Theorem cdlemg17a 37789

Description: TODO: FIX COMMENT. (Contributed by NM, 8-May-2013.)

Hypotheses

Ref	Expression
cdlemg12.l	⊢ ≤ = (le‘𝐾)
cdlemg12.j	⊢ ∨ = (join‘𝐾)
cdlemg12.m	⊢ ∧ = (meet‘𝐾)
cdlemg12.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg12.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg12.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg12b.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemg17a	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ≤ (𝑃 ∨ 𝑄))

Proof of Theorem cdlemg17a

Step	Hyp	Ref	Expression
1		eqid 2819	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		cdlemg12.l	. 2 ⊢ ≤ = (le‘𝐾)
3		simp1l 1191	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
4	3	hllatd 36492	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat)
5		simp1 1130	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		simp3l 1195	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐺 ∈ 𝑇)
7		simp2ll 1234	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
8		cdlemg12.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdlemg12.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
10		cdlemg12.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	2, 8, 9, 10	ltrnat 37268	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
12	5, 6, 7, 11	syl3anc 1365	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ∈ 𝐴)
13	1, 8	atbase 36417	. . 3 ⊢ ((𝐺‘𝑃) ∈ 𝐴 → (𝐺‘𝑃) ∈ (Base‘𝐾))
14	12, 13	syl 17	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ∈ (Base‘𝐾))
15		cdlemg12.j	. . . 4 ⊢ ∨ = (join‘𝐾)
16	1, 15, 8	hlatjcl 36495	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
17	3, 7, 12, 16	syl3anc 1365	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
18		simp2rl 1236	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
19	1, 15, 8	hlatjcl 36495	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
20	3, 7, 18, 19	syl3anc 1365	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
21	2, 15, 8	hlatlej2 36504	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝐺‘𝑃) ≤ (𝑃 ∨ (𝐺‘𝑃)))
22	3, 7, 12, 21	syl3anc 1365	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ≤ (𝑃 ∨ (𝐺‘𝑃)))
23		simp2l 1193	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
24		cdlemg12.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
25		eqid 2819	. . . . 5 ⊢ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)
26	2, 15, 24, 8, 9, 25	cdleme0cp 37342	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐺‘𝑃) ∈ 𝐴)) → (𝑃 ∨ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐺‘𝑃)))
27	5, 23, 12, 26	syl12anc 834	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐺‘𝑃)))
28	2, 15, 8	hlatlej1 36503	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄))
29	3, 7, 18, 28	syl3anc 1365	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≤ (𝑃 ∨ 𝑄))
30		cdlemg12b.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
31	2, 15, 24, 8, 9, 10, 30	trlval2 37291	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊))
32	5, 6, 23, 31	syl3anc 1365	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊))
33		simp3r 1196	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))
34	32, 33	eqbrtrrd 5081	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
35	1, 8	atbase 36417	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
36	7, 35	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ (Base‘𝐾))
37		simp1r 1192	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ 𝐻)
38	1, 9	lhpbase 37126	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
39	37, 38	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ (Base‘𝐾))
40	1, 24	latmcl 17654	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
41	4, 17, 39, 40	syl3anc 1365	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
42	1, 2, 15	latjle12 17664	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑃 ∨ 𝑄) ∧ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) ≤ (𝑃 ∨ 𝑄)))
43	4, 36, 41, 20, 42	syl13anc 1366	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ≤ (𝑃 ∨ 𝑄) ∧ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) ≤ (𝑃 ∨ 𝑄)))
44	29, 34, 43	mpbi2and 710	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)) ≤ (𝑃 ∨ 𝑄))
45	27, 44	eqbrtrrd 5081	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ (𝐺‘𝑃)) ≤ (𝑃 ∨ 𝑄))
46	1, 2, 4, 14, 17, 20, 22, 45	lattrd 17660	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ≤ (𝑃 ∨ 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   class class class wbr 5057  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Latclat 17647  Atomscatm 36391  HLchlt 36478  LHypclh 37112  LTrncltrn 37229  trLctrl 37286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-psubsp 36631  df-pmap 36632  df-padd 36924  df-lhyp 37116  df-laut 37117  df-ldil 37232  df-ltrn 37233  df-trl 37287

This theorem is referenced by:  cdlemg17b  37790