Theorem relexpaddg 15089

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 12526	. . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		elnn0 12526	. . . . . . 7 ⊢ (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
3		relexpaddnn 15087	. . . . . . . . . . 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 1118	. . . . . . . . 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 12965	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))
8		coires1 6286	. . . . . . . . . . . . . 14 ⊢ ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
9		simpll 767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 = 1)
10		simplr 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 = 0)
11	9, 10	oveq12d 7449	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (1 + 0))
12		1p0e1 12388	. . . . . . . . . . . . . . . . . 18 ⊢ (1 + 0) = 1
13	11, 12	eqtrdi 2791	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 1)
14		simprr 773	. . . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1))
17		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅 ∈ 𝑉)
18		relexp1g 15062	. . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟𝑁) = 𝑅)
21	20	releqd 5791	. . . . . . . . . . . . . . . 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 12275	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ
24	9, 23	eqeltrdi 2847	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℕ)
25		relexpnndm 15077	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
26	24, 17, 25	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
27		ssun1 4188	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
28	26, 27	sstrdi 4008	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
29		relssres 6042	. . . . . . . . . . . . . . 15 ⊢ ((Rel (𝑅↑𝑟𝑁) ∧ dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
30	22, 28, 29	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
31	8, 30	eqtrid 2787	. . . . . . . . . . . . 13 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁))
32	10	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
33		relexp0g 15058	. . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36	35	coeq2d 5876	. . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (𝑁 + 0))
38		ax-1cn 11211	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
39	9, 38	eqeltrdi 2847	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℂ)
40	39	addridd 11459	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 0) = 𝑁)
41	37, 40	eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 𝑁)
42	41	oveq2d 7447	. . . . . . . . . . . . 13 ⊢ (((𝑁 = 1 ∧ 𝑀 = 0) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁))
43	31, 36, 42	3eqtr4d 2785	. . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ (ℤ≥‘2))
46		simp3 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
47		relexpuzrel 15088	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
48	45, 46, 47	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
49		eluz2nn 12922	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
52	51, 27	sstrdi 4008	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
53	48, 52, 29	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
54	8, 53	eqtrid 2787	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁))
55		simp2 1136	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
56	55	oveq2d 7447	. . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
59	58	coeq2d 5876	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
60	55	oveq2d 7447	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
61		eluzelcn 12888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℂ)
62	45, 61	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ)
63	62	addridd 11459	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 0) = 𝑁)
64	60, 63	eqtrd 2775	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑁)
65	64	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁))
66	54, 59, 65	3eqtr4d 2785	. . . . . . . . . . . . 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 1118	. . . . . . . . . . 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 857	. . . . . . . . . 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 217	. . . . . . . . 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 857	. . . . . . 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 217	. . . . . 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 1347	. . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
76		elnn1uz2 12965	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2)))
77		coires1 6286	. . . . . . . . . . . . . . 15 ⊢ (◡𝑅 ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))) = (◡𝑅 ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
78		relcnv 6125	. . . . . . . . . . . . . . . . 17 ⊢ Rel ◡𝑅
79		ssun1 4188	. . . . . . . . . . . . . . . . 17 ⊢ dom ◡𝑅 ⊆ (dom ◡𝑅 ∪ ran ◡𝑅)
80	78, 79	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (Rel ◡𝑅 ∧ dom ◡𝑅 ⊆ (dom ◡𝑅 ∪ ran ◡𝑅))
81		relssres 6042	. . . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅 ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))) = ◡𝑅)
84		cnvco 5899	. . . . . . . . . . . . . . 15 ⊢ ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (◡(𝑅↑𝑟𝑀) ∘ ◡(𝑅↑𝑟𝑁))
85		simplr 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 = 1)
86		1nn0 12540	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ0
87	85, 86	eqeltrdi 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑀 ∈ ℕ0)
88		simprl 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑅 ∈ 𝑉)
89		relexpcnv 15071	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
90	87, 88, 89	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
91	85	oveq2d 7447	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅↑𝑟𝑀) = (◡𝑅↑𝑟1))
92		cnvexg 7947	. . . . . . . . . . . . . . . . . . 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 15062	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟𝑀) = ◡𝑅)
97		simpll 767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 = 0)
98		0nn0 12539	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℕ0
99	97, 98	eqeltrdi 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → 𝑁 ∈ ℕ0)
100		relexpcnv 15071	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
101	99, 88, 100	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
102	97	oveq2d 7447	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅↑𝑟𝑁) = (◡𝑅↑𝑟0))
103		relexp0g 15058	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
106	96, 105	coeq12d 5878	. . . . . . . . . . . . . . 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 2787	. . . . . . . . . . . . . 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 12589	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) ∈ ℕ0)
109		relexpcnv 15071	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 + 𝑀) ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟(𝑁 + 𝑀)))
110	108, 88, 109	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟(𝑁 + 𝑀)))
111	97, 85	oveq12d 7449	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = (0 + 1))
112		0p1e1 12386	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 1) = 1
113	111, 112	eqtrdi 2791	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑁 + 𝑀) = 1)
114	113	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (◡𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟1))
115	110, 114, 95	3eqtrd 2779	. . . . . . . . . . . . . 14 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡(𝑅↑𝑟(𝑁 + 𝑀)) = ◡𝑅)
