Theorem relexpaddnn 15204

Description: Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.)

Assertion

Ref	Expression
relexpaddnn	⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))

Proof of Theorem relexpaddnn

Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7428	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
2	1	coeq1d 5838	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)))
3		oveq1 7427	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
4	3	oveq2d 7436	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
5	2, 4	eqeq12d 2776	. . . 4 ⊢ (𝑛 = 1 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀))))
6	5	imbi2d 343	. . 3 ⊢ (𝑛 = 1 → (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀)))))
7		oveq2 7428	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑘))
8	7	coeq1d 5838	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
9		oveq1 7427	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
10	9	oveq2d 7436	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀)))
11	8, 10	eqeq12d 2776	. . . 4 ⊢ (𝑛 = 𝑘 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))))
12	11	imbi2d 343	. . 3 ⊢ (𝑛 = 𝑘 → (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀)))))
13		oveq2 7428	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑘 + 1)))
14	13	coeq1d 5838	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)))
15		oveq1 7427	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
16	15	oveq2d 7436	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))
17	14, 16	eqeq12d 2776	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀))))
18	17	imbi2d 343	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))))
19		oveq2 7428	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
20	19	coeq1d 5838	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)))
21		oveq1 7427	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
22	21	oveq2d 7436	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
23	20, 22	eqeq12d 2776	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
24	23	imbi2d 343	. . 3 ⊢ (𝑛 = 𝑁 → (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀))) ↔ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))))
25		relexp1g 15179	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	adantl 487	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
27	26	coeq1d 5838	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
28		relexpsucnnl 15183	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ ℕ) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
29	28	ancoms 464	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
30		simpl 488	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ)
31	30	nncnd 12351	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
32		1cnd 11302	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
33	31, 32	addcomd 11512	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑀 + 1) = (1 + 𝑀))
34	33	oveq2d 7436	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅↑𝑟(1 + 𝑀)))
35	27, 29, 34	3eqtr2d 2801	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
36		simp2r 1219	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑅 ∈ 𝑉)
37		simp1 1154	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℕ)
38		relexpsucnnl 15183	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑘)))
39	36, 37, 38	syl2anc 596	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑅↑𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑘)))
40	39	coeq1d 5838	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)) = ((𝑅 ∘ (𝑅↑𝑟𝑘)) ∘ (𝑅↑𝑟𝑀)))
41		coass 6267	. . . . . . 7 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑘)) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
42	40, 41	eqtrdi 2811	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀))))
43		simp3 1156	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀)))
44	43	coeq2d 5839	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑅 ∘ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀))) = (𝑅 ∘ (𝑅↑𝑟(𝑘 + 𝑀))))
45	37	nncnd 12351	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46		1cnd 11302	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 1 ∈ ℂ)
47	31	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℂ)
48	45, 46, 47	add32d 11538	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
49	48	oveq2d 7436	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑅↑𝑟((𝑘 + 1) + 𝑀)) = (𝑅↑𝑟((𝑘 + 𝑀) + 1)))
50	30	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℕ)
51	37, 50	nnaddcld 12390	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52		relexpsucnnl 15183	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑘 + 𝑀) ∈ ℕ) → (𝑅↑𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑘 + 𝑀))))
53	36, 51, 52	syl2anc 596	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑅↑𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑘 + 𝑀))))
54	49, 53	eqtr2d 2796	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑅 ∘ (𝑅↑𝑟(𝑘 + 𝑀))) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))
55	42, 44, 54	3eqtrd 2799	. . . . 5 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))
56	55	3exp 1137	. . . 4 ⊢ (𝑘 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀)) → ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))))
57	56	a2d 30	. . 3 ⊢ (𝑘 ∈ ℕ → (((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))))
58	6, 12, 18, 24, 35, 57	nnind 12353	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
59	58	3impib 1134	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ∘ ccom 5654  (class class class)co 7420  ℂcc 11198  1c1 11201   + caddc 11203  ℕcn 12335  ↑_𝑟crelexp 15172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-seq 14145  df-relexp 15173

This theorem is used by:  relexpaddg  15206  iunrelexpmin1  44708  relexpmulnn  44709  iunrelexpmin2  44712  relexpaddss  44718