Theorem llnexchb2lem 40067

Description: Lemma for llnexchb2 40068. (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 1249	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ HL)
2		simpl21 1252	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ 𝐴)
3		simpl12 1250	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑋 ∈ 𝑁)
4		eqid 2734	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		llnexch.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
6	4, 5	llnbase 39708	. . . . . . 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 39563	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ Lat)
9		simpl13 1251	. . . . . . . 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 39708	. . . . . . . 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 18361	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
14	8, 7, 11, 13	syl3anc 1373	. . . . . 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 18385	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
17	8, 7, 11, 16	syl3anc 1373	. . . . . 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 40061	. . . . . 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 1385	. . . . 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 39488	. . . . . . . . 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 18384	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑃) = (𝑃 ∧ 𝑋))
25	8, 7, 23, 24	syl3anc 1373	. . . . . . 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 1254	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑃 ≤ 𝑋)
27		hlatl 39559	. . . . . . . . . 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 2734	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
30	4, 15, 12, 29, 19	atnle 39516	. . . . . . . . 9 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ (Base‘𝐾)) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = (0.‘𝐾)))
31	28, 2, 7, 30	syl3anc 1373	. . . . . . . 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 2769	. . . . . 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 7371	. . . . 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 39558	. . . . . . . . 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 1194	. . . . . . . 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 1253	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) ∧ (𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
40		breq1 5099	. . . . . . . . . . . 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 2944	. . . . . . . . . 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 2985	. . . . . . . 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 39533	. . . . . . . 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 1385	. . . . . . 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 7372	. . . . 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 2778	. . . 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 39560	. . . . . 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 39425	. . . . 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 2770	. . 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 1204	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ HL)
57	56	hllatd 39563	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ Lat)
58		simp12 1205	. . . . 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 1207	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑃 ∈ 𝐴)
61		simp22 1208	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝑄 ∈ 𝐴)
62	4, 18, 19	hlatjcl 39566	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
63	56, 60, 61, 62	syl3anc 1373	. . . 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 18386	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
65	57, 59, 63, 64	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ≤ (𝑃 ∨ 𝑄))
66		breq1 5099	. . 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 1541   ∈ wcel 2113   ≠ wne 2930   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  lecple 17182  joincjn 18232  meetcmee 18233  0.cp0 18342  Latclat 18352  OLcol 39373  Atomscatm 39462  AtLatcal 39463  CvLatclc 39464  HLchlt 39549  LLinesclln 39690

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

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-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-proset 18215  df-poset 18234  df-plt 18249  df-lub 18265  df-glb 18266  df-join 18267  df-meet 18268  df-p0 18344  df-lat 18353  df-clat 18420  df-oposet 39375  df-ol 39377  df-oml 39378  df-covers 39465  df-ats 39466  df-atl 39497  df-cvlat 39521  df-hlat 39550  df-llines 39697  df-psubsp 39702  df-pmap 39703  df-padd 39995

This theorem is referenced by:  llnexchb2  40068