Theorem relexpaddss 41980

Description: The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where 𝑅 is a relation as shown by relexpaddd 14939 or when the sum of the powers isn't 1 as shown by relexpaddg 14938. (Contributed by RP, 3-Jun-2020.)

Assertion

Ref	Expression
relexpaddss	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))

Proof of Theorem relexpaddss

Step	Hyp	Ref	Expression
1		elnn0 12415	. . 3 ⊢ (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2		elnn0 12415	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
3	2	biimpi 215	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4		relexpaddnn 14936	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
5		eqimss 4000	. . . . . . . 8 ⊢ (((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
6	4, 5	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
7	6	3exp 1119	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
8		elnn1uz2 12850	. . . . . . 7 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2)))
9		relco 6060	. . . . . . . . . . . . . 14 ⊢ Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)
10		dfrel2 6141	. . . . . . . . . . . . . . 15 ⊢ (Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ↔ ◡◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
11	10	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) → ◡◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
12	9, 11	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ◡◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)
13		cnvco 5841	. . . . . . . . . . . . . . . . 17 ⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
14		cnvresid 6580	. . . . . . . . . . . . . . . . . 18 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
15	14	coeq2i 5816	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
16		coires1 6216	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
17	13, 15, 16	3eqtri 2768	. . . . . . . . . . . . . . . 16 ⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
18		eqimss 4000	. . . . . . . . . . . . . . . 16 ⊢ (◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
19	17, 18	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
20		cnvss 5828	. . . . . . . . . . . . . . 15 ⊢ (◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) → ◡◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ ◡(◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
21	19, 20	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ◡◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ ◡(◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
22		resss 5962	. . . . . . . . . . . . . . 15 ⊢ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡𝑅
23		cnvss 5828	. . . . . . . . . . . . . . 15 ⊢ ((◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡𝑅 → ◡(◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡◡𝑅)
24	22, 23	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ◡(◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡◡𝑅
25	21, 24	sstri 3953	. . . . . . . . . . . . 13 ⊢ ◡◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅
26	12, 25	eqsstrri 3979	. . . . . . . . . . . 12 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅
27		cnvcnvss 6146	. . . . . . . . . . . 12 ⊢ ◡◡𝑅 ⊆ 𝑅
28	26, 27	sstri 3953	. . . . . . . . . . 11 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
29	28	a1i 11	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅)
30		simp1 1136	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
31	30	oveq2d 7373	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
32		relexp0g 14907	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
33	32	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
34	31, 33	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
35		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 1)
36	35	oveq2d 7373	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟1))
37		relexp1g 14911	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
38	37	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
39	36, 38	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = 𝑅)
40	34, 39	coeq12d 5820	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
41	30, 35	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 1))
42		1cnd 11150	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
43	42	addid2d 11356	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (0 + 1) = 1)
44	41, 43	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1)
45	44	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1))
46	45, 38	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅)
47	29, 40, 46	3sstr4d 3991	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
48	47	3exp 1119	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑀 = 1 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
49		coires1 6216	. . . . . . . . . . . . . 14 ⊢ ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅))
50		simp2 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ (ℤ≥‘2))
51		cnvexg 7861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V)
52	51	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡𝑅 ∈ V)
53		relexpuzrel 14937	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ ◡𝑅 ∈ V) → Rel (◡𝑅↑𝑟𝑀))
54	50, 52, 53	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (◡𝑅↑𝑟𝑀))
55		eluz2nn 12809	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℕ)
56	50, 55	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ)
57		relexpnndm 14926	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ ℕ ∧ ◡𝑅 ∈ V) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅)
58	56, 52, 57	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅)
59		df-rn 5644	. . . . . . . . . . . . . . . . 17 ⊢ ran 𝑅 = dom ◡𝑅
60		ssun2 4133	. . . . . . . . . . . . . . . . 17 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
61	59, 60	eqsstrri 3979	. . . . . . . . . . . . . . . 16 ⊢ dom ◡𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
62	58, 61	sstrdi 3956	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅))
63		relssres 5978	. . . . . . . . . . . . . . 15 ⊢ ((Rel (◡𝑅↑𝑟𝑀) ∧ dom (◡𝑅↑𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (◡𝑅↑𝑟𝑀))
64	54, 62, 63	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (◡𝑅↑𝑟𝑀))
65	49, 64	eqtrid 2788	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅↑𝑟𝑀))
66		cnvco 5841	. . . . . . . . . . . . . 14 ⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
67		simp3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
68		eluzge2nn0 12812	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℕ0)
69	50, 68	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ0)
70	67, 69	relexpcnvd 14921	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
71	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
72	70, 71	coeq12d 5820	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
73	66, 72	eqtrid 2788	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
74	65, 73, 70	3eqtr4d 2786	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀))
75		relco 6060	. . . . . . . . . . . . 13 ⊢ Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀))
76		relexpuzrel 14937	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀))
77	76	3adant1 1130	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀))
78		cnveqb 6148	. . . . . . . . . . . . 13 ⊢ ((Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) ∧ Rel (𝑅↑𝑟𝑀)) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀) ↔ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀)))
79	75, 77, 78	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀) ↔ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀)))
80	74, 79	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀))
81		simp1 1136	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
82	81	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
83	32	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
