Theorem trclrelexplem 44164

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 7372	. . . 4 ⊢ (𝑥 = 1 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟1))
2	1	iuneq2d 4965	. . 3 ⊢ (𝑥 = 1 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1))
3		oveq2 7372	. . 3 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1))
4	2, 3	sseq12d 3956	. 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 7372	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑦))
6	5	iuneq2d 4965	. . 3 ⊢ (𝑥 = 𝑦 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
7		oveq2 7372	. . 3 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦))
8	6, 7	sseq12d 3956	. 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 7372	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
10	9	iuneq2d 4965	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)))
11		oveq2 7372	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)))
12	10, 11	sseq12d 3956	. 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 7372	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ((𝐷↑𝑟𝑘)↑𝑟𝑁))
14	13	iuneq2d 4965	. . 3 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑥) = ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁))
15		oveq2 7372	. . 3 ⊢ (𝑥 = 𝑁 → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))
16	14, 15	sseq12d 3956	. 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 7372	. . . . . 6 ⊢ (𝑘 = 𝑙 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑙))
18	17	cbviunv 4982	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙)
19		oveq2 7372	. . . . . 6 ⊢ (𝑙 = 𝑗 → (𝐷↑𝑟𝑙) = (𝐷↑𝑟𝑗))
20	19	cbviunv 4982	. . . . 5 ⊢ ∪ 𝑙 ∈ ℕ (𝐷↑𝑟𝑙) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
21	18, 20	eqtri 2760	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)
22		ovex 7397	. . . . . 6 ⊢ (𝐷↑𝑟𝑘) ∈ V
23		relexp1g 14985	. . . . . 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 4956	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝐷↑𝑟𝑘)
26		nnex 12177	. . . . . 6 ⊢ ℕ ∈ V
27		ovex 7397	. . . . . 6 ⊢ (𝐷↑𝑟𝑗) ∈ V
28	26, 27	iunex 7918	. . . . 5 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) ∈ V
29		relexp1g 14985	. . . . 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 2770	. . 3 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) = (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
32	31	eqimssi 3983	. 2 ⊢ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟1) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟1)
33		oveq2 7372	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝐷↑𝑟𝑘) = (𝐷↑𝑟𝑚))
34	33	oveq1d 7379	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → ((𝐷↑𝑟𝑘)↑𝑟𝑦) = ((𝐷↑𝑟𝑚)↑𝑟𝑦))
35	34, 33	coeq12d 5817	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)))
36	35	cbviunv 4982	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) = ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
37		ss2iun 4953	. . . . . . . 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 4992	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → ((𝐷↑𝑟𝑚)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦))
39		coss1 5808	. . . . . . . . 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 3058	. . . . . . 7 ⊢ ∪ 𝑚 ∈ ℕ (((𝐷↑𝑟𝑚)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
42	36, 41	eqsstri 3969	. . . . . 6 ⊢ ∪ 𝑘 ∈ ℕ (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)) ⊆ ∪ 𝑚 ∈ ℕ (∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
43		coss1 5808	. . . . . . . 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 3134	. . . . . . 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 4953	. . . . . . 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 3934	. . . . 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 14984	. . . . . . 7 ⊢ (((𝐷↑𝑟𝑘) ∈ V ∧ 𝑦 ∈ ℕ) → ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
50	22, 49	mpan 691	. . . . . 6 ⊢ (𝑦 ∈ ℕ → ((𝐷↑𝑟𝑘)↑𝑟(𝑦 + 1)) = (((𝐷↑𝑟𝑘)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑘)))
51	50	iuneq2d 4965	. . . . 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 14984	. . . . . . 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 691	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)))
55		oveq2 7372	. . . . . . . . 9 ⊢ (𝑗 = 𝑚 → (𝐷↑𝑟𝑗) = (𝐷↑𝑟𝑚))
56	55	cbviunv 4982	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗) = ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)
57	56	coeq2i 5813	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚))
58		coiun 6219	. . . . . . 7 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑚 ∈ ℕ (𝐷↑𝑟𝑚)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
59	57, 58	eqtri 2760	. . . . . 6 ⊢ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)) = ∪ 𝑚 ∈ ℕ ((∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑦) ∘ (𝐷↑𝑟𝑚))
60	54, 59	eqtrdi 2788	. . . . 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 3978	. . 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 12189	1 ⊢ (𝑁 ∈ ℕ → ∪ 𝑘 ∈ ℕ ((𝐷↑𝑟𝑘)↑𝑟𝑁) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890  ∪ ciun 4934   ∘ ccom 5632  (class class class)co 7364  1c1 11036   + caddc 11038  ℕcn 12171  ↑_𝑟crelexp 14978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686  ax-cnex 11091  ax-resscn 11092  ax-1cn 11093  ax-icn 11094  ax-addcl 11095  ax-addrcl 11096  ax-mulcl 11097  ax-mulrcl 11098  ax-mulcom 11099  ax-addass 11100  ax-mulass 11101  ax-distr 11102  ax-i2m1 11103  ax-1ne0 11104  ax-1rid 11105  ax-rnegex 11106  ax-rrecex 11107  ax-cnre 11108  ax-pre-lttri 11109  ax-pre-lttrn 11110  ax-pre-ltadd 11111  ax-pre-mulgt0 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5523  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5581  df-we 5583  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7321  df-ov 7367  df-oprab 7368  df-mpo 7369  df-om 7815  df-2nd 7940  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11178  df-mnf 11179  df-xr 11180  df-ltxr 11181  df-le 11182  df-sub 11376  df-neg 11377  df-nn 12172  df-n0 12435  df-z 12522  df-uz 12786  df-seq 13961  df-relexp 14979

This theorem is referenced by: (None)