Theorem nntopi 7954

Description: Mapping from ℕ to N. (Contributed by Jim Kingdon, 13-Jul-2021.)

Hypothesis

Ref	Expression
nntopi.n	⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}

Assertion

Ref	Expression
nntopi	⊢ (𝐴 ∈ 𝑁 → ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴)

Distinct variable groups:   𝑥,𝑦   𝑧,𝐴   𝑧,𝑁,𝑦,𝑥   𝑢,𝑙,𝑧,𝑦,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑢,𝑙)   𝑁(𝑢,𝑙)

Proof of Theorem nntopi

Dummy variables 𝑤 𝑘 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nntopi.n	. 2 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
2		eqeq2 2203	. . 3 ⊢ (𝑤 = 1 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1))
3	2	rexbidv 2495	. 2 ⊢ (𝑤 = 1 → (∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1))
4		eqeq2 2203	. . 3 ⊢ (𝑤 = 𝑘 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘))
5	4	rexbidv 2495	. 2 ⊢ (𝑤 = 𝑘 → (∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘))
6		eqeq2 2203	. . 3 ⊢ (𝑤 = (𝑘 + 1) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
7	6	rexbidv 2495	. 2 ⊢ (𝑤 = (𝑘 + 1) → (∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
8		eqeq2 2203	. . 3 ⊢ (𝑤 = 𝐴 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴))
9	8	rexbidv 2495	. 2 ⊢ (𝑤 = 𝐴 → (∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑤 ↔ ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴))
10		1pi 7375	. . 3 ⊢ 1o ∈ N
11		eqid 2193	. . 3 ⊢ 1 = 1
12		opeq1 3804	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 1o → ⟨𝑧, 1o⟩ = ⟨1o, 1o⟩)
13	12	eceq1d 6623	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
14		df-1nqqs 7411	. . . . . . . . . . . . . . . 16 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
15	13, 14	eqtr4di 2244	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = 1Q)
16	15	breq2d 4041	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → (𝑙 <Q [⟨𝑧, 1o⟩] ~Q ↔ 𝑙 <Q 1Q))
17	16	abbidv 2311	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1o → {𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q 1Q})
18	15	breq1d 4039	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → ([⟨𝑧, 1o⟩] ~Q <Q 𝑢 ↔ 1Q <Q 𝑢))
19	18	abbidv 2311	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1o → {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
20	17, 19	opeq12d 3812	. . . . . . . . . . . 12 ⊢ (𝑧 = 1o → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩)
21		df-i1p 7527	. . . . . . . . . . . 12 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
22	20, 21	eqtr4di 2244	. . . . . . . . . . 11 ⊢ (𝑧 = 1o → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ = 1P)
23	22	oveq1d 5933	. . . . . . . . . 10 ⊢ (𝑧 = 1o → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (1P +P 1P))
24	23	opeq1d 3810	. . . . . . . . 9 ⊢ (𝑧 = 1o → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(1P +P 1P), 1P⟩)
25	24	eceq1d 6623	. . . . . . . 8 ⊢ (𝑧 = 1o → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R )
26		df-1r 7792	. . . . . . . 8 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
27	25, 26	eqtr4di 2244	. . . . . . 7 ⊢ (𝑧 = 1o → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R)
28	27	opeq1d 3810	. . . . . 6 ⊢ (𝑧 = 1o → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨1R, 0R⟩)
29		df-1 7880	. . . . . 6 ⊢ 1 = ⟨1R, 0R⟩
30	28, 29	eqtr4di 2244	. . . . 5 ⊢ (𝑧 = 1o → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
31	30	eqeq1d 2202	. . . 4 ⊢ (𝑧 = 1o → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1 ↔ 1 = 1))
32	31	rspcev 2864	. . 3 ⊢ ((1o ∈ N ∧ 1 = 1) → ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
33	10, 11, 32	mp2an 426	. 2 ⊢ ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
34		simplr 528	. . . . . . 7 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑧 ∈ N) ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → 𝑧 ∈ N)
35		addclpi 7387	. . . . . . 7 ⊢ ((𝑧 ∈ N ∧ 1o ∈ N) → (𝑧 +N 1o) ∈ N)
36	34, 10, 35	sylancl 413	. . . . . 6 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑧 ∈ N) ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → (𝑧 +N 1o) ∈ N)
37		pitonnlem2 7907	. . . . . . . 8 ⊢ (𝑧 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
38	34, 37	syl 14	. . . . . . 7 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑧 ∈ N) ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
39		simpr 110	. . . . . . . 8 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑧 ∈ N) ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘)
40	39	oveq1d 5933	. . . . . . 7 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑧 ∈ N) ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = (𝑘 + 1))
