Theorem relexpaddg 14997

Description: Relation composition becomes addition under exponentiation except when the exponents total to one and the class isn't a relation. (Contributed by RP, 30-May-2020.)

Assertion

Ref	Expression
relexpaddg	⊢ ((𝑁 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))

Proof of Theorem relexpaddg

Step	Hyp	Ref	Expression
1		elnn0 12471	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		elnn0 12471	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
3		relexpaddnn 14995	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
4	3	a1d 25	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
5	4	3exp 1116	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
6	5	com12 32	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
7		elnn1uz2 12906	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))
8		coires1 6253	. . . . . . . . . . . . . 14 ⊢ ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
9		simpll 764	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 = 1)
10		simplr 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 = 0)
11	9, 10	oveq12d 7419	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (1 + 0))
12		1p0e1 12333	. . . . . . . . . . . . . . . . . 18 ⊢ (1 + 0) = 1
13	11, 12	eqtrdi 2780	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 1)
14		simprr 770	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑁 + 𝑀) = 1 → Rel 𝑅))
15	13, 14	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel 𝑅)
16	9	oveq2d 7417	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1))
17		simprl 768	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅 ∈ 𝑉)
18		relexp1g 14970	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
19	17, 18	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟1) = 𝑅)
20	16, 19	eqtrd 2764	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟𝑁) = 𝑅)
21	20	releqd 5768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (Rel (𝑅↑𝑟𝑁) ↔ Rel 𝑅))
22	15, 21	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel (𝑅↑𝑟𝑁))
23		1nn 12220	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ
24	9, 23	eqeltrdi 2833	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℕ)
25		relexpnndm 14985	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
26	24, 17, 25	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
27		ssun1 4164	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
28	26, 27	sstrdi 3986	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
29		relssres 6012	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑅↑𝑟𝑁) ∧ dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
30	22, 28, 29	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
31	8, 30	eqtrid 2776	. . . . . . . . . . . . 13 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁))
32	10	oveq2d 7417	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
33		relexp0g 14966	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
34	17, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
35	32, 34	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36	35	coeq2d 5852	. . . . . . . . . . . . 13 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
37	10	oveq2d 7417	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (𝑁 + 0))
38		ax-1cn 11164	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
39	9, 38	eqeltrdi 2833	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℂ)
40	39	addridd 11411	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 0) = 𝑁)
41	37, 40	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 𝑁)
42	41	oveq2d 7417	. . . . . . . . . . . . 13 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁))
43	31, 36, 42	3eqtr4d 2774	. . . . . . . . . . . 12 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
44	43	exp43 436	. . . . . . . . . . 11 ⊢ (𝑁 = 1 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
45		simp1 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ (ℤ≥‘2))
46		simp3 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
47		relexpuzrel 14996	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
48	45, 46, 47	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
49		eluz2nn 12865	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ)
50	45, 49	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ)
51	50, 46, 25	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
52	51, 27	sstrdi 3986	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
53	48, 52, 29	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
54	8, 53	eqtrid 2776	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁))
55		simp2 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
56	55	oveq2d 7417	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
57	46, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
58	56, 57	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
59	58	coeq2d 5852	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
60	55	oveq2d 7417	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
61		eluzelcn 12831	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℂ)
62	45, 61	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ)
63	62	addridd 11411	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 0) = 𝑁)
64	60, 63	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑁)
65	64	oveq2d 7417	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁))
66	54, 59, 65	3eqtr4d 2774	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
67	66	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
68	67	3exp 1116	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
69	44, 68	jaoi 854	. . . . . . . . . 10 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
70	7, 69	sylbi 216	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑀 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
71	70	com12 32	. . . . . . . 8 ⊢ (𝑀 = 0 → (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
72	6, 71	jaoi 854	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
73	2, 72	sylbi 216	. . . . . 6 ⊢ (𝑀 ∈ ℕ0 → (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
74	73	com12 32	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
75	74	3impd 1345	. . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
76		elnn1uz2 12906	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2)))
77		coires1 6253	. . . . . . . . . . . . . . 15 ⊢ (◡𝑅 ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))) = (◡𝑅 ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
78		relcnv 6093	. . . . . . . . . . . . . . . . 17 ⊢ Rel ◡𝑅
79		ssun1 4164	. . . . . . . . . . . . . . . . 17 ⊢ dom ◡𝑅 ⊆ (dom ◡𝑅 ∪ ran ◡𝑅)
