cdlemg10 - Mathbox for Norm Megill

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Theorem cdlemg10 40025

Description: TODO: FIX COMMENT. (Contributed by NM, 4-May-2013.)

**Hypotheses**
Ref	Expression
cdlemg8.l	⊢ ≤ = (le‘𝐾)
cdlemg8.j	⊢ ∨ = (join‘𝐾)
cdlemg8.m	⊢ ∧ = (meet‘𝐾)
cdlemg8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg10.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemg10	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ 𝑊)

**Proof of Theorem cdlemg10**
Step	Hyp	Ref	Expression
1		eqid 2726	. 2 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		cdlemg8.l	. 2 ⊢ ≤ = (le‘𝐾)
3		simp11l 1281	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
4	3	hllatd 38747	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat)
5		simp12l 1283	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
6		simp11 1200	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		simp21 1203	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐹 ∈ 𝑇)
8		simp22 1204	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝐺 ∈ 𝑇)
9		cdlemg8.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
10		cdlemg8.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
11		cdlemg8.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12	2, 9, 10, 11	ltrnat 39524	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐺‘𝑃) ∈ 𝐴)
13	6, 8, 5, 12	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑃) ∈ 𝐴)
14	2, 9, 10, 11	ltrnat 39524	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝐺‘𝑃) ∈ 𝐴) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴)
15	6, 7, 13, 14	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐹‘(𝐺‘𝑃)) ∈ 𝐴)
16		cdlemg8.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
17	1, 16, 9	hlatjcl 38750	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑃)) ∈ 𝐴) → (𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∈ (Base‘𝐾))
18	3, 5, 15, 17	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∈ (Base‘𝐾))
19		simp13l 1285	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
20	2, 9, 10, 11	ltrnat 39524	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐺‘𝑄) ∈ 𝐴)
21	6, 8, 19, 20	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐺‘𝑄) ∈ 𝐴)
22	2, 9, 10, 11	ltrnat 39524	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝐺‘𝑄) ∈ 𝐴) → (𝐹‘(𝐺‘𝑄)) ∈ 𝐴)
23	6, 7, 21, 22	syl3anc 1368	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝐹‘(𝐺‘𝑄)) ∈ 𝐴)
24	1, 16, 9	hlatjcl 38750	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘(𝐺‘𝑄)) ∈ 𝐴) → (𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∈ (Base‘𝐾))
25	3, 19, 23, 24	syl3anc 1368	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∈ (Base‘𝐾))
26		cdlemg8.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
27	1, 26	latmcl 18405	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∈ (Base‘𝐾) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ∈ (Base‘𝐾))
28	4, 18, 25, 27	syl3anc 1368	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ∈ (Base‘𝐾))
29		cdlemg10.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
30	1, 10, 11, 29	trlcl 39548	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
31	6, 7, 30	syl2anc 583	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐹) ∈ (Base‘𝐾))
32	1, 10, 11, 29	trlcl 39548	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ∈ (Base‘𝐾))
33	6, 8, 32	syl2anc 583	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) ∈ (Base‘𝐾))
34	1, 16	latjcl 18404	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾)) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
35	4, 31, 33, 34	syl3anc 1368	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ∈ (Base‘𝐾))
36		simp11r 1282	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ 𝐻)
37	1, 10	lhpbase 39382	. . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
38	36, 37	syl 17	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ (Base‘𝐾))
39	2, 16, 26, 9, 10, 11, 29	cdlemg10a 40024	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
40	2, 10, 11, 29	trlle 39568	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
41	6, 7, 40	syl2anc 583	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐹) ≤ 𝑊)
42	2, 10, 11, 29	trlle 39568	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) ≤ 𝑊)
43	6, 8, 42	syl2anc 583	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐺) ≤ 𝑊)
44	1, 2, 16	latjle12 18415	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑅‘𝐺) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑅‘𝐹) ≤ 𝑊 ∧ (𝑅‘𝐺) ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ≤ 𝑊))
45	4, 31, 33, 38, 44	syl13anc 1369	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → (((𝑅‘𝐹) ≤ 𝑊 ∧ (𝑅‘𝐺) ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ≤ 𝑊))
46	41, 43, 45	mpbi2and 709	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑅‘𝐹) ∨ (𝑅‘𝐺)) ≤ 𝑊)
47	1, 2, 4, 28, 35, 38, 39, 46	lattrd 18411	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 lecple 17213 joincjn 18276 meetcmee 18277 Latclat 18396 Atomscatm 38646 HLchlt 38733 LHypclh 39368 LTrncltrn 39485 trLctrl 39542

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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-riotaBAD 38336

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-undef 8259 df-map 8824 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543

This theorem is referenced by: cdlemg12f 40032

W3C validator