84	82, 83	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
85	84	coeq1d 5817	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)))
86	81	oveq1d 7372	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
87		eluzelcn 12775	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℂ)
88	50, 87	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
89	88	addid2d 11356	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (0 + 𝑀) = 𝑀)
90	86, 89	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑀)
91	90	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑀))
92	80, 85, 91	3eqtr4d 2786	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
93	92, 5	syl 17	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
94	93	3exp 1119	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑀 ∈ (ℤ≥‘2) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
95	48, 94	jaod 857	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2)) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
96	8, 95	biimtrid 241	. . . . . 6 ⊢ (𝑁 = 0 → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
97	7, 96	jaoi 855	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
98	3, 97	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
99		elnn1uz2 12850	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))
100	99	biimpi 215	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))
101		coires1 6216	. . . . . . . . . . . 12 ⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
102		resss 5962	. . . . . . . . . . . 12 ⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
103	101, 102	eqsstri 3978	. . . . . . . . . . 11 ⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅
104	103	a1i 11	. . . . . . . . . 10 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅)
105		simp1 1136	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 1)
106	105	oveq2d 7373	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1))
107	37	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
108	106, 107	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = 𝑅)
109		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
110	109	oveq2d 7373	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
111	32	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
112	110, 111	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
113	108, 112	coeq12d 5820	. . . . . . . . . 10 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
114	105, 109	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (1 + 0))
115		1cnd 11150	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
116	115	addid1d 11355	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (1 + 0) = 1)
117	114, 116	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1)
118	117	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1))
119	118, 107	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅)
120	104, 113, 119	3sstr4d 3991	. . . . . . . . 9 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
121	120	3exp 1119	. . . . . . . 8 ⊢ (𝑁 = 1 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
122		coires1 6216	. . . . . . . . . . . 12 ⊢ ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
123		relexpuzrel 14937	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
124	123	3adant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
125		simp1 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ (ℤ≥‘2))
126		eluz2nn 12809	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ)
127	125, 126	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ)
128		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
129		relexpnndm 14926	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
130	127, 128, 129	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
131		ssun1 4132	. . . . . . . . . . . . . 14 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
132	130, 131	sstrdi 3956	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
133		relssres 5978	. . . . . . . . . . . . 13 ⊢ ((Rel (𝑅↑𝑟𝑁) ∧ dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
134	124, 132, 133	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
135	122, 134	eqtrid 2788	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁))
136		simp2 1137	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
137	136	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
138	32	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
139	137, 138	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
140	139	coeq2d 5818	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
141	136	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
142		eluzelcn 12775	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℂ)
143	125, 142	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ)
144	143	addid1d 11355	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 0) = 𝑁)
145	141, 144	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑁)
146	145	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁))
147	135, 140, 146	3eqtr4d 2786	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
148	147, 5	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
149	148	3exp 1119	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
150	121, 149	jaoi 855	. . . . . . 7 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
151	100, 150	syl 17	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
152		coires1 6216	. . . . . . . . . 10 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅))
153		resres 5950	. . . . . . . . . 10 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)))
154		inidm 4178	. . . . . . . . . . 11 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
155	154	reseq2i 5934	. . . . . . . . . 10 ⊢ ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
156	152, 153, 155	3eqtri 2768	. . . . . . . . 9 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
157		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
158	157	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
159	32	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
160	158, 159	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
161		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
162	161	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
163	162, 159	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
164	160, 163	coeq12d 5820	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
165	157, 161	oveq12d 7375	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 0))
166		00id 11330	. . . . . . . . . . . . 13 ⊢ (0 + 0) = 0
167	166	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (0 + 0) = 0)
168	165, 167	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 0)
169	168	oveq2d 7373	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟0))
170	169, 159	eqtrd 2776	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
171	156, 164, 170	3eqtr4a 2802	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
172	171, 5	syl 17	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
173	172	3exp 1119	. . . . . 6 ⊢ (𝑁 = 0 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
174	151, 173	jaoi 855	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
175	3, 174	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
176	98, 175	jaod 857	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
177	1, 176	biimtrid 241	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
178	177	3imp 1111	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910   I cid 5530  ^◡ccnv 5632  dom cdm 5633  ran crn 5634   ↾ cres 5635   ∘ ccom 5637  Rel wrel 5638  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  0cc0 11051  1c1 11052   + caddc 11054  ℕcn 12153  2c2 12208  ℕ₀cn0 12413  ℤ_≥cuz 12763  ↑_𝑟crelexp 14904

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-seq 13907  df-relexp 14905

This theorem is referenced by:  iunrelexpuztr  41981  cotrclrcl  42004