Theorem relexpaddss 41326

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 14765 or when the sum of the powers isn't 1 as shown by relexpaddg 14764. (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 12235	. . 3 ⊢ (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2		elnn0 12235	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
3	2	biimpi 215	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4		relexpaddnn 14762	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
5		eqimss 3977	. . . . . . . 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 1118	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
8		elnn1uz2 12665	. . . . . . 7 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2)))
9		relco 6148	. . . . . . . . . . . . . 14 ⊢ Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)
10		dfrel2 6092	. . . . . . . . . . . . . . 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 5794	. . . . . . . . . . . . . . . . 17 ⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
14		cnvresid 6513	. . . . . . . . . . . . . . . . . 18 ⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
15	14	coeq2i 5769	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
16		coires1 6168	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
17	13, 15, 16	3eqtri 2770	. . . . . . . . . . . . . . . 16 ⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
18		eqimss 3977	. . . . . . . . . . . . . . . 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 5781	. . . . . . . . . . . . . . 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 5916	. . . . . . . . . . . . . . 15 ⊢ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡𝑅
23		cnvss 5781	. . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . 13 ⊢ ◡◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅
26	12, 25	eqsstrri 3956	. . . . . . . . . . . 12 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅
27		cnvcnvss 6097	. . . . . . . . . . . 12 ⊢ ◡◡𝑅 ⊆ 𝑅
28	26, 27	sstri 3930	. . . . . . . . . . 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 1135	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
31	30	oveq2d 7291	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
32		relexp0g 14733	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
33	32	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
34	31, 33	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
35		simp2 1136	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 1)
36	35	oveq2d 7291	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟1))
37		relexp1g 14737	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
38	37	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
39	36, 38	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = 𝑅)
40	34, 39	coeq12d 5773	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
41	30, 35	oveq12d 7293	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 1))
42		1cnd 10970	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
43	42	addid2d 11176	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (0 + 1) = 1)
44	41, 43	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1)
45	44	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1))
46	45, 38	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅)
47	29, 40, 46	3sstr4d 3968	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
48	47	3exp 1118	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑀 = 1 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
49		coires1 6168	. . . . . . . . . . . . . 14 ⊢ ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅))
50		simp2 1136	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ (ℤ≥‘2))
51		cnvexg 7771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V)
52	51	3ad2ant3 1134	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡𝑅 ∈ V)
53		relexpuzrel 14763	. . . . . . . . . . . . . . . 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 12624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℕ)
56	50, 55	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ)
57		relexpnndm 14752	. . . . . . . . . . . . . . . . 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 5600	. . . . . . . . . . . . . . . . 17 ⊢ ran 𝑅 = dom ◡𝑅
60		ssun2 4107	. . . . . . . . . . . . . . . . 17 ⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
61	59, 60	eqsstrri 3956	. . . . . . . . . . . . . . . 16 ⊢ dom ◡𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
62	58, 61	sstrdi 3933	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅))
63		relssres 5932	. . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅↑𝑟𝑀))
66		cnvco 5794	. . . . . . . . . . . . . 14 ⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)))
67		eluzge2nn0 12627	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℕ0)
68	50, 67	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ0)
69		simp3 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
70		relexpcnv 14746	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
71	68, 69, 70	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀))
72	14	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
73	71, 72	coeq12d 5773	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
74	66, 73	eqtrid 2790	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
75	65, 74, 71	3eqtr4d 2788	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀))
76		relco 6148	. . . . . . . . . . . . 13 ⊢ Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀))
77		relexpuzrel 14763	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀))
78	77	3adant1 1129	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀))
79		cnveqb 6099	. . . . . . . . . . . . 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)))
80	76, 78, 79	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)))
81	75, 80	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀))
82		simp1 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
83	82	oveq2d 7291	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
84	32	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
85	83, 84	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
86	85	coeq1d 5770	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)))
87	82	oveq1d 7290	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
88		eluzelcn 12594	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ≥‘2) → 𝑀 ∈ ℂ)
89	50, 88	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ)
90	89	addid2d 11176	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (0 + 𝑀) = 𝑀)
91	87, 90	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑀)
92	91	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑀))
93	81, 86, 92	3eqtr4d 2788	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
94	93, 5	syl 17	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
95	94	3exp 1118	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑀 ∈ (ℤ≥‘2) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
96	48, 95	jaod 856	. . . . . . 7 ⊢ (𝑁 = 0 → ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2)) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
