Theorem trclrelexplem 43693

Description: The union of relational powers to positive multiples of 𝑁 is a subset to the transitive closure raised to the power of 𝑁. (Contributed by RP, 15-Jun-2020.)

Assertion

Ref	Expression
trclrelexplem	⊢ (𝑁 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))

Distinct variable groups:   𝐷,𝑗   𝐷,𝑘   𝑘,𝑁

Allowed substitution hint:   𝑁(𝑗)

Proof of Theorem trclrelexplem

Dummy variables 𝑥 𝑦 𝑙 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7377	. . . 4 ⊢ (𝑥 = 1 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟1))
2	1	iuneq2d 4982	. . 3 ⊢ (𝑥 = 1 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1))
3		oveq2 7377	. . 3 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1))
4	2, 3	sseq12d 3977	. 2 ⊢ (𝑥 = 1 → (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) ↔ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)))
5		oveq2 7377	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑦))
6	5	iuneq2d 4982	. . 3 ⊢ (𝑥 = 𝑦 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
7		oveq2 7377	. . 3 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦))
8	6, 7	sseq12d 3977	. 2 ⊢ (𝑥 = 𝑦 → (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) ↔ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦)))
9		oveq2 7377	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
10	9	iuneq2d 4982	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
11		oveq2 7377	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)))
12	10, 11	sseq12d 3977	. 2 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) ↔ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1))))
13		oveq2 7377	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑁))
14	13	iuneq2d 4982	. . 3 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁))
15		oveq2 7377	. . 3 ⊢ (𝑥 = 𝑁 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))
16	14, 15	sseq12d 3977	. 2 ⊢ (𝑥 = 𝑁 → (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) ↔ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁)))
17		oveq2 7377	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑙))
18	17	cbviunv 4999	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙)
19		oveq2 7377	. . . . . 6 ⊢ (𝑙 = 𝑗 → (𝐷↑𝑟𝑙) = (𝐷↑𝑟𝑗))
20	19	cbviunv 4999	. . . . 5 ⊢ ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
21	18, 20	eqtri 2752	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
22		ovex 7402	. . . . . 6 ⊢ (𝐷↑𝑟𝑘) ∈ V
23		relexp1g 14968	. . . . . 6 ⊢ ((𝐷↑𝑟𝑘) ∈ V → ((𝐷↑𝑟𝑘)↑𝑟1) = (𝐷↑𝑟𝑘))
24	22, 23	mp1i 13	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((𝐷↑𝑟𝑘)↑𝑟1) = (𝐷↑𝑟𝑘))
25	24	iuneq2i 4973	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘)
26		nnex 12168	. . . . . 6 ⊢ ℕ ∈ V
27		ovex 7402	. . . . . 6 ⊢ (𝐷↑𝑟𝑗) ∈ V
28	26, 27	iunex 7926	. . . . 5 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) ∈ V
29		relexp1g 14968	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) ∈ V → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗))
30	28, 29	ax-mp 5	. . . 4 ⊢ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
31	21, 25, 30	3eqtr4i 2762	. . 3 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
32	31	eqimssi 4004	. 2 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
33		oveq2 7377	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑚))
34	33	oveq1d 7384	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐷↑𝑟𝑘)↑𝑟𝑦) = ((𝐷↑𝑟𝑚)↑𝑟𝑦))
35	34, 33	coeq12d 5818	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
36	35	cbviunv 4999	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
37		ss2iun 4970	. . . . . . . 8 ⊢ (∀𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) → ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
38	34	ssiun2s 5007	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝐷↑𝑟𝑚)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
39		coss1 5809	. . . . . . . . 9 ⊢ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) → (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
40	38, 39	syl 17	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
41	37, 40	mprg 3050	. . . . . . 7 ⊢ ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
42	36, 41	eqsstri 3990	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
43		coss1 5809	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) → (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
44	43	ralrimivw 3129	. . . . . . 7 ⊢ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) → ∀𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
45		ss2iun 4970	. . . . . . 7 ⊢ (∀𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) → ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
46	44, 45	syl 17	. . . . . 6 ⊢ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) → ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
47	42, 46	sstrid 3955	. . . . 5 ⊢ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) → ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) ⊆ ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
48	47	adantl 481	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦)) → ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) ⊆ ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
49		relexpsucnnr 14967	. . . . . . 7 ⊢ (((𝐷↑𝑟𝑘) ∈ V ∧ 𝑦 ∈ ℕ) → ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
50	22, 49	mpan 690	. . . . . 6 ⊢ (𝑦 ∈ ℕ → ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
51	50	iuneq2d 4982	. . . . 5 ⊢ (𝑦 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
52	51	adantr 480	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦)) → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
53		relexpsucnnr 14967	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)))
54	28, 53	mpan 690	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)))
55		oveq2 7377	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → (𝐷↑𝑟𝑗) = (𝐷↑𝑟𝑚))
56	55	cbviunv 4999	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) = ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)
57	56	coeq2i 5814	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚))
58		coiun 6217	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
59	57, 58	eqtri 2752	. . . . . 6 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
60	54, 59	eqtrdi 2780	. . . . 5 ⊢ (𝑦 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
61	60	adantr 480	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦)) → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
62	48, 52, 61	3sstr4d 3999	. . 3 ⊢ ((𝑦 ∈ ℕ ∧ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦)) → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)))
63	62	ex 412	. 2 ⊢ (𝑦 ∈ ℕ → (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1))))
64	4, 8, 12, 16, 32, 63	nnind 12180	1 ⊢ (𝑁 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ⊆ wss 3911  ∪ ciun 4951   ∘ ccom 5635  (class class class)co 7369  1c1 11045   + caddc 11047  ℕcn 12162  ↑_𝑟crelexp 14961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770  df-seq 13943  df-relexp 14962

This theorem is referenced by: (None)