Theorem nninfsellemdc 11559

Description: Lemma for nninfself 11562. Showing that the selection function is well defined. (Contributed by Jim Kingdon, 8-Aug-2022.)

Assertion

Ref	Expression
nninfsellemdc	⊢ ((𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑁 ∈ ω) → DECID ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)

Distinct variable groups:   𝑘,𝑁   𝑄,𝑘   𝑖,𝑘

Allowed substitution hints:   𝑄(𝑖)   𝑁(𝑖)

Proof of Theorem nninfsellemdc

Dummy variables 𝑤 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceq 4220	. . . . . 6 ⊢ (𝑤 = ∅ → suc 𝑤 = suc ∅)
2	1	raleqdv 2568	. . . . 5 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc ∅(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
3	2	dcbid 786	. . . 4 ⊢ (𝑤 = ∅ → (DECID ∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ DECID ∀𝑘 ∈ suc ∅(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
4	3	imbi2d 228	. . 3 ⊢ (𝑤 = ∅ → ((𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜) ↔ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc ∅(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)))
5		suceq 4220	. . . . . 6 ⊢ (𝑤 = 𝑗 → suc 𝑤 = suc 𝑗)
6	5	raleqdv 2568	. . . . 5 ⊢ (𝑤 = 𝑗 → (∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
7	6	dcbid 786	. . . 4 ⊢ (𝑤 = 𝑗 → (DECID ∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ DECID ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
8	7	imbi2d 228	. . 3 ⊢ (𝑤 = 𝑗 → ((𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜) ↔ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)))
9		suceq 4220	. . . . . 6 ⊢ (𝑤 = suc 𝑗 → suc 𝑤 = suc suc 𝑗)
10	9	raleqdv 2568	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
11	10	dcbid 786	. . . 4 ⊢ (𝑤 = suc 𝑗 → (DECID ∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ DECID ∀𝑘 ∈ suc suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
12	11	imbi2d 228	. . 3 ⊢ (𝑤 = suc 𝑗 → ((𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜) ↔ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)))
13		suceq 4220	. . . . . 6 ⊢ (𝑤 = 𝑁 → suc 𝑤 = suc 𝑁)
14	13	raleqdv 2568	. . . . 5 ⊢ (𝑤 = 𝑁 → (∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
15	14	dcbid 786	. . . 4 ⊢ (𝑤 = 𝑁 → (DECID ∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ DECID ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
16	15	imbi2d 228	. . 3 ⊢ (𝑤 = 𝑁 → ((𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc 𝑤(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜) ↔ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)))
17		elmapi 6407	. . . . . . 7 ⊢ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → 𝑄:ℕ∞⟶2𝑜)
18		peano1 4399	. . . . . . . 8 ⊢ ∅ ∈ ω
19		nnnninf 6785	. . . . . . . 8 ⊢ (∅ ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅)) ∈ ℕ∞)
20	18, 19	mp1i 10	. . . . . . 7 ⊢ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅)) ∈ ℕ∞)
21	17, 20	ffvelrnd 5419	. . . . . 6 ⊢ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) ∈ 2𝑜)
22		2onn 6260	. . . . . 6 ⊢ 2𝑜 ∈ ω
23		elnn 4410	. . . . . 6 ⊢ (((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) ∈ 2𝑜 ∧ 2𝑜 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) ∈ ω)
24	21, 22, 23	sylancl 404	. . . . 5 ⊢ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) ∈ ω)
25		1onn 6259	. . . . 5 ⊢ 1𝑜 ∈ ω
26		nndceq 6242	. . . . 5 ⊢ (((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) ∈ ω ∧ 1𝑜 ∈ ω) → DECID (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) = 1𝑜)
27	24, 25, 26	sylancl 404	. . . 4 ⊢ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) = 1𝑜)
28		suc0 4229	. . . . . . 7 ⊢ suc ∅ = {∅}
29	28	raleqi 2566	. . . . . 6 ⊢ (∀𝑘 ∈ suc ∅(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ {∅} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
30		0ex 3958	. . . . . . 7 ⊢ ∅ ∈ V
31		eleq2 2151	. . . . . . . . . . 11 ⊢ (𝑘 = ∅ → (𝑖 ∈ 𝑘 ↔ 𝑖 ∈ ∅))
32	31	ifbid 3408	. . . . . . . . . 10 ⊢ (𝑘 = ∅ → if(𝑖 ∈ 𝑘, 1𝑜, ∅) = if(𝑖 ∈ ∅, 1𝑜, ∅))
33	32	mpteq2dv 3921	. . . . . . . . 9 ⊢ (𝑘 = ∅ → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅)))
34	33	fveq2d 5293	. . . . . . . 8 ⊢ (𝑘 = ∅ → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))))
35	34	eqeq1d 2096	. . . . . . 7 ⊢ (𝑘 = ∅ → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) = 1𝑜))
36	30, 35	ralsn 3481	. . . . . 6 ⊢ (∀𝑘 ∈ {∅} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) = 1𝑜)
37	29, 36	bitri 182	. . . . 5 ⊢ (∀𝑘 ∈ suc ∅(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) = 1𝑜)
38	37	dcbii 785	. . . 4 ⊢ (DECID ∀𝑘 ∈ suc ∅(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ DECID (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1𝑜, ∅))) = 1𝑜)
39	27, 38	sylibr 132	. . 3 ⊢ (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc ∅(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
40	17	adantl 271	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → 𝑄:ℕ∞⟶2𝑜)
41		peano2 4400	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
42	41	adantr 270	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → suc 𝑗 ∈ ω)
43		nnnninf 6785	. . . . . . . . . . . . 13 ⊢ (suc 𝑗 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅)) ∈ ℕ∞)