41	38, 40	eqtr3d 2228	. . . . . 6 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑧 ∈ N) ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1))
42		opeq1 3804	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑧 +N 1o) → ⟨𝑣, 1o⟩ = ⟨(𝑧 +N 1o), 1o⟩)
43	42	eceq1d 6623	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 +N 1o) → [⟨𝑣, 1o⟩] ~Q = [⟨(𝑧 +N 1o), 1o⟩] ~Q )
44	43	breq2d 4041	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 +N 1o) → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q ↔ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q ))
45	44	abbidv 2311	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 +N 1o) → {𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q })
46	43	breq1d 4039	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 +N 1o) → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢))
47	46	abbidv 2311	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 +N 1o) → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢})
48	45, 47	opeq12d 3812	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑧 +N 1o) → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩)
49	48	oveq1d 5933	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑧 +N 1o) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
50	49	opeq1d 3810	. . . . . . . . . 10 ⊢ (𝑣 = (𝑧 +N 1o) → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
51	50	eceq1d 6623	. . . . . . . . 9 ⊢ (𝑣 = (𝑧 +N 1o) → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
52	51	opeq1d 3810	. . . . . . . 8 ⊢ (𝑣 = (𝑧 +N 1o) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
53	52	eqeq1d 2202	. . . . . . 7 ⊢ (𝑣 = (𝑧 +N 1o) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1) ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
54	53	rspcev 2864	. . . . . 6 ⊢ (((𝑧 +N 1o) ∈ N ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)) → ∃𝑣 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1))
55	36, 41, 54	syl2anc 411	. . . . 5 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑧 ∈ N) ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘) → ∃𝑣 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1))
56	55	ex 115	. . . 4 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑧 ∈ N) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘 → ∃𝑣 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
57	56	rexlimdva 2611	. . 3 ⊢ (𝑘 ∈ 𝑁 → (∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘 → ∃𝑣 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
58		opeq1 3804	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ⟨𝑣, 1o⟩ = ⟨𝑧, 1o⟩)
59	58	eceq1d 6623	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → [⟨𝑣, 1o⟩] ~Q = [⟨𝑧, 1o⟩] ~Q )
60	59	breq2d 4041	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q ))
61	60	abbidv 2311	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → {𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q })
62	59	breq1d 4039	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑧, 1o⟩] ~Q <Q 𝑢))
63	62	abbidv 2311	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢})
64	61, 63	opeq12d 3812	. . . . . . . . 9 ⊢ (𝑣 = 𝑧 → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩)
65	64	oveq1d 5933	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
66	65	opeq1d 3810	. . . . . . 7 ⊢ (𝑣 = 𝑧 → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
67	66	eceq1d 6623	. . . . . 6 ⊢ (𝑣 = 𝑧 → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
68	67	opeq1d 3810	. . . . 5 ⊢ (𝑣 = 𝑧 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
69	68	eqeq1d 2202	. . . 4 ⊢ (𝑣 = 𝑧 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1) ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
70	69	cbvrexv 2727	. . 3 ⊢ (∃𝑣 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1) ↔ ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1))
71	57, 70	imbitrdi 161	. 2 ⊢ (𝑘 ∈ 𝑁 → (∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝑘 → ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = (𝑘 + 1)))
72	1, 3, 5, 7, 9, 33, 71	nnindnn 7953	1 ⊢ (𝐴 ∈ 𝑁 → ∃𝑧 ∈ N ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  ⟨cop 3621  ∩ cint 3870   class class class wbr 4029  (class class class)co 5918  1_oc1o 6462  [cec 6585  Ncnpi 7332   +_N cpli 7333   ~_Q ceq 7339  1_Qc1q 7341   <_Q cltq 7345  1_Pc1p 7352   +_P cpp 7353   ~_R cer 7356  0_Rc0r 7358  1_Rc1r 7359  1c1 7873   + caddc 7875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-i1p 7527  df-iplp 7528  df-enr 7786  df-nr 7787  df-plr 7788  df-0r 7791  df-1r 7792  df-c 7878  df-1 7880  df-r 7882  df-add 7883

This theorem is referenced by:  axcaucvglemres  7959