Theorem relexpaddnn 13639

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 6557	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
2	1	coeq1d 5205	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)))
3		oveq1 6556	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
4	3	oveq2d 6565	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
5	2, 4	eqeq12d 2625	. . . 4 ⊢ (𝑛 = 1 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀))))
6	5	imbi2d 329	. . 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 6557	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑘))
8	7	coeq1d 5205	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
9		oveq1 6556	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
10	9	oveq2d 6565	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀)))
11	8, 10	eqeq12d 2625	. . . 4 ⊢ (𝑛 = 𝑘 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))))
12	11	imbi2d 329	. . 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 6557	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑘 + 1)))
14	13	coeq1d 5205	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)))
15		oveq1 6556	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
16	15	oveq2d 6565	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))
17	14, 16	eqeq12d 2625	. . . 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 329	. . 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 6557	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
20	19	coeq1d 5205	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)))
21		oveq1 6556	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
22	21	oveq2d 6565	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
23	20, 22	eqeq12d 2625	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
24	23	imbi2d 329	. . 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 13614	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	adantl 481	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
27	26	coeq1d 5205	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
28		relexpsucnnl 13620	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ ℕ) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
29	28	ancoms 468	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
30		simpl 472	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ)
31	30	nncnd 10913	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
32		1cnd 9935	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
33	31, 32	addcomd 10117	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑀 + 1) = (1 + 𝑀))
34	33	oveq2d 6565	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅↑𝑟(1 + 𝑀)))
35	27, 29, 34	3eqtr2d 2650	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
36		simp2r 1081	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑅 ∈ 𝑉)
37		simp1 1054	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℕ)
38		relexpsucnnl 13620	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑘)))
39	36, 37, 38	syl2anc 691	. . . . . . . 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 5205	. . . . . . 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 5571	. . . . . . 7 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑘)) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
42	40, 41	syl6eq 2660	. . . . . 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 1056	. . . . . . 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 5206	. . . . . 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 10913	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46		1cnd 9935	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 1 ∈ ℂ)
47	31	3ad2ant2 1076	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℂ)
48	45, 46, 47	add32d 10142	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
49	48	oveq2d 6565	. . . . . . 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 1076	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℕ)
51	37, 50	nnaddcld 10944	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52		relexpsucnnl 13620	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑘 + 𝑀) ∈ ℕ) → (𝑅↑𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑘 + 𝑀))))
53	36, 51, 52	syl2anc 691	. . . . . . 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 2645	. . . . . 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 2648	. . . . 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 1256	. . . 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 10915	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
59	58	3impib 1254	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∘ ccom 5042  (class class class)co 6549  ℂcc 9813  1c1 9816   + caddc 9818  ℕcn 10897  ↑_𝑟crelexp 13608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-relexp 13609

This theorem is referenced by:  relexpaddg  13641  iunrelexpmin1  37019  relexpmulnn  37020  iunrelexpmin2  37023  relexpaddss  37029