Theorem llnexchb2lem 39851

Description: Lemma for llnexchb2 39852. (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 1247	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ HL)
2		simpl21 1250	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ 𝐴)
3		simpl12 1248	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑋 ∈ 𝑁)
4		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		llnexch.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
6	4, 5	llnbase 39492	. . . . . . 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 39346	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ Lat)
9		simpl13 1249	. . . . . . . 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 39492	. . . . . . . 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 18498	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
14	8, 7, 11, 13	syl3anc 1370	. . . . . 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 18522	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
17	8, 7, 11, 16	syl3anc 1370	. . . . . 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 39845	. . . . . 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 1382	. . . . 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 39271	. . . . . . . . 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 18521	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋))
25	8, 7, 23, 24	syl3anc 1370	. . . . . . 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 1252	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑃 ≤ 𝑋)
27		hlatl 39342	. . . . . . . . . 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 2735	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
30	4, 15, 12, 29, 19	atnle 39299	. . . . . . . . 9 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = (0.‘𝐾)))
31	28, 2, 7, 30	syl3anc 1370	. . . . . . . 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 232	. . . . . . 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 2775	. . . . . 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 7446	. . . . 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 39341	. . . . . . . . 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 1192	. . . . . . . 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 1251	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
40		breq1 5151	. . . . . . . . . . . 12 ⊢ (𝑃 = (𝑋 ∧ 𝑌) → (𝑃 ≤ 𝑋 ↔ (𝑋 ∧ 𝑌) ≤ 𝑋))
41	17, 40	syl5ibrcom 247	. . . . . . . . . . 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 2952	. . . . . . . . . 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 2994	. . . . . . . 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 39316	. . . . . . . 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 1382	. . . . . . 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 232	. . . . . 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 7447	. . . . 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 2784	. . . 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 39343	. . . . . 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 39208	. . . . 5 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ (𝑋 ∧ 𝑌)) = (𝑋 ∧ 𝑌))
53	51, 14, 52	syl2anc 584	. . . 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 2776	. . 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 1202	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
57	56	hllatd 39346	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58		simp12 1203	. . . . 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 1205	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑃 ∈ 𝐴)
61		simp22 1206	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑄 ∈ 𝐴)
62	4, 18, 19	hlatjcl 39349	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
63	56, 60, 61, 62	syl3anc 1370	. . . 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 18523	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
65	57, 59, 63, 64	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
66		breq1 5151	. . 3 ⊢ ((𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄)) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) ↔ (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄)))
67	65, 66	syl5ibrcom 247	. 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 212	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  0.cp0 18481  Latclat 18489  OLcol 39156  Atomscatm 39245  AtLatcal 39246  CvLatclc 39247  HLchlt 39332  LLinesclln 39474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-psubsp 39486  df-pmap 39487  df-padd 39779

This theorem is referenced by:  llnexchb2  39852