Theorem cdlemg17a 41855

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 2760	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		cdlemg12.l	. 2 ⊢ ≤ = (le‘𝐾)
3		simp1l 1216	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
4	3	hllatd 40558	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat)
5		simp1 1154	. . . 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 1220	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐺 ∈ 𝑇)
7		simp2ll 1259	. . . 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 41334	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
12	5, 6, 7, 11	syl3anc 1398	. . 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 40483	. . 3 ⊢ ((𝐺‘𝑃) ∈ 𝐴 → (𝐺‘𝑃) ∈ (Base‘𝐾))
14	12, 13	syl 18	. 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 40561	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾))
17	3, 7, 12, 16	syl3anc 1398	. 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 1261	. . 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 40561	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
20	3, 7, 18, 19	syl3anc 1398	. 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 40570	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝐺‘𝑃) ≤ (𝑃 ∨ (𝐺‘𝑃)))
22	3, 7, 12, 21	syl3anc 1398	. 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 1218	. . . 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 2760	. . . . 5 ⊢ ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊)
26	2, 15, 24, 8, 9, 25	cdleme0cp 41408	. . . 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 850	. . 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 40569	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑄))
29	3, 7, 18, 28	syl3anc 1398	. . . 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 41357	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊))
32	5, 6, 23, 31	syl3anc 1398	. . . . 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 1221	. . . . 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 5128	. . . 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 40483	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
36	7, 35	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ (Base‘𝐾))
37		simp1r 1217	. . . . . . 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 41192	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
39	37, 38	syl 18	. . . . . 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 18623	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐺‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐺‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
41	4, 17, 39, 40	syl3anc 1398	. . . . 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 18633	. . . . 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 1399	. . . 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 725	. . 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 5128	. 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 18629	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ≤ (𝑃 ∨ 𝑄))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  lecple 17444  joincjn 18494  meetcmee 18495  Latclat 18614  Atomscatm 40457  HLchlt 40544  LHypclh 41178  LTrncltrn 41295  trLctrl 41352

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-map 8857  df-proset 18477  df-poset 18496  df-plt 18511  df-lub 18527  df-glb 18528  df-join 18529  df-meet 18530  df-p0 18606  df-p1 18607  df-lat 18615  df-clat 18682  df-oposet 40370  df-ol 40372  df-oml 40373  df-covers 40460  df-ats 40461  df-atl 40492  df-cvlat 40516  df-hlat 40545  df-psubsp 40697  df-pmap 40698  df-padd 40990  df-lhyp 41182  df-laut 41183  df-ldil 41298  df-ltrn 41299  df-trl 41353

This theorem is used by:  cdlemg17b  41856