  0.cp0 18477  Latclat 18487  OLcol 39873  Atomscatm 39962  AtLatcal 39963  CvLatclc 39964  HLchlt 40049  LLinesclln 40190

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-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

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-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-lat 18488  df-clat 18555  df-oposet 39875  df-ol 39877  df-oml 39878  df-covers 39965  df-ats 39966  df-atl 39997  df-cvlat 40021  df-hlat 40050  df-llines 40197  df-psubsp 40202  df-pmap 40203  df-padd 40495

This theorem is referenced by:  llnexchb2  40568