trlval4 - Mathbox for Norm Megill

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Theorem trlval4 39571

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 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 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 39523	. . . . . . . . 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 38757	. . . . . . . 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 39549	. . . . . . . . 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 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 2766	. . . . . . 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 5169	. . . . . 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 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 2726	. . . . . . . . . . . . . 14 ⊢ (0.‘𝐾) = (0.‘𝐾)
33	11, 32, 12, 13, 14, 21	trl0 39553	. . . . . . . . . . . . 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 38742	. . . . . . . . . . . . . . 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 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 2726	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
40	39, 17, 12	hlatjcl 38749	. . . . . . . . . . . . . . 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 38686	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (0.‘𝐾) ≤ (𝑃 ∨ 𝑄))
43	36, 41, 42	syl2anc 583	. . . . . . . . . . . . 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 5163	. . . . . . . . . . 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 2948	. . . . . . . . 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 39552	. . . . . . . 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 2990	. . . . . . 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 38778	. . . . . . 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 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 2948	. . 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 39570	. 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))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 class class class wbr 5141 ‘cfv 6536 (class class class)co 7404 Basecbs 17150 lecple 17210 joincjn 18273 meetcmee 18274 0.cp0 18385 Atomscatm 38645 AtLatcal 38646 HLchlt 38732 LHypclh 39367 LTrncltrn 39484 trLctrl 39541

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 7721

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 df-psubsp 38886 df-pmap 38887 df-padd 39179 df-lhyp 39371 df-laut 39372 df-ldil 39487 df-ltrn 39488 df-trl 39542

This theorem is referenced by: cdlemg10a 40023 cdlemg12d 40029

W3C validator