80	78, 79	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (Rel ◡𝑅 ∧ dom ◡𝑅 ⊆ (dom ◡𝑅 ∪ ran ◡𝑅))
81		relssres 6012	. . . . . . . . . . . . . . . 16 ⊢ ((Rel ◡𝑅 ∧ dom ◡𝑅 ⊆ (dom ◡𝑅 ∪ ran ◡𝑅)) → (◡𝑅 ↾ (dom ◡𝑅 ∪ ran ◡𝑅)) = ◡𝑅)
82	80, 81	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅 ↾ (dom ◡𝑅 ∪ ran ◡𝑅)) = ◡𝑅)
83	77, 82	eqtrid 2776	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅 ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))) = ◡𝑅)
84		cnvco 5875	. . . . . . . . . . . . . . 15 ⊢ ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (◡(𝑅↑𝑟𝑀) ∘ ◡(𝑅↑𝑟𝑁))
85		simplr 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 = 1)
86		1nn0 12485	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ0
87	85, 86	eqeltrdi 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 ∈ ℕ0)
88		simprl 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅 ∈ 𝑉)
89		relexpcnv 14979	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
90	87, 88, 89	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
91	85	oveq2d 7417	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅↑𝑟𝑀) = (◡𝑅↑𝑟1))
92		cnvexg 7908	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V)
93	88, 92	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡𝑅 ∈ V)
94		relexp1g 14970	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟1) = ◡𝑅)
95	93, 94	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅↑𝑟1) = ◡𝑅)
96	90, 91, 95	3eqtrd 2768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟𝑀) = ◡𝑅)
97		simpll 764	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 = 0)
98		0nn0 12484	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℕ0
99	97, 98	eqeltrdi 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℕ0)
100		relexpcnv 14979	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
101	99, 88, 100	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
102	97	oveq2d 7417	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
103		relexp0g 14966	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑅 ∈ V → (◡𝑅↑𝑟0) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
104	93, 103	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅↑𝑟0) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
105	101, 102, 104	3eqtrd 2768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
106	96, 105	coeq12d 5854	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡(𝑅↑𝑟𝑀) ∘ ◡(𝑅↑𝑟𝑁)) = (◡𝑅 ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))))
107	84, 106	eqtrid 2776	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (◡𝑅 ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))))
108	99, 87	nn0addcld 12533	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) ∈ ℕ0)
109		relexpcnv 14979	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 + 𝑀) ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟(𝑁 + 𝑀)))
110	108, 88, 109	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟(𝑁 + 𝑀)))
111	97, 85	oveq12d 7419	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (0 + 1))
112		0p1e1 12331	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 1) = 1
113	111, 112	eqtrdi 2780	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 1)
114	113	oveq2d 7417	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟1))
115	110, 114, 95	3eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟(𝑁 + 𝑀)) = ◡𝑅)
116	83, 107, 115	3eqtr4d 2774	. . . . . . . . . . . . 13 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟(𝑁 + 𝑀)))
117		relco 6097	. . . . . . . . . . . . . 14 ⊢ Rel ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀))
118		simprr 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑁 + 𝑀) = 1 → Rel 𝑅))
119	113, 118	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel 𝑅)
120	113	oveq2d 7417	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1))
121	88, 18	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟1) = 𝑅)
122	120, 121	eqtrd 2764	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅)
123	122	releqd 5768	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (Rel (𝑅↑𝑟(𝑁 + 𝑀)) ↔ Rel 𝑅))
124	119, 123	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → Rel (𝑅↑𝑟(𝑁 + 𝑀)))
125		cnveqb 6185	. . . . . . . . . . . . . 14 ⊢ ((Rel ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ∧ Rel (𝑅↑𝑟(𝑁 + 𝑀))) → (((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)) ↔ ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟(𝑁 + 𝑀))))
126	117, 124, 125	sylancr 586	. . . . . . . . . . . . 13 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)) ↔ ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟(𝑁 + 𝑀))))
127	116, 126	mpbird 257	. . . . . . . . . . . 12 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
128	127	exp43 436	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝑀 = 1 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
129	128	com12 32	. . . . . . . . . 10 ⊢ (𝑀 = 1 → (𝑁 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
130		coires1 6253	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))) = ((◡𝑅↑𝑟𝑀) ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
131		simp2 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ (ℤ≥‘2))
132		simp3 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
133	132, 92	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡𝑅 ∈ V)
134		relexpuzrel 14996	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ ◡𝑅 ∈ V) → Rel (◡𝑅↑𝑟𝑀))
135	131, 133, 134	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (◡𝑅↑𝑟𝑀))
136		eluz2nn 12865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℕ)
137	131, 136	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ)
138		relexpnndm 14985	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℕ ∧ ◡𝑅 ∈ V) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅)
139	137, 133, 138	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅)
140	139, 79	sstrdi 3986	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ (dom ◡𝑅 ∪ ran ◡𝑅))
141		relssres 6012	. . . . . . . . . . . . . . . . . 18 ⊢ ((Rel (◡𝑅↑𝑟𝑀) ∧ dom (◡𝑅↑𝑟𝑀) ⊆ (dom ◡𝑅 ∪ ran ◡𝑅)) → ((◡𝑅↑𝑟𝑀) ↾ (dom ◡𝑅 ∪ ran ◡𝑅)) = (◡𝑅↑𝑟𝑀))
142	135, 140, 141	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ↾ (dom ◡𝑅 ∪ ran ◡𝑅)) = (◡𝑅↑𝑟𝑀))
143	130, 142	eqtrid 2776	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))) = (◡𝑅↑𝑟𝑀))
144		simp1 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
145	144	oveq2d 7417	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
146	133, 103	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟0) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
