Theorem trlval4 40189

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 1136	. 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 1207	. 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 1208	. 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 1209	. 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 1203	. . 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 1225	. . . . . . . 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 1295	. . . . . . . . 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 480	. . . . . . . 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 1192	. . . . . . . . 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 1252	. . . . . . . . 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 40141	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴)
16	9, 10, 8, 15	syl3anc 1373	. . . . . . . 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 39375	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ (𝐹‘𝑄)))
19	6, 8, 16, 18	syl3anc 1373	. . . . . . 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 1253	. . . . . . . . 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 40167	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
23	9, 10, 20, 22	syl3anc 1373	. . . . . . . 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 484	. . . . . . . 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 2765	. . . . . . 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 5138	. . . . . 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 1230	. . . . . . . . 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 1213	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 14 ⊢ (0.‘𝐾) = (0.‘𝐾)
33	11, 32, 12, 13, 14, 21	trl0 40171	. . . . . . . . . . . . 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 1376	. . . . . . . . . . . 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 39360	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
36	6, 35	syl 17	. . . . . . . . . . . . . 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 1293	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
40	39, 17, 12	hlatjcl 39367	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
41	6, 38, 8, 40	syl3anc 1373	. . . . . . . . . . . . . 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 39304	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
43	36, 41, 42	syl2anc 584	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 5132	. . . . . . . . . . 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 412	. . . . . . . . . 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 2940	. . . . . . . . 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 15	. . . . . . . 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 40170	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
50	9, 20, 10, 48, 49	syl112anc 1376	. . . . . . 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 1229	. . . . . . . 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 2981	. . . . . . 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 39396	. . . . . . 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 1385	. . . . . 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 15	. . . . 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 412	. . . 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 2940	. . 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 15	. 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 40188	. 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 1384	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))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  meetcmee 18280  0.cp0 18389  Atomscatm 39263  AtLatcal 39264  HLchlt 39350  LHypclh 39985  LTrncltrn 40102  trLctrl 40159

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-p1 18392  df-lat 18398  df-clat 18465  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-llines 39499  df-psubsp 39504  df-pmap 39505  df-padd 39797  df-lhyp 39989  df-laut 39990  df-ldil 40105  df-ltrn 40106  df-trl 40160

This theorem is referenced by:  cdlemg10a  40641  cdlemg12d  40647