Theorem trclrelexplem 40076

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 7164	. . . 4 ⊢ (𝑥 = 1 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟1))
2	1	iuneq2d 4948	. . 3 ⊢ (𝑥 = 1 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1))
3		oveq2 7164	. . 3 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1))
4	2, 3	sseq12d 4000	. 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 7164	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑦))
6	5	iuneq2d 4948	. . 3 ⊢ (𝑥 = 𝑦 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
7		oveq2 7164	. . 3 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦))
8	6, 7	sseq12d 4000	. 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 7164	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
10	9	iuneq2d 4948	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
11		oveq2 7164	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)))
12	10, 11	sseq12d 4000	. 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 7164	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑁))
14	13	iuneq2d 4948	. . 3 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁))
15		oveq2 7164	. . 3 ⊢ (𝑥 = 𝑁 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))
16	14, 15	sseq12d 4000	. 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 7164	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑙))
18	17	cbviunv 4965	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙)
19		oveq2 7164	. . . . . 6 ⊢ (𝑙 = 𝑗 → (𝐷↑𝑟𝑙) = (𝐷↑𝑟𝑗))
20	19	cbviunv 4965	. . . . 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 7189	. . . . . 6 ⊢ (𝐷↑𝑟𝑘) ∈ V
23		relexp1g 14385	. . . . . 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 4940	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘)
26		nnex 11644	. . . . . 6 ⊢ ℕ ∈ V
27		ovex 7189	. . . . . 6 ⊢ (𝐷↑𝑟𝑗) ∈ V
28	26, 27	iunex 7669	. . . . 5 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) ∈ V
29		relexp1g 14385	. . . . 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 4025	. 2 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
33		oveq2 7164	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑚))
34	33	oveq1d 7171	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐷↑𝑟𝑘)↑𝑟𝑦) = ((𝐷↑𝑟𝑚)↑𝑟𝑦))
35	34, 33	coeq12d 5735	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
36	35	cbviunv 4965	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
37		ss2iun 4937	. . . . . . . 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 4972	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝐷↑𝑟𝑚)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
39		coss1 5726	. . . . . . . . 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 4001	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
43		coss1 5726	. . . . . . . 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 4937	. . . . . . 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 3978	. . . . 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 14384	. . . . . . 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 4948	. . . . 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 14384	. . . . . . 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 7164	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → (𝐷↑𝑟𝑗) = (𝐷↑𝑟𝑚))
56	55	cbviunv 4965	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) = ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)
57	56	coeq2i 5731	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚))
58		coiun 6109	. . . . . . 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 4014	. . 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 11656	1 ⊢ (𝑁 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ⊆ wss 3936  ∪ ciun 4919   ∘ ccom 5559  (class class class)co 7156  1c1 10538   + caddc 10540  ℕcn 11638  ↑_𝑟crelexp 14379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-seq 13371  df-relexp 14380

This theorem is referenced by: (None)