Theorem relexpaddnn 14500

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 7178	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
2	1	coeq1d 5704	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)))
3		oveq1 7177	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
4	3	oveq2d 7186	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
5	2, 4	eqeq12d 2754	. . . 4 ⊢ (𝑛 = 1 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀))))
6	5	imbi2d 344	. . 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 7178	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑘))
8	7	coeq1d 5704	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
9		oveq1 7177	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
10	9	oveq2d 7186	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀)))
11	8, 10	eqeq12d 2754	. . . 4 ⊢ (𝑛 = 𝑘 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))))
12	11	imbi2d 344	. . 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 7178	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑘 + 1)))
14	13	coeq1d 5704	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)))
15		oveq1 7177	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
16	15	oveq2d 7186	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))
17	14, 16	eqeq12d 2754	. . . 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 344	. . 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 7178	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
20	19	coeq1d 5704	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)))
21		oveq1 7177	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
22	21	oveq2d 7186	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
23	20, 22	eqeq12d 2754	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑛 + 𝑀)) ↔ ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
24	23	imbi2d 344	. . 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 14475	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	adantl 485	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
27	26	coeq1d 5704	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
28		relexpsucnnl 14479	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ ℕ) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
29	28	ancoms 462	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
30		simpl 486	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ)
31	30	nncnd 11732	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
32		1cnd 10714	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
33	31, 32	addcomd 10920	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑀 + 1) = (1 + 𝑀))
34	33	oveq2d 7186	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅↑𝑟(1 + 𝑀)))
35	27, 29, 34	3eqtr2d 2779	. . 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 14479	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑘 ∈ ℕ) → (𝑅↑𝑟(𝑘 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑘)))
39	36, 37, 38	syl2anc 587	. . . . . . . 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 5704	. . . . . . 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 6098	. . . . . . 7 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑘)) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
42	40, 41	eqtrdi 2789	. . . . . 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 5705	. . . . . 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 11732	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46		1cnd 10714	. . . . . . . . 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 10945	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
49	48	oveq2d 7186	. . . . . . 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 11768	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52		relexpsucnnl 14479	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑘 + 𝑀) ∈ ℕ) → (𝑅↑𝑟((𝑘 + 𝑀) + 1)) = (𝑅 ∘ (𝑅↑𝑟(𝑘 + 𝑀))))
53	36, 51, 52	syl2anc 587	. . . . . . 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 2774	. . . . . 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 2777	. . . . 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 11734	. 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 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ∘ ccom 5529  (class class class)co 7170  ℂcc 10613  1c1 10616   + caddc 10618  ℕcn 11716  ↑_𝑟crelexp 14468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-seq 13461  df-relexp 14469

This theorem is referenced by:  relexpaddg  14502  iunrelexpmin1  40862  relexpmulnn  40863  iunrelexpmin2  40866  relexpaddss  40872