Theorem trlval4 41437

Description: The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)

Hypotheses

Ref	Expression
trlval3.l	⊢ ≤ = (le‘𝐾)
trlval3.j	⊢ ∨ = (join‘𝐾)
trlval3.m	⊢ ∧ = (meet‘𝐾)
trlval3.a	⊢ 𝐴 = (Atoms‘𝐾)
trlval3.h	⊢ 𝐻 = (LHyp‘𝐾)
trlval3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlval3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
trlval4	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))))

Proof of Theorem trlval4

Step	Hyp	Ref	Expression
1		simp1 1154	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp21 1225	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → 𝐹 ∈ 𝑇)
3		simp22 1226	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		simp23 1227	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
5		simp3r 1221	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))
6		simpl1l 1243	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝐾 ∈ HL)
7		simp23l 1313	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
8	7	adantr 486	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝑄 ∈ 𝐴)
9		simpl1 1210	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		simpl21 1270	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝐹 ∈ 𝑇)
11		trlval3.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
12		trlval3.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
13		trlval3.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
14		trlval3.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
15	11, 12, 13, 14	ltrnat 41389	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴)
16	9, 10, 8, 15	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝐹‘𝑄) ∈ 𝐴)
17		trlval3.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
18	11, 17, 12	hlatlej1 40624	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ (𝐹‘𝑄)))
19	6, 8, 16, 18	syl3anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝑄 ≤ (𝑄 ∨ (𝐹‘𝑄)))
20		simpl22 1271	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
21		trlval3.r	. . . . . . . . . 10 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
22	11, 17, 12, 13, 14, 21	trljat1 41415	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
23	9, 10, 20, 22	syl3anc 1399	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
24		simpr 490	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄)))
25	23, 24	eqtrd 2798	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑄 ∨ (𝐹‘𝑄)))
26	19, 25	breqtrrd 5136	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)))
27		simpl3r 1248	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))
28		simpll1 1231	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29	20	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
30	10	adantr 486	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) ∧ (𝐹‘𝑃) = 𝑃) → 𝐹 ∈ 𝑇)
31		simpr 490	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐹‘𝑃) = 𝑃)
32		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (0.‘𝐾) = (0.‘𝐾)
33	11, 32, 12, 13, 14, 21	trl0 41419	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = (0.‘𝐾))
34	28, 29, 30, 31, 33	syl112anc 1402	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑅‘𝐹) = (0.‘𝐾))
35		hlatl 40609	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
36	6, 35	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝐾 ∈ AtLat)
37		simp22l 1311	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
38	37	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝑃 ∈ 𝐴)
39		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
40	39, 17, 12	hlatjcl 40616	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
41	6, 38, 8, 40	syl3anc 1399	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
42	39, 11, 32	atl0le 40553	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
43	36, 41, 42	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
44	43	adantr 486	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) ∧ (𝐹‘𝑃) = 𝑃) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
45	34, 44	eqbrtrd 5130	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))
46	45	ex 418	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → ((𝐹‘𝑃) = 𝑃 → (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄)))
47	46	necon3bd 2973	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) → (𝐹‘𝑃) ≠ 𝑃))
48	27, 47	mpd 16	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝐹‘𝑃) ≠ 𝑃)
49	11, 12, 13, 14, 21	trlat 41418	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
50	9, 20, 10, 48, 49	syl112anc 1402	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝑅‘𝐹) ∈ 𝐴)
51		simpl3l 1247	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝑃 ≠ 𝑄)
52	51	necomd 3014	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → 𝑄 ≠ 𝑃)
53	11, 17, 12	hlatexch1 40644	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑄 ≠ 𝑃) → (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) → (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄)))
54	6, 8, 50, 38, 52, 53	syl131anc 1411	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝑄 ≤ (𝑃 ∨ (𝑅‘𝐹)) → (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄)))
55	26, 54	mpd 16	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) ∧ (𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄))) → (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))
56	55	ex 418	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘𝑃)) = (𝑄 ∨ (𝐹‘𝑄)) → (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄)))
57	56	necon3bd 2973	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → (¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) → (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄))))
58	5, 57	mpd 16	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))
59		trlval3.m	. . 3 ⊢ ∧ = (meet‘𝐾)
60	11, 17, 59, 12, 13, 14, 21	trlval3 41436	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))))
61	1, 2, 3, 4, 58, 60	syl113anc 1410	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄))) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  lecple 17464  joincjn 18514  meetcmee 18515  0.cp0 18624  Atomscatm 40512  AtLatcal 40513  HLchlt 40599  LHypclh 41233  LTrncltrn 41350  trLctrl 41407

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877  df-proset 18497  df-poset 18516  df-plt 18531  df-lub 18547  df-glb 18548  df-join 18549  df-meet 18550  df-p0 18626  df-p1 18627  df-lat 18635  df-clat 18702  df-oposet 40425  df-ol 40427  df-oml 40428  df-covers 40515  df-ats 40516  df-atl 40547  df-cvlat 40571  df-hlat 40600  df-llines 40747  df-psubsp 40752  df-pmap 40753  df-padd 41045  df-lhyp 41237  df-laut 41238  df-ldil 41353  df-ltrn 41354  df-trl 41408

This theorem is used by:  cdlemg10a  41889  cdlemg12d  41895