trlval3 - Mathbox for Norm Megill

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Theorem trlval3 38128

Description: The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (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
trlval3	⊢ (((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))))

**Proof of Theorem trlval3**
Step	Hyp	Ref	Expression
1		simpl1 1189	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl31 1252	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simpl2 1190	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → 𝐹 ∈ 𝑇)
4		simpr 484	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐹‘𝑃) = 𝑃)
5		trlval3.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
6		eqid 2738	. . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾)
7		trlval3.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8		trlval3.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
9		trlval3.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10		trlval3.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
11	5, 6, 7, 8, 9, 10	trl0 38111	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑃)) → (𝑅‘𝐹) = (0.‘𝐾))
12	1, 2, 3, 4, 11	syl112anc 1372	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑅‘𝐹) = (0.‘𝐾))
13		simpl33 1254	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))
14		simpl1l 1222	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → 𝐾 ∈ HL)
15		hlatl 37301	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
16	14, 15	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → 𝐾 ∈ AtLat)
17	4	oveq2d 7271	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑃 ∨ (𝐹‘𝑃)) = (𝑃 ∨ 𝑃))
18		simp31l 1294	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → 𝑃 ∈ 𝐴)
19	18	adantr 480	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → 𝑃 ∈ 𝐴)
20		trlval3.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
21	20, 7	hlatjidm 37310	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑃 ∨ 𝑃) = 𝑃)
22	14, 19, 21	syl2anc 583	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑃 ∨ 𝑃) = 𝑃)
23	17, 22	eqtrd 2778	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑃 ∨ (𝐹‘𝑃)) = 𝑃)
24	23, 19	eqeltrd 2839	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑃 ∨ (𝐹‘𝑃)) ∈ 𝐴)
25		simp1 1134	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26		simp2 1135	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → 𝐹 ∈ 𝑇)
27		simp31 1207	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
28		simp32 1208	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
29	5, 7, 8, 9	ltrn2ateq 38121	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) = 𝑃 ↔ (𝐹‘𝑄) = 𝑄))
30	25, 26, 27, 28, 29	syl13anc 1370	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → ((𝐹‘𝑃) = 𝑃 ↔ (𝐹‘𝑄) = 𝑄))
31	30	biimpa 476	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐹‘𝑄) = 𝑄)
32	31	oveq2d 7271	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑄 ∨ (𝐹‘𝑄)) = (𝑄 ∨ 𝑄))
33		simp32l 1296	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → 𝑄 ∈ 𝐴)
34	33	adantr 480	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → 𝑄 ∈ 𝐴)
35	20, 7	hlatjidm 37310	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
36	14, 34, 35	syl2anc 583	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑄 ∨ 𝑄) = 𝑄)
37	32, 36	eqtrd 2778	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑄 ∨ (𝐹‘𝑄)) = 𝑄)
38	37, 34	eqeltrd 2839	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑄 ∨ (𝐹‘𝑄)) ∈ 𝐴)
39		trlval3.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
40	39, 6, 7	atnem0 37259	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ 𝐴 ∧ (𝑄 ∨ (𝐹‘𝑄)) ∈ 𝐴) → ((𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)) ↔ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) = (0.‘𝐾)))
41	16, 24, 38, 40	syl3anc 1369	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → ((𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)) ↔ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) = (0.‘𝐾)))
42	13, 41	mpbid 231	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) = (0.‘𝐾))
43	12, 42	eqtr4d 2781	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) = 𝑃) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))))
44		simpl1 1189	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45		simpl2 1190	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝐹 ∈ 𝑇)
46		simpl31 1252	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
47	5, 20, 39, 7, 8, 9, 10	trlval2 38104	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
48	44, 45, 46, 47	syl3anc 1369	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
49		simpl1l 1222	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝐾 ∈ HL)
50	49	hllatd 37305	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝐾 ∈ Lat)
51	18	adantr 480	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝑃 ∈ 𝐴)
52	5, 7, 8, 9	ltrnat 38081	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴)
53	44, 45, 51, 52	syl3anc 1369	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹‘𝑃) ∈ 𝐴)
54		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
55	54, 20, 7	hlatjcl 37308	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
56	49, 51, 53, 55	syl3anc 1369	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
57		simpl1r 1223	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝑊 ∈ 𝐻)
58	54, 8	lhpbase 37939	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
