Theorem trlval2 37314

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 36514	. . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	anim1i 616	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻))
3		eqid 2821	. . . . 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 37313	. . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = (℩𝑥 ∈ (Base‘𝐾)∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
12	11	3adant3 1128	. . 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 1193	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat)
14		simp3l 1197	. . . . . . 7 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
15	3, 7	atbase 36440	. . . . . . 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 37276	. . . . . . 7 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
18	16, 17	syld3an3 1405	. . . . . 6 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
19	3, 5	latjcl 17661	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
20	13, 16, 18, 19	syl3anc 1367	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
21		simp1r 1194	. . . . . 6 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
22	3, 8	lhpbase 37149	. . . . . 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 17662	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
25	13, 20, 23, 24	syl3anc 1367	. . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
26		simpl3l 1224	. . . . . 6 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑃 ∈ 𝐴)
27		simpl3r 1225	. . . . . 6 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → ¬ 𝑃 ≤ 𝑊)
28		breq1 5069	. . . . . . . . . 10 ⊢ (𝑞 = 𝑃 → (𝑞 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊))
29	28	notbid 320	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑃 ≤ 𝑊))
30		id 22	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑃 → 𝑞 = 𝑃)
31		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑃 → (𝐹‘𝑞) = (𝐹‘𝑃))
32	30, 31	oveq12d 7174	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑃 → (𝑞 ∨ (𝐹‘𝑞)) = (𝑃 ∨ (𝐹‘𝑃)))
33	32	oveq1d 7171	. . . . . . . . . 10 ⊢ (𝑞 = 𝑃 → ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
34	33	eqeq2d 2832	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) ↔ 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
35	29, 34	imbi12d 347	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ (¬ 𝑃 ≤ 𝑊 → 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))))
36	35	rspcv 3618	. . . . . . 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 1199	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → (𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻))
40		simp12 1200	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝐹 ∈ 𝑇)
41		simp13l 1284	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝑃 ∈ 𝐴)
42		simp13r 1285	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → ¬ 𝑃 ≤ 𝑊)
43		simp3 1134	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
44		simp2 1133	. . . . . . . . . . 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 37272	. . . . . . . . . . 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 1382	. . . . . . . . . 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 2833	. . . . . . . . . . 11 ⊢ (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ↔ 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
48	47	biimpd 231	. . . . . . . . . 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 1115	. . . . . . . 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 3188	. . . . . 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 483	. . . . 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 214	. . . 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 7143	. . 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 2856	. 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
57	2, 56	syl3an1 1159	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138   class class class wbr 5066  ‘cfv 6355  ℩crio 7113  (class class class)co 7156  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  Latclat 17655  Atomscatm 36414  HLchlt 36501  LHypclh 37135  LTrncltrn 37252  trLctrl 37309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-lat 17656  df-ats 36418  df-atl 36449  df-cvlat 36473  df-hlat 36502  df-lhyp 37139  df-laut 37140  df-ldil 37255  df-ltrn 37256  df-trl 37310

This theorem is referenced by:  trlcl  37315  trlcnv  37316  trljat1  37317  trljat2  37318  trlat  37320  trl0  37321  trlle  37335  trlval3  37338  trlval5  37340  cdlemd6  37354  cdlemf  37714  cdlemg4a  37759  cdlemg4b1  37760  cdlemg4b2  37761  cdlemg4  37768  cdlemg11b  37793  cdlemg13a  37802  cdlemg13  37803  cdlemg17a  37812  cdlemg17dN  37814  cdlemg17e  37816  cdlemg17f  37817  trlcoabs2N  37873  trlcolem  37877  cdlemg42  37880  cdlemg43  37881  cdlemi1  37969  cdlemk4  37985  cdlemk39  38067  dia2dimlem1  38215  dia2dimlem2  38216  dia2dimlem3  38217  cdlemm10N  38269  cdlemn2  38346  cdlemn10  38357  dihjatcclem3  38571