Theorem trlval4 37764

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 1133	. 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 1203	. 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 1204	. 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 1205	. 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 1199	. . 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 1221	. . . . . . . 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 1291	. . . . . . . . 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 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))) → 𝑄 ∈ 𝐴)
9		simpl1 1188	. . . . . . . . 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 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))) → 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 37716	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴)
16	9, 10, 8, 15	syl3anc 1368	. . . . . . . 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 36951	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ (𝐹‘𝑄)))
19	6, 8, 16, 18	syl3anc 1368	. . . . . . 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 1249	. . . . . . . . 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 37742	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
23	9, 10, 20, 22	syl3anc 1368	. . . . . . . 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 488	. . . . . . . 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 2793	. . . . . . 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 5060	. . . . . 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 1226	. . . . . . . . 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 1209	. . . . . . . . . . . . 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 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) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
30	10	adantr 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 ∈ 𝑇)
31		simpr 488	. . . . . . . . . . . . 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 2758	. . . . . . . . . . . . . 14 ⊢ (0.‘𝐾) = (0.‘𝐾)
33	11, 32, 12, 13, 14, 21	trl0 37746	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 36936	. . . . . . . . . . . . . . 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 1289	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 2758	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
40	39, 17, 12	hlatjcl 36943	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
41	6, 38, 8, 40	syl3anc 1368	. . . . . . . . . . . . . 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 36880	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
43	36, 41, 42	syl2anc 587	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 5054	. . . . . . . . . . 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 416	. . . . . . . . . 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 2965	. . . . . . . . 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 37745	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
50	9, 20, 10, 48, 49	syl112anc 1371	. . . . . . 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 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))) → 𝑃 ≠ 𝑄)
52	51	necomd 3006	. . . . . . 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 36971	. . . . . . 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 1380	. . . . . 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 416	. . . 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 2965	. . 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 37763	. 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 1379	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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   class class class wbr 5032  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  lecple 16630  joincjn 17620  meetcmee 17621  0.cp0 17713  Atomscatm 36839  AtLatcal 36840  HLchlt 36926  LHypclh 37560  LTrncltrn 37677  trLctrl 37734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418  df-proset 17604  df-poset 17622  df-plt 17634  df-lub 17650  df-glb 17651  df-join 17652  df-meet 17653  df-p0 17715  df-p1 17716  df-lat 17722  df-clat 17784  df-oposet 36752  df-ol 36754  df-oml 36755  df-covers 36842  df-ats 36843  df-atl 36874  df-cvlat 36898  df-hlat 36927  df-llines 37074  df-psubsp 37079  df-pmap 37080  df-padd 37372  df-lhyp 37564  df-laut 37565  df-ldil 37680  df-ltrn 37681  df-trl 37735

This theorem is referenced by:  cdlemg10a  38216  cdlemg12d  38222