Theorem trlval2 40282

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 39482	. . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	anim1i 615	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻))
3		eqid 2733	. . . . 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 40281	. . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) = (℩𝑥 ∈ (Base‘𝐾)∀𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊))))
12	11	3adant3 1132	. . 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 1198	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐾 ∈ Lat)
14		simp3l 1202	. . . . . . 7 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑃 ∈ 𝐴)
15	3, 7	atbase 39408	. . . . . . 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 40244	. . . . . . 7 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
18	16, 17	syld3an3 1411	. . . . . 6 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
19	3, 5	latjcl 18347	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘𝑃) ∈ (Base‘𝐾)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
20	13, 16, 18, 19	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
21		simp1r 1199	. . . . . 6 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
22	3, 8	lhpbase 40117	. . . . . 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 18348	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
25	13, 20, 23, 24	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∈ (Base‘𝐾))
26		simpl3l 1229	. . . . . 6 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑃 ∈ 𝐴)
27		simpl3r 1230	. . . . . 6 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘𝐾)) → ¬ 𝑃 ≤ 𝑊)
28		breq1 5096	. . . . . . . . . 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 6828	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑃 → (𝐹‘𝑞) = (𝐹‘𝑃))
32	30, 31	oveq12d 7370	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑃 → (𝑞 ∨ (𝐹‘𝑞)) = (𝑃 ∨ (𝐹‘𝑃)))
33	32	oveq1d 7367	. . . . . . . . . 10 ⊢ (𝑞 = 𝑃 → ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
34	33	eqeq2d 2744	. . . . . . . . 9 ⊢ (𝑞 = 𝑃 → (𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) ↔ 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
35	29, 34	imbi12d 344	. . . . . . . 8 ⊢ (𝑞 = 𝑃 → ((¬ 𝑞 ≤ 𝑊 → 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)) ↔ (¬ 𝑃 ≤ 𝑊 → 𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))))
36	35	rspcv 3569	. . . . . . 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 1204	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → (𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻))
40		simp12 1205	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝐹 ∈ 𝑇)
41		simp13l 1289	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝑃 ∈ 𝐴)
42		simp13r 1290	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → ¬ 𝑃 ≤ 𝑊)
43		simp3 1138	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
44		simp2 1137	. . . . . . . . . . 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 40240	. . . . . . . . . . 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 1388	. . . . . . . . . 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 2745	. . . . . . . . . . 11 ⊢ (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊) → (𝑥 = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ↔ 𝑥 = ((𝑞 ∨ (𝐹‘𝑞)) ∧ 𝑊)))
48	47	biimpd 229	. . . . . . . . . 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 1119	. . . . . . . 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 3131	. . . . . 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 480	. . . . 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 212	. . . 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 7338	. . 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 2768	. 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
57	2, 56	syl3an1 1163	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048   class class class wbr 5093  ‘cfv 6486  ℩crio 7308  (class class class)co 7352  Basecbs 17122  lecple 17170  joincjn 18219  meetcmee 18220  Latclat 18339  Atomscatm 39382  HLchlt 39469  LHypclh 40103  LTrncltrn 40220  trLctrl 40277

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8758  df-lub 18252  df-glb 18253  df-join 18254  df-meet 18255  df-lat 18340  df-ats 39386  df-atl 39417  df-cvlat 39441  df-hlat 39470  df-lhyp 40107  df-laut 40108  df-ldil 40223  df-ltrn 40224  df-trl 40278

This theorem is referenced by:  trlcl  40283  trlcnv  40284  trljat1  40285  trljat2  40286  trlat  40288  trl0  40289  trlle  40303  trlval3  40306  trlval5  40308  cdlemd6  40322  cdlemf  40682  cdlemg4a  40727  cdlemg4b1  40728  cdlemg4b2  40729  cdlemg4  40736  cdlemg11b  40761  cdlemg13a  40770  cdlemg13  40771  cdlemg17a  40780  cdlemg17dN  40782  cdlemg17e  40784  cdlemg17f  40785  trlcoabs2N  40841  trlcolem  40845  cdlemg42  40848  cdlemg43  40849  cdlemi1  40937  cdlemk4  40953  cdlemk39  41035  dia2dimlem1  41183  dia2dimlem2  41184  dia2dimlem3  41185  cdlemm10N  41237  cdlemn2  41314  cdlemn10  41325  dihjatcclem3  41539