Theorem trclrelexplem 44711

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 7428	. . . 4 ⊢ (𝑥 = 1 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟1))
2	1	iuneq2d 4980	. . 3 ⊢ (𝑥 = 1 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1))
3		oveq2 7428	. . 3 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1))
4	2, 3	sseq12d 3941	. 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 7428	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑦))
6	5	iuneq2d 4980	. . 3 ⊢ (𝑥 = 𝑦 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
7		oveq2 7428	. . 3 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦))
8	6, 7	sseq12d 3941	. 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 7428	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
10	9	iuneq2d 4980	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
11		oveq2 7428	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)))
12	10, 11	sseq12d 3941	. 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 7428	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑁))
14	13	iuneq2d 4980	. . 3 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁))
15		oveq2 7428	. . 3 ⊢ (𝑥 = 𝑁 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))
16	14, 15	sseq12d 3941	. 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 7428	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑙))
18	17	cbviunv 4996	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙)
19		oveq2 7428	. . . . . 6 ⊢ (𝑙 = 𝑗 → (𝐷↑𝑟𝑙) = (𝐷↑𝑟𝑗))
20	19	cbviunv 4996	. . . . 5 ⊢ ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
21	18, 20	eqtri 2783	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
22		ovex 7453	. . . . . 6 ⊢ (𝐷↑𝑟𝑘) ∈ V
23		relexp1g 15179	. . . . . 6 ⊢ ((𝐷↑𝑟𝑘) ∈ V → ((𝐷↑𝑟𝑘)↑𝑟1) = (𝐷↑𝑟𝑘))
24	22, 23	mp1i 14	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((𝐷↑𝑟𝑘)↑𝑟1) = (𝐷↑𝑟𝑘))
25	24	iuneq2i 4972	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘)
26		nnex 12341	. . . . . 6 ⊢ ℕ ∈ V
27		ovex 7453	. . . . . 6 ⊢ (𝐷↑𝑟𝑗) ∈ V
28	26, 27	iunex 7980	. . . . 5 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) ∈ V
29		relexp1g 15179	. . . . 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 2793	. . 3 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
32	31	eqimssi 3968	. 2 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
33		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑚))
34	33	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐷↑𝑟𝑘)↑𝑟𝑦) = ((𝐷↑𝑟𝑚)↑𝑟𝑦))
35	34, 33	coeq12d 5841	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
36	35	cbviunv 4996	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
37		ss2iun 4969	. . . . . . . 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 5006	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝐷↑𝑟𝑚)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
39		coss1 5832	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
41	37, 40	mprg 3082	. . . . . . 7 ⊢ ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
42	36, 41	eqsstri 3954	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
43		coss1 5832	. . . . . . . 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 3158	. . . . . . 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 4969	. . . . . . 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 18	. . . . . 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 3919	. . . . 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 487	. . . 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 15178	. . . . . . 7 ⊢ (((𝐷↑𝑟𝑘) ∈ V ∧ 𝑦 ∈ ℕ) → ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
50	22, 49	mpan 703	. . . . . 6 ⊢ (𝑦 ∈ ℕ → ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
51	50	iuneq2d 4980	. . . . 5 ⊢ (𝑦 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
52	51	adantr 486	. . . 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 15178	. . . . . . 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 703	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)))
55		oveq2 7428	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → (𝐷↑𝑟𝑗) = (𝐷↑𝑟𝑚))
56	55	cbviunv 4996	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) = ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)
57	56	coeq2i 5837	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚))
58		coiun 6258	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
59	57, 58	eqtri 2783	. . . . . 6 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
60	54, 59	eqtrdi 2811	. . . . 5 ⊢ (𝑦 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
61	60	adantr 486	. . . 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 3963	. . 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 418	. 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 12353	1 ⊢ (𝑁 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ∪ ciun 4950   ∘ ccom 5654  (class class class)co 7420  1c1 11201   + caddc 11203  ℕcn 12335  ↑_𝑟crelexp 15172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-seq 14145  df-relexp 15173

This theorem is used by: (None)