59	57, 58	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝑊 ∈ (Base‘𝐾))
60	54, 5, 39	latmle1 18097	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ≤ (𝑃 ∨ (𝐹‘𝑃)))
61	50, 56, 59, 60	syl3anc 1369	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ≤ (𝑃 ∨ (𝐹‘𝑃)))
62	48, 61	eqbrtrd 5092	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) ≤ (𝑃 ∨ (𝐹‘𝑃)))
63		simpl32 1253	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
64	5, 20, 39, 7, 8, 9, 10	trlval2 38104	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
65	44, 45, 63, 64	syl3anc 1369	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) = ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊))
66	33	adantr 480	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝑄 ∈ 𝐴)
67	5, 7, 8, 9	ltrnat 38081	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴) → (𝐹‘𝑄) ∈ 𝐴)
68	44, 45, 66, 67	syl3anc 1369	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹‘𝑄) ∈ 𝐴)
69	54, 20, 7	hlatjcl 37308	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴) → (𝑄 ∨ (𝐹‘𝑄)) ∈ (Base‘𝐾))
70	49, 66, 68, 69	syl3anc 1369	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑄 ∨ (𝐹‘𝑄)) ∈ (Base‘𝐾))
71	54, 5, 39	latmle1 18097	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ (𝐹‘𝑄)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊) ≤ (𝑄 ∨ (𝐹‘𝑄)))
72	50, 70, 59, 71	syl3anc 1369	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → ((𝑄 ∨ (𝐹‘𝑄)) ∧ 𝑊) ≤ (𝑄 ∨ (𝐹‘𝑄)))
73	65, 72	eqbrtrd 5092	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) ≤ (𝑄 ∨ (𝐹‘𝑄)))
74	54, 8, 9, 10	trlcl 38105	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ (Base‘𝐾))
75	44, 45, 74	syl2anc 583	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) ∈ (Base‘𝐾))
76	54, 5, 39	latlem12 18099	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝐹) ∈ (Base‘𝐾) ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 ∨ (𝐹‘𝑄)) ∈ (Base‘𝐾))) → (((𝑅‘𝐹) ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑅‘𝐹) ≤ (𝑄 ∨ (𝐹‘𝑄))) ↔ (𝑅‘𝐹) ≤ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄)))))
77	50, 75, 56, 70, 76	syl13anc 1370	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (((𝑅‘𝐹) ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑅‘𝐹) ≤ (𝑄 ∨ (𝐹‘𝑄))) ↔ (𝑅‘𝐹) ≤ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄)))))
78	62, 73, 77	mpbi2and 708	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) ≤ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))))
79	49, 15	syl 17	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → 𝐾 ∈ AtLat)
80		simpr 484	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹‘𝑃) ≠ 𝑃)
81	5, 7, 8, 9, 10	trlat 38110	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
82	44, 46, 45, 80, 81	syl112anc 1372	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) ∈ 𝐴)
83	54, 39	latmcl 18073	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ (𝑄 ∨ (𝐹‘𝑄)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ∈ (Base‘𝐾))
84	50, 56, 70, 83	syl3anc 1369	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ∈ (Base‘𝐾))
85	54, 5, 6, 7	atlen0 37251	. . . . . . 7 ⊢ (((𝐾 ∈ AtLat ∧ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ∈ (Base‘𝐾) ∧ (𝑅‘𝐹) ∈ 𝐴) ∧ (𝑅‘𝐹) ≤ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄)))) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ≠ (0.‘𝐾))
86	79, 84, 82, 78, 85	syl31anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ≠ (0.‘𝐾))
87	86	neneqd 2947	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → ¬ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) = (0.‘𝐾))
88		simpl33 1254	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))
89	20, 39, 6, 7	2atmat0 37467	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) ∧ (𝑄 ∈ 𝐴 ∧ (𝐹‘𝑄) ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) → (((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ∈ 𝐴 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) = (0.‘𝐾)))
90	49, 51, 53, 66, 68, 88, 89	syl33anc 1383	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ∈ 𝐴 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) = (0.‘𝐾)))
91	90	ord 860	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (¬ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ∈ 𝐴 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) = (0.‘𝐾)))
92	87, 91	mt3d 148	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ∈ 𝐴)
93	5, 7	atcmp 37252	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ∈ 𝐴) → ((𝑅‘𝐹) ≤ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ↔ (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄)))))
94	79, 82, 92, 93	syl3anc 1369	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → ((𝑅‘𝐹) ≤ ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))) ↔ (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄)))))
95	78, 94	mpbid 231	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∨ (𝐹‘𝑃)) ≠ (𝑄 ∨ (𝐹‘𝑄)))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑄 ∨ (𝐹‘𝑄))))
96	43, 95	pm2.61dane 3031	1 ⊢ (((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))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 lecple 16895 joincjn 17944 meetcmee 17945 0.cp0 18056 Latclat 18064 Atomscatm 37204 AtLatcal 37205 HLchlt 37291 LHypclh 37925 LTrncltrn 38042 trLctrl 38099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100

This theorem is referenced by: trlval4 38129

W3C validator