Theorem llnexchb2lem 37809

Description: Lemma for llnexchb2 37810. (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 1246	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ HL)
2		simpl21 1249	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ 𝐴)
3		simpl12 1247	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑋 ∈ 𝑁)
4		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		llnexch.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
6	4, 5	llnbase 37450	. . . . . . 7 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾))
7	3, 6	syl 17	. . . . . 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 37305	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ Lat)
9		simpl13 1248	. . . . . . . 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 37450	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾))
11	9, 10	syl 17	. . . . . . 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 18073	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
14	8, 7, 11, 13	syl3anc 1369	. . . . . 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 18097	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
17	8, 7, 11, 16	syl3anc 1369	. . . . . 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 37803	. . . . . 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 1381	. . . . 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 37230	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
23	2, 22	syl 17	. . . . . . . 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 18096	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋))
25	8, 7, 23, 24	syl3anc 1369	. . . . . . 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 1251	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑃 ≤ 𝑋)
27		hlatl 37301	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28	1, 27	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ AtLat)
29		eqid 2738	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
30	4, 15, 12, 29, 19	atnle 37258	. . . . . . . . 9 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = (0.‘𝐾)))
31	28, 2, 7, 30	syl3anc 1369	. . . . . . . 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 231	. . . . . . 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 2778	. . . . . 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 7270	. . . . 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 484	. . . . . . 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 37300	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
37	1, 36	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ CvLat)
38		simpl3 1191	. . . . . . . 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 1250	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
40		breq1 5073	. . . . . . . . . . . 12 ⊢ (𝑃 = (𝑋 ∧ 𝑌) → (𝑃 ≤ 𝑋 ↔ (𝑋 ∧ 𝑌) ≤ 𝑋))
41	17, 40	syl5ibrcom 246	. . . . . . . . . . 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 2956	. . . . . . . . . 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 15	. . . . . . . . 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 2998	. . . . . . . 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 37275	. . . . . . . 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 1381	. . . . . . 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 231	. . . . . 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 7271	. . . . 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 2787	. . . 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 37302	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
51	1, 50	syl 17	. . . . 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 37167	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ (𝑋 ∧ 𝑌)) = (𝑋 ∧ 𝑌))
53	51, 14, 52	syl2anc 583	. . . 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 2779	. . 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 412	. 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 1201	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
57	56	hllatd 37305	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58		simp12 1202	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑋 ∈ 𝑁)
59	58, 6	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑋 ∈ (Base‘𝐾))
60		simp21 1204	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑃 ∈ 𝐴)
61		simp22 1205	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑄 ∈ 𝐴)
62	4, 18, 19	hlatjcl 37308	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
63	56, 60, 61, 62	syl3anc 1369	. . . 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 18098	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
65	57, 59, 63, 64	syl3anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
66		breq1 5073	. . 3 ⊢ ((𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄)) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) ↔ (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄)))
67	65, 66	syl5ibrcom 246	. 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 211	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  joincjn 17944  meetcmee 17945  0.cp0 18056  Latclat 18064  OLcol 37115  Atomscatm 37204  AtLatcal 37205  CvLatclc 37206  HLchlt 37291  LLinesclln 37432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-psubsp 37444  df-pmap 37445  df-padd 37737

This theorem is referenced by:  llnexchb2  37810