Theorem relexpaddnn 15090

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 7439	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
2	1	coeq1d 5872	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)))
3		oveq1 7438	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
4	3	oveq2d 7447	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
5	2, 4	eqeq12d 2753	. . . 4 ⊢ (𝑛 = 1 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀))))
6	5	imbi2d 340	. . 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 7439	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑘))
8	7	coeq1d 5872	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
9		oveq1 7438	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
10	9	oveq2d 7447	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀)))
11	8, 10	eqeq12d 2753	. . . 4 ⊢ (𝑛 = 𝑘 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))))
12	11	imbi2d 340	. . 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 7439	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑘 + 1)))
14	13	coeq1d 5872	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)))
15		oveq1 7438	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
16	15	oveq2d 7447	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))
17	14, 16	eqeq12d 2753	. . . 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 340	. . 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 7439	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
20	19	coeq1d 5872	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)))
21		oveq1 7438	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
22	21	oveq2d 7447	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
23	20, 22	eqeq12d 2753	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
24	23	imbi2d 340	. . 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 15065	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
27	26	coeq1d 5872	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
28		relexpsucnnl 15069	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ ℕ) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
29	28	ancoms 458	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
30		simpl 482	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ)
31	30	nncnd 12282	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
32		1cnd 11256	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
33	31, 32	addcomd 11463	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑀 + 1) = (1 + 𝑀))
34	33	oveq2d 7447	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅↑𝑟(1 + 𝑀)))
35	27, 29, 34	3eqtr2d 2783	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
36		simp2r 1201	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑅 ∈ 𝑉)
37		simp1 1137	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℕ)
38		relexpsucnnl 15069	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑘)))
39	36, 37, 38	syl2anc 584	. . . . . . . 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 5872	. . . . . . 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 6285	. . . . . . 7 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑘)) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
42	40, 41	eqtrdi 2793	. . . . . 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 1139	. . . . . . 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 5873	. . . . . 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 12282	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46		1cnd 11256	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 1 ∈ ℂ)
47	31	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℂ)
48	45, 46, 47	add32d 11489	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
49	48	oveq2d 7447	. . . . . . 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 1135	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℕ)
51	37, 50	nnaddcld 12318	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52		relexpsucnnl 15069	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑘 + 𝑀) ∈ ℕ) → (𝑅↑𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑘 + 𝑀))))
53	36, 51, 52	syl2anc 584	. . . . . . 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 2778	. . . . . 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 2781	. . . . 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 1120	. . . 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 29	. . 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 12284	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
59	58	3impib 1117	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ∘ ccom 5689  (class class class)co 7431  ℂcc 11153  1c1 11156   + caddc 11158  ℕcn 12266  ↑_𝑟crelexp 15058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-seq 14043  df-relexp 15059

This theorem is referenced by:  relexpaddg  15092  iunrelexpmin1  43721  relexpmulnn  43722  iunrelexpmin2  43725  relexpaddss  43731