116	83, 107, 115	3eqtr4d 2785	. . . . . . . . . . . . 13 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟(𝑁 + 𝑀)))
117		relco 6129	. . . . . . . . . . . . . 14 ⊢ Rel ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀))
118		simprr 773	. . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 = 0 ∧ 𝑀 = 1) ∧ (𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅)
123	122	releqd 5791	. . . . . . . . . . . . . . 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 6218	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 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 6286	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))) = ((◡𝑅↑𝑟𝑀) ↾ (dom ◡𝑅 ∪ ran ◡𝑅))
131		simp2 1136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ (ℤ≥‘2))
132		simp3 1137	. . . . . . . . . . . . . . . . . . . 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 15088	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ ◡𝑅 ∈ V) → Rel (◡𝑅↑𝑟𝑀))
135	131, 133, 134	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (◡𝑅↑𝑟𝑀))
136		eluz2nn 12922	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℕ)
137	131, 136	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ)
138		relexpnndm 15077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℕ ∧ ◡𝑅 ∈ V) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅)
139	137, 133, 138	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅)
140	139, 79	sstrdi 4008	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ (dom ◡𝑅 ∪ ran ◡𝑅))
141		relssres 6042	. . . . . . . . . . . . . . . . . 18 ⊢ ((Rel (◡𝑅↑𝑟𝑀) ∧ dom (◡𝑅↑𝑟𝑀) ⊆ (dom ◡𝑅 ∪ ran ◡𝑅)) → ((◡𝑅↑𝑟𝑀) ↾ (dom ◡𝑅 ∪ ran ◡𝑅)) = (◡𝑅↑𝑟𝑀))
142	135, 140, 141	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ↾ (dom ◡𝑅 ∪ ran ◡𝑅)) = (◡𝑅↑𝑟𝑀))
143	130, 142	eqtrid 2787	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))) = (◡𝑅↑𝑟𝑀))
144		simp1 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
145	144	oveq2d 7447	. . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟𝑁) = ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅)))
148	147	coeq2d 5876	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ (◡𝑅↑𝑟𝑁)) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom ◡𝑅 ∪ ran ◡𝑅))))
149	144	oveq1d 7446	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
150		eluzelcn 12888	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℂ)
151	131, 150	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
152	151	addlidd 11460	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (0 + 𝑀) = 𝑀)
153	149, 152	eqtrd 2775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑀)
154	153	oveq2d 7447	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟𝑀))
155	143, 148, 154	3eqtr4d 2785	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ (◡𝑅↑𝑟𝑁)) = (◡𝑅↑𝑟(𝑁 + 𝑀)))
156		nnnn0 12531	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
159	144, 98	eqeltrdi 2847	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ0)
160	159, 132, 100	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))
161	158, 160	coeq12d 5878	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡(𝑅↑𝑟𝑀) ∘ ◡(𝑅↑𝑟𝑁)) = ((◡𝑅↑𝑟𝑀) ∘ (◡𝑅↑𝑟𝑁)))
162	84, 161	eqtrid 2787	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((◡𝑅↑𝑟𝑀) ∘ (◡𝑅↑𝑟𝑁)))
163	159, 157	nn0addcld 12589	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) ∈ ℕ0)
164	163, 132, 109	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟(𝑁 + 𝑀)) = (◡𝑅↑𝑟(𝑁 + 𝑀)))
165	155, 162, 164	3eqtr4d 2785	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟(𝑁 + 𝑀)))
166	159	nn0cnd 12587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ)
167	151, 166	addcomd 11461	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑀 + 𝑁) = (𝑁 + 𝑀))
168		uzaddcl 12944	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝑀 + 𝑁) ∈ (ℤ≥‘2))
169	131, 159, 168	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑀 + 𝑁) ∈ (ℤ≥‘2))
170	167, 169	eqeltrrd 2840	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) ∈ (ℤ≥‘2))
171		relexpuzrel 15088	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 + 𝑀) ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟(𝑁 + 𝑀)))
172	170, 132, 171	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟(𝑁 + 𝑀)))
173	117, 172, 125	sylancr 587	. . . . . . . . . . . . . 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 1118	. . . . . . . . . . 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 857	. . . . . . . . 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 217	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (𝑁 = 0 → (𝑅 ∈ 𝑉 → (((𝑁 + 𝑀) = 1 → Rel 𝑅) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))))
180		coires1 6286	. . . . . . . . . . . . 13 ⊢ ((𝑅↑𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅))
181		simp3 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
182		relexp0rel 15073	. . . . . . . . . . . . . . 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 5919	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟0) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
186		dmresi 6072	. . . . . . . . . . . . . . . 16 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
187	185, 186	eqtrdi 2791	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟0) = (dom 𝑅 ∪ ran 𝑅))
188		eqimss 4054	. . . . . . . . . . . . . . 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 6042	. . . . . . . . . . . . . 14 ⊢ ((Rel (𝑅↑𝑟0) ∧ dom (𝑅↑𝑟0) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅↑𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟0))
191	183, 189, 190	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟0) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟0))
192	180, 191	eqtrid 2787	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟0))
193		simp1 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
194	193	oveq2d 7447	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
195		simp2 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
196	195	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
197	196, 184	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
198	194, 197	coeq12d 5878	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟0) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
199	193, 195	oveq12d 7449	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 0))
200		00id 11434	. . . . . . . . . . . . . 14 ⊢ (0 + 0) = 0
201	199, 200	eqtrdi 2791	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 0)
202	201	oveq2d 7447	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟0))
203	192, 198, 202	3eqtr4d 2785	. . . . . . . . . . 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 1118	. . . . . . . . 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 857	. . . . . . 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 217	. . . . . 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 1347	. . . 4 ⊢ (𝑁 = 0 → ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅)) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))))
211	75, 210	jaoi 857	. . 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 217	. 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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963   I cid 5582  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   ↾ cres 5691   ∘ ccom 5693  Rel wrel 5694  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  0cc0 11153  1c1 11154   + caddc 11156  ℕcn 12264  2c2 12319  ℕ₀cn0 12524  ℤ_≥cuz 12876  ↑_𝑟crelexp 15055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-relexp 15056

This theorem is referenced by:  relexpaddd  15090