97	8, 96	syl5bi 241	. . . . . 6 ⊢ (𝑁 = 0 → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
98	7, 97	jaoi 854	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
99	3, 98	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
100		elnn1uz2 12665	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))
101	100	biimpi 215	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)))
102		coires1 6168	. . . . . . . . . . . 12 ⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
103		resss 5916	. . . . . . . . . . . 12 ⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
104	102, 103	eqsstri 3955	. . . . . . . . . . 11 ⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅
105	104	a1i 11	. . . . . . . . . 10 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅)
106		simp1 1135	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 1)
107	106	oveq2d 7291	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1))
108	37	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅)
109	107, 108	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = 𝑅)
110		simp2 1136	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
111	110	oveq2d 7291	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
112	32	3ad2ant3 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
113	111, 112	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
114	109, 113	coeq12d 5773	. . . . . . . . . 10 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
115	106, 110	oveq12d 7293	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (1 + 0))
116		1cnd 10970	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ)
117	116	addid1d 11175	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (1 + 0) = 1)
118	115, 117	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1)
119	118	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1))
120	119, 108	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅)
121	105, 114, 120	3sstr4d 3968	. . . . . . . . 9 ⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
122	121	3exp 1118	. . . . . . . 8 ⊢ (𝑁 = 1 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
123		coires1 6168	. . . . . . . . . . . 12 ⊢ ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
124		relexpuzrel 14763	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
125	124	3adant2 1130	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁))
126		simp1 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ (ℤ≥‘2))
127		eluz2nn 12624	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ)
128	126, 127	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ)
129		simp3 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
130		relexpnndm 14752	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
131	128, 129, 130	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅)
132		ssun1 4106	. . . . . . . . . . . . . 14 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
133	131, 132	sstrdi 3933	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
134		relssres 5932	. . . . . . . . . . . . 13 ⊢ ((Rel (𝑅↑𝑟𝑁) ∧ dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
135	125, 133, 134	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁))
136	123, 135	eqtrid 2790	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁))
137		simp2 1136	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
138	137	oveq2d 7291	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
139	32	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
140	138, 139	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
141	140	coeq2d 5771	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
142	137	oveq2d 7291	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
143		eluzelcn 12594	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℂ)
144	126, 143	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ)
145	144	addid1d 11175	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 0) = 𝑁)
146	142, 145	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑁)
147	146	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁))
148	136, 141, 147	3eqtr4d 2788	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
149	148, 5	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
150	149	3exp 1118	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
151	122, 150	jaoi 854	. . . . . . 7 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
152	101, 151	syl 17	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
153		coires1 6168	. . . . . . . . . 10 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅))
154		resres 5904	. . . . . . . . . 10 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)))
155		inidm 4152	. . . . . . . . . . 11 ⊢ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
156	155	reseq2i 5888	. . . . . . . . . 10 ⊢ ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
157	153, 154, 156	3eqtri 2770	. . . . . . . . 9 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
158		simp1 1135	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
159	158	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0))
160	32	3ad2ant3 1134	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
161	159, 160	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
162		simp2 1136	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0)
163	162	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0))
164	163, 160	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
165	161, 164	coeq12d 5773	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
166	158, 162	oveq12d 7293	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 0))
167		00id 11150	. . . . . . . . . . . . 13 ⊢ (0 + 0) = 0
168	167	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (0 + 0) = 0)
169	166, 168	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 0)
170	169	oveq2d 7291	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟0))
171	170, 160	eqtrd 2778	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
172	157, 165, 171	3eqtr4a 2804	. . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)))
173	172, 5	syl 17	. . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
174	173	3exp 1118	. . . . . 6 ⊢ (𝑁 = 0 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
175	152, 174	jaoi 854	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
176	3, 175	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
177	99, 176	jaod 856	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
178	1, 177	syl5bi 241	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))))
179	178	3imp 1110	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887   I cid 5488  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   ∘ ccom 5593  Rel wrel 5594  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  0cc0 10871  1c1 10872   + caddc 10874  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ℤ_≥cuz 12582  ↑_𝑟crelexp 14730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-seq 13722  df-relexp 14731

This theorem is referenced by:  iunrelexpuztr  41327  cotrclrcl  41350