Theorem llnexchb2lem 39870

Description: Lemma for llnexchb2 39871. (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 2737	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		llnexch.n	. . . . . . . 8 ⊢ 𝑁 = (LLines‘𝐾)
6	4, 5	llnbase 39511	. . . . . . 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 39365	. . . . . . 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 39511	. . . . . . . 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 18485	. . . . . . 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 18509	. . . . . . 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 39864	. . . . . 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 39290	. . . . . . . . 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 18508	. . . . . . . 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 39361	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
30	4, 15, 12, 29, 19	atnle 39318	. . . . . . . . 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 2777	. . . . . 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 39360	. . . . . . . . 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 5146	. . . . . . . . . . . 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 2954	. . . . . . . . . 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 2996	. . . . . . . 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 39335	. . . . . . . 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 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 2786	. . . 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 39362	. . . . . 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 39227	. . . . 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 2778	. . 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 39365	. . . 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 39368	. . . . 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 18510	. . . 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 5146	. . 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 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  0.cp0 18468  Latclat 18476  OLcol 39175  Atomscatm 39264  AtLatcal 39265  CvLatclc 39266  HLchlt 39351  LLinesclln 39493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-psubsp 39505  df-pmap 39506  df-padd 39798

This theorem is referenced by:  llnexchb2  39871