Theorem trclrelexplem 40049

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 7158	. . . 4 ⊢ (𝑥 = 1 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟1))
2	1	iuneq2d 4940	. . 3 ⊢ (𝑥 = 1 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1))
3		oveq2 7158	. . 3 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1))
4	2, 3	sseq12d 3999	. 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 7158	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑦))
6	5	iuneq2d 4940	. . 3 ⊢ (𝑥 = 𝑦 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
7		oveq2 7158	. . 3 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦))
8	6, 7	sseq12d 3999	. 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 7158	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
10	9	iuneq2d 4940	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
11		oveq2 7158	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)))
12	10, 11	sseq12d 3999	. 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 7158	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑁))
14	13	iuneq2d 4940	. . 3 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁))
15		oveq2 7158	. . 3 ⊢ (𝑥 = 𝑁 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))
16	14, 15	sseq12d 3999	. 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 7158	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑙))
18	17	cbviunv 4957	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙)
19		oveq2 7158	. . . . . 6 ⊢ (𝑙 = 𝑗 → (𝐷↑𝑟𝑙) = (𝐷↑𝑟𝑗))
20	19	cbviunv 4957	. . . . 5 ⊢ ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
21	18, 20	eqtri 2844	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
22		ovex 7183	. . . . . 6 ⊢ (𝐷↑𝑟𝑘) ∈ V
23		relexp1g 14379	. . . . . 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 4932	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘)
26		nnex 11638	. . . . . 6 ⊢ ℕ ∈ V
27		ovex 7183	. . . . . 6 ⊢ (𝐷↑𝑟𝑗) ∈ V
28	26, 27	iunex 7663	. . . . 5 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) ∈ V
29		relexp1g 14379	. . . . 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 2854	. . 3 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
32	31	eqimssi 4024	. 2 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
33		oveq2 7158	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑚))
34	33	oveq1d 7165	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐷↑𝑟𝑘)↑𝑟𝑦) = ((𝐷↑𝑟𝑚)↑𝑟𝑦))
35	34, 33	coeq12d 5729	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
36	35	cbviunv 4957	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
37		ss2iun 4929	. . . . . . . 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 4964	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝐷↑𝑟𝑚)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
39		coss1 5720	. . . . . . . . 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 3152	. . . . . . 7 ⊢ ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
42	36, 41	eqsstri 4000	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
43		coss1 5720	. . . . . . . 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 3183	. . . . . . 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 4929	. . . . . . 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 3977	. . . . 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 484	. . . 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 14378	. . . . . . 7 ⊢ (((𝐷↑𝑟𝑘) ∈ V ∧ 𝑦 ∈ ℕ) → ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
50	22, 49	mpan 688	. . . . . 6 ⊢ (𝑦 ∈ ℕ → ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
51	50	iuneq2d 4940	. . . . 5 ⊢ (𝑦 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
52	51	adantr 483	. . . 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 14378	. . . . . . 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 688	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)))
55		oveq2 7158	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → (𝐷↑𝑟𝑗) = (𝐷↑𝑟𝑚))
56	55	cbviunv 4957	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) = ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)
57	56	coeq2i 5725	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚))
58		coiun 6103	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
59	57, 58	eqtri 2844	. . . . . 6 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
60	54, 59	syl6eq 2872	. . . . 5 ⊢ (𝑦 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
61	60	adantr 483	. . . 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 4013	. . 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 415	. 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 11650	1 ⊢ (𝑁 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ⊆ wss 3935  ∪ ciun 4911   ∘ ccom 5553  (class class class)co 7150  1c1 10532   + caddc 10534  ℕcn 11632  ↑_𝑟crelexp 14373

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-seq 13364  df-relexp 14374

This theorem is referenced by: (None)