Theorem trlval2 38177

Description: The value of the trace of a lattice translation, given any atom 𝑃 not under the fiducial co-atom 𝑊. Note: this requires only the weaker assumption 𝐾 ∈ Lat; we use 𝐾 ∈ HL for convenience. (Contributed by NM, 20-May-2012.)

Hypotheses

Ref	Expression
trlval2.l	⊢ ≤ = (le‘𝐾)
trlval2.j	⊢ ∨ = (join‘𝐾)
trlval2.m	⊢ ∧ = (meet‘𝐾)
trlval2.a	⊢ 𝐴 = (Atoms‘𝐾)
trlval2.h	⊢ 𝐻 = (LHyp‘𝐾)
trlval2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlval2.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
trlval2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))

Proof of Theorem trlval2

Dummy variables 𝑥 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hllat 37377	. . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	anim1i 615	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻))
3		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		trlval2.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5		trlval2.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
6		trlval2.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
7		trlval2.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8		trlval2.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
9		trlval2.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10		trlval2.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
11	3, 4, 5, 6, 7, 8, 9, 10	trlval 38176	. . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = (℩𝑥 ∈ (Base‘𝐾)∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
12	11	3adant3 1131	. . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = (℩𝑥 ∈ (Base‘𝐾)∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
13		simp1l 1196	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat)
14		simp3l 1200	. . . . . . 7 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
15	3, 7	atbase 37303	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
16	14, 15	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ (Base‘𝐾))
17	3, 8, 9	ltrncl 38139	. . . . . . 7 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
18	16, 17	syld3an3 1408	. . . . . 6 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
19	3, 5	latjcl 18157	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
20	13, 16, 18, 19	syl3anc 1370	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
21		simp1r 1197	. . . . . 6 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
22	3, 8	lhpbase 38012	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
23	21, 22	syl 17	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ (Base‘𝐾))
24	3, 6	latmcl 18158	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
25	13, 20, 23, 24	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
26		simpl3l 1227	. . . . . 6 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑃 ∈ 𝐴)
27		simpl3r 1228	. . . . . 6 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → ¬ 𝑃 ≤ 𝑊)
28		breq1 5077	. . . . . . . . . 10 ⊢ (𝑞 = 𝑃 → (𝑞 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊))
29	28	notbid 318	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊))
30		id 22	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑃 → 𝑞 = 𝑃)
31		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑃 → (𝐹‘𝑞) = (𝐹‘𝑃))
32	30, 31	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑃 → (𝑞 ∨ (𝐹‘𝑞)) = (𝑃 ∨ (𝐹‘𝑃)))
33	32	oveq1d 7290	. . . . . . . . . 10 ⊢ (𝑞 = 𝑃 → ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
34	33	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) ↔ 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
35	29, 34	imbi12d 345	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ (¬ 𝑃 ≤ 𝑊 → 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))))
36	35	rspcv 3557	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → (∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) → (¬ 𝑃 ≤ 𝑊 → 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))))
37	36	com23 86	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → (¬ 𝑃 ≤ 𝑊 → (∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) → 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))))
38	26, 27, 37	sylc 65	. . . . 5 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) → 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
39		simp11 1202	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → (𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻))
40		simp12 1203	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝐹 ∈ 𝑇)
41		simp13l 1287	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝑃 ∈ 𝐴)
42		simp13r 1288	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → ¬ 𝑃 ≤ 𝑊)
43		simp3 1137	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
44		simp2 1136	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → ¬ 𝑞 ≤ 𝑊)
45	4, 5, 6, 7, 8, 9	ltrnu 38135	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))
46	39, 40, 41, 42, 43, 44, 45	syl222anc 1385	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))
47		eqeq2 2750	. . . . . . . . . . 11 ⊢ (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ↔ 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
48	47	biimpd 228	. . . . . . . . . 10 ⊢ (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
49	46, 48	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
50	49	3exp 1118	. . . . . . . 8 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (¬ 𝑞 ≤ 𝑊 → (𝑞 ∈ 𝐴 → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))))
51	50	com24 95	. . . . . . 7 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) → (𝑞 ∈ 𝐴 → (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))))
52	51	ralrimdv 3105	. . . . . 6 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) → ∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
53	52	adantr 481	. . . . 5 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) → ∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
54	38, 53	impbid 211	. . . 4 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
55	25, 54	riota5 7262	. . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (℩𝑥 ∈ (Base‘𝐾)∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
56	12, 55	eqtrd 2778	. 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
57	2, 56	syl3an1 1162	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   class class class wbr 5074  ‘cfv 6433  ℩crio 7231  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029  meetcmee 18030  Latclat 18149  Atomscatm 37277  HLchlt 37364  LHypclh 37998  LTrncltrn 38115  trLctrl 38172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-lat 18150  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-lhyp 38002  df-laut 38003  df-ldil 38118  df-ltrn 38119  df-trl 38173

This theorem is referenced by:  trlcl  38178  trlcnv  38179  trljat1  38180  trljat2  38181  trlat  38183  trl0  38184  trlle  38198  trlval3  38201  trlval5  38203  cdlemd6  38217  cdlemf  38577  cdlemg4a  38622  cdlemg4b1  38623  cdlemg4b2  38624  cdlemg4  38631  cdlemg11b  38656  cdlemg13a  38665  cdlemg13  38666  cdlemg17a  38675  cdlemg17dN  38677  cdlemg17e  38679  cdlemg17f  38680  trlcoabs2N  38736  trlcolem  38740  cdlemg42  38743  cdlemg43  38744  cdlemi1  38832  cdlemk4  38848  cdlemk39  38930  dia2dimlem1  39078  dia2dimlem2  39079  dia2dimlem3  39080  cdlemm10N  39132  cdlemn2  39209  cdlemn10  39220  dihjatcclem3  39434