Theorem relexpaddnn 14402

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 7143	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
2	1	coeq1d 5696	. . . . 5 ⊢ (𝑛 = 1 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)))
3		oveq1 7142	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛 + 𝑀) = (1 + 𝑀))
4	3	oveq2d 7151	. . . . 5 ⊢ (𝑛 = 1 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
5	2, 4	eqeq12d 2814	. . . 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 7143	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑘))
8	7	coeq1d 5696	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
9		oveq1 7142	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 + 𝑀) = (𝑘 + 𝑀))
10	9	oveq2d 7151	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀)))
11	8, 10	eqeq12d 2814	. . . 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 7143	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑘 + 1)))
14	13	coeq1d 5696	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟(𝑘 + 1)) ∘ (𝑅↑𝑟𝑀)))
15		oveq1 7142	. . . . . 6 ⊢ (𝑛 = (𝑘 + 1) → (𝑛 + 𝑀) = ((𝑘 + 1) + 𝑀))
16	15	oveq2d 7151	. . . . 5 ⊢ (𝑛 = (𝑘 + 1) → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟((𝑘 + 1) + 𝑀)))
17	14, 16	eqeq12d 2814	. . . 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 7143	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟𝑁))
20	19	coeq1d 5696	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑅↑𝑟𝑛) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)))
21		oveq1 7142	. . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛 + 𝑀) = (𝑁 + 𝑀))
22	21	oveq2d 7151	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑅↑𝑟(𝑛 + 𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
23	20, 22	eqeq12d 2814	. . . 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 14377	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
26	25	adantl 485	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
27	26	coeq1d 5696	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ (𝑅↑𝑟𝑀)))
28		relexpsucnnl 14381	. . . . 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 11641	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
32		1cnd 10625	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
33	31, 32	addcomd 10831	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑀 + 1) = (1 + 𝑀))
34	33	oveq2d 7151	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑀 + 1)) = (𝑅↑𝑟(1 + 𝑀)))
35	27, 29, 34	3eqtr2d 2839	. . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟1) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(1 + 𝑀)))
36		simp2r 1197	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑅 ∈ 𝑉)
37		simp1 1133	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℕ)
38		relexpsucnnl 14381	. . . . . . . . 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 5696	. . . . . . 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 6085	. . . . . . 7 ⊢ ((𝑅 ∘ (𝑅↑𝑟𝑘)) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)))
42	40, 41	eqtrdi 2849	. . . . . 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 1135	. . . . . . 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 5697	. . . . . 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 11641	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑘 ∈ ℂ)
46		1cnd 10625	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 1 ∈ ℂ)
47	31	3ad2ant2 1131	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℂ)
48	45, 46, 47	add32d 10856	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → ((𝑘 + 1) + 𝑀) = ((𝑘 + 𝑀) + 1))
49	48	oveq2d 7151	. . . . . . 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 1131	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → 𝑀 ∈ ℕ)
51	37, 50	nnaddcld 11677	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) ∧ ((𝑅↑𝑟𝑘) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑘 + 𝑀))) → (𝑘 + 𝑀) ∈ ℕ)
52		relexpsucnnl 14381	. . . . . . . 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 2834	. . . . . 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 2837	. . . . 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 1116	. . . 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 11643	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
59	58	3impib 1113	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∘ ccom 5523  (class class class)co 7135  ℂcc 10524  1c1 10527   + caddc 10529  ℕcn 11625  ↑_𝑟crelexp 14370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-relexp 14371

This theorem is referenced by:  relexpaddg  14404  iunrelexpmin1  40409  relexpmulnn  40410  iunrelexpmin2  40413  relexpaddss  40419