44	42, 43	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅)) ∈ ℕ∞)
45	40, 44	ffvelrnd 5419	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) ∈ 2𝑜)
46		elnn 4410	. . . . . . . . . . 11 ⊢ (((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) ∈ 2𝑜 ∧ 2𝑜 ∈ ω) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) ∈ ω)
47	45, 22, 46	sylancl 404	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) ∈ ω)
48		nndceq 6242	. . . . . . . . . 10 ⊢ (((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) ∈ ω ∧ 1𝑜 ∈ ω) → DECID (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) = 1𝑜)
49	47, 25, 48	sylancl 404	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → DECID (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) = 1𝑜)
50		eleq2 2151	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = suc 𝑗 → (𝑖 ∈ 𝑘 ↔ 𝑖 ∈ suc 𝑗))
51	50	ifbid 3408	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = suc 𝑗 → if(𝑖 ∈ 𝑘, 1𝑜, ∅) = if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))
52	51	mpteq2dv 3921	. . . . . . . . . . . . . 14 ⊢ (𝑘 = suc 𝑗 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅)))
53	52	fveq2d 5293	. . . . . . . . . . . . 13 ⊢ (𝑘 = suc 𝑗 → (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))))
54	53	eqeq1d 2096	. . . . . . . . . . . 12 ⊢ (𝑘 = suc 𝑗 → ((𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) = 1𝑜))
55	54	ralsng 3478	. . . . . . . . . . 11 ⊢ (suc 𝑗 ∈ ω → (∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) = 1𝑜))
56	42, 55	syl 14	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → (∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) = 1𝑜))
57	56	dcbid 786	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → (DECID ∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ DECID (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑗, 1𝑜, ∅))) = 1𝑜))
58	49, 57	mpbird 165	. . . . . . . 8 ⊢ ((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) → DECID ∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
59		dcan 880	. . . . . . . 8 ⊢ (DECID ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 → (DECID ∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 → DECID (∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ∧ ∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)))
60	58, 59	mpan9 275	. . . . . . 7 ⊢ (((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) ∧ DECID ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜) → DECID (∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ∧ ∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
61		ralunb 3179	. . . . . . . 8 ⊢ (∀𝑘 ∈ (suc 𝑗 ∪ {suc 𝑗})(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ (∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ∧ ∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
62	61	dcbii 785	. . . . . . 7 ⊢ (DECID ∀𝑘 ∈ (suc 𝑗 ∪ {suc 𝑗})(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ DECID (∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ∧ ∀𝑘 ∈ {suc 𝑗} (𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
63	60, 62	sylibr 132	. . . . . 6 ⊢ (((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) ∧ DECID ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜) → DECID ∀𝑘 ∈ (suc 𝑗 ∪ {suc 𝑗})(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
64		df-suc 4189	. . . . . . . 8 ⊢ suc suc 𝑗 = (suc 𝑗 ∪ {suc 𝑗})
65	64	raleqi 2566	. . . . . . 7 ⊢ (∀𝑘 ∈ suc suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ ∀𝑘 ∈ (suc 𝑗 ∪ {suc 𝑗})(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
66	65	dcbii 785	. . . . . 6 ⊢ (DECID ∀𝑘 ∈ suc suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 ↔ DECID ∀𝑘 ∈ (suc 𝑗 ∪ {suc 𝑗})(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
67	63, 66	sylibr 132	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ 𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞)) ∧ DECID ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜) → DECID ∀𝑘 ∈ suc suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)
68	67	exp31 356	. . . 4 ⊢ (𝑗 ∈ ω → (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → (DECID ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜 → DECID ∀𝑘 ∈ suc suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)))
69	68	a2d 26	. . 3 ⊢ (𝑗 ∈ ω → ((𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜) → (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc suc 𝑗(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)))
70	4, 8, 12, 16, 39, 69	finds 4405	. 2 ⊢ (𝑁 ∈ ω → (𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) → DECID ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜))
71	70	impcom 123	1 ⊢ ((𝑄 ∈ (2𝑜 ↑𝑚 ℕ∞) ∧ 𝑁 ∈ ω) → DECID ∀𝑘 ∈ suc 𝑁(𝑄‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑘, 1𝑜, ∅))) = 1𝑜)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  DECID wdc 780   = wceq 1289   ∈ wcel 1438  ∀wral 2359   ∪ cun 2995  ∅c0 3284  ifcif 3389  {csn 3441   ↦ cmpt 3891  suc csuc 4183  ωcom 4395  ⟶wf 4998  ‘cfv 5002  (class class class)co 5634  1_𝑜c1o 6156  2_𝑜c2o 6157   ↑_𝑚 cmap 6385  ℕ_∞xnninf 6768

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-nninf 6770

This theorem is referenced by:  nninfsellemcl  11560  nninfsellemsuc  11561