147	145, 146	eqtrd 2764	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
148	147	coeq2d 5852	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ (◡𝑅↑𝑟𝑁)) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))))
149	144	oveq1d 7416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
150		eluzelcn 12831	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℂ)
151	131, 150	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
152	151	addlidd 11412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (0 + 𝑀) = 𝑀)
153	149, 152	eqtrd 2764	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑀)
154	153	oveq2d 7417	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟𝑀))
155	143, 148, 154	3eqtr4d 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ (◡𝑅↑𝑟𝑁)) = (◡𝑅↑𝑟(𝑁 + 𝑀)))
156		nnnn0 12476	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
157	131, 136, 156	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ0)
158	157, 132, 89	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
159	144, 98	eqeltrdi 2833	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ0)
160	159, 132, 100	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
161	158, 160	coeq12d 5854	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡(𝑅↑𝑟𝑀) ∘ ◡(𝑅↑𝑟𝑁)) = ((◡𝑅↑𝑟𝑀) ∘ (◡𝑅↑𝑟𝑁)))
162	84, 161	eqtrid 2776	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((◡𝑅↑𝑟𝑀) ∘ (◡𝑅↑𝑟𝑁)))
163	159, 157	nn0addcld 12533	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) ∈ ℕ0)
164	163, 132, 109	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟(𝑁 + 𝑀)))
165	155, 162, 164	3eqtr4d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟(𝑁 + 𝑀)))
166	159	nn0cnd 12531	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ)
167	151, 166	addcomd 11413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑀 + 𝑁) = (𝑁 + 𝑀))
168		uzaddcl 12885	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ (ℤ≥‘2))
169	131, 159, 168	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑀 + 𝑁) ∈ (ℤ≥‘2))
170	167, 169	eqeltrrd 2826	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) ∈ (ℤ≥‘2))
171		relexpuzrel 14996	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 + 𝑀) ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟(𝑁 + 𝑀)))
172	170, 132, 171	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟(𝑁 + 𝑀)))
173	117, 172, 125	sylancr 586	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)) ↔ ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟(𝑁 + 𝑀))))
174	165, 173	mpbird 257	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
175	174	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
176	175	3exp 1116	. . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝑀 ∈ (ℤ≥‘2) → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
177	176	com12 32	. . . . . . . . . 10 ⊢ (𝑀 ∈ (ℤ≥‘2) → (𝑁 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
178	129, 177	jaoi 854	. . . . . . . . 9 ⊢ ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2)) → (𝑁 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
179	76, 178	sylbi 216	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑁 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
180		coires1 6253	. . . . . . . . . . . . 13 ⊢ ((𝑅↑𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅))
181		simp3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
182		relexp0rel 14981	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0))
183	181, 182	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟0))
184	181, 33	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
185	184	dmeqd 5895	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
186		dmresi 6041	. . . . . . . . . . . . . . . 16 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
187	185, 186	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟0) = (dom 𝑅 ∪ ran 𝑅))
188		eqimss 4032	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑅↑𝑟0) = (dom 𝑅 ∪ ran 𝑅) → dom (𝑅↑𝑟0) ⊆ (dom 𝑅 ∪ ran 𝑅))
189	187, 188	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟0) ⊆ (dom 𝑅 ∪ ran 𝑅))
190		relssres 6012	. . . . . . . . . . . . . 14 ⊢ ((Rel (𝑅↑𝑟0) ∧ dom (𝑅↑𝑟0) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅↑𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟0))
191	183, 189, 190	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟0))
192	180, 191	eqtrid 2776	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟0))
193		simp1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
194	193	oveq2d 7417	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
195		simp2 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
196	195	oveq2d 7417	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
197	196, 184	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
198	194, 197	coeq12d 5854	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
199	193, 195	oveq12d 7419	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 0))
200		00id 11386	. . . . . . . . . . . . . 14 ⊢ (0 + 0) = 0
201	199, 200	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 0)
202	201	oveq2d 7417	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟0))
203	192, 198, 202	3eqtr4d 2774	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
204	203	a1d 25	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
205	204	3exp 1116	. . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
206	205	com12 32	. . . . . . . 8 ⊢ (𝑀 = 0 → (𝑁 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
207	179, 206	jaoi 854	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑁 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
208	2, 207	sylbi 216	. . . . . 6 ⊢ (𝑀 ∈ ℕ0 → (𝑁 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
209	208	com12 32	. . . . 5 ⊢ (𝑁 = 0 → (𝑀 ∈ ℕ0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
210	209	3impd 1345	. . . 4 ⊢ (𝑁 = 0 → ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
211	75, 210	jaoi 854	. . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
212	1, 211	sylbi 216	. 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
213	212	imp 406	1 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∪ cun 3938   ⊆ wss 3940   I cid 5563  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668   ∘ ccom 5670  Rel wrel 5671  ‘cfv 6533  (class class class)co 7401  ℂcc 11104  0cc0 11106  1c1 11107   + caddc 11109  ℕcn 12209  2c2 12264  ℕ₀cn0 12469  ℤ_≥cuz 12819  ↑_𝑟crelexp 14963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-seq 13964  df-relexp 14964

This theorem is referenced by:  relexpaddd  14998