Theorem nntopi 7978

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 2206	. . 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 2498	. 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 2206	. . 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 2498	. 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 2206	. . 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 2498	. 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 2206	. . 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 2498	. 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 7399	. . 3 ⊢ 1o ∈ N
11		eqid 2196	. . 3 ⊢ 1 = 1
12		opeq1 3809	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 1o → ⟨𝑧, 1o⟩ = ⟨1o, 1o⟩)
13	12	eceq1d 6637	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
14		df-1nqqs 7435	. . . . . . . . . . . . . . . 16 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
15	13, 14	eqtr4di 2247	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = 1Q)
16	15	breq2d 4046	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → (𝑙 <Q [⟨𝑧, 1o⟩] ~Q ↔ 𝑙 <Q 1Q))
17	16	abbidv 2314	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1o → {𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q 1Q})
18	15	breq1d 4044	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → ([⟨𝑧, 1o⟩] ~Q <Q 𝑢 ↔ 1Q <Q 𝑢))
19	18	abbidv 2314	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1o → {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
20	17, 19	opeq12d 3817	. . . . . . . . . . . 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 7551	. . . . . . . . . . . 12 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
22	20, 21	eqtr4di 2247	. . . . . . . . . . 11 ⊢ (𝑧 = 1o → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ = 1P)
23	22	oveq1d 5940	. . . . . . . . . 10 ⊢ (𝑧 = 1o → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (1P +P 1P))
24	23	opeq1d 3815	. . . . . . . . 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 6637	. . . . . . . 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 7816	. . . . . . . 8 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
27	25, 26	eqtr4di 2247	. . . . . . 7 ⊢ (𝑧 = 1o → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R)
28	27	opeq1d 3815	. . . . . 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 7904	. . . . . 6 ⊢ 1 = ⟨1R, 0R⟩
30	28, 29	eqtr4di 2247	. . . . 5 ⊢ (𝑧 = 1o → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
31	30	eqeq1d 2205	. . . 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 2868	. . 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 7411	. . . . . . 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 7931	. . . . . . . 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 5940	. . . . . . 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 2231	. . . . . 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 3809	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑧 +N 1o) → ⟨𝑣, 1o⟩ = ⟨(𝑧 +N 1o), 1o⟩)
43	42	eceq1d 6637	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 +N 1o) → [⟨𝑣, 1o⟩] ~Q = [⟨(𝑧 +N 1o), 1o⟩] ~Q )
44	43	breq2d 4046	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 +N 1o) → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q ↔ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q ))
45	44	abbidv 2314	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 +N 1o) → {𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q })
46	43	breq1d 4044	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 +N 1o) → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢))
47	46	abbidv 2314	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 +N 1o) → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢})
48	45, 47	opeq12d 3817	. . . . . . . . . . . 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 5940	. . . . . . . . . . 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 3815	. . . . . . . . . 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 6637	. . . . . . . . 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 3815	. . . . . . . 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 2205	. . . . . . 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 2868	. . . . . 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 2614	. . 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 3809	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ⟨𝑣, 1o⟩ = ⟨𝑧, 1o⟩)
59	58	eceq1d 6637	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → [⟨𝑣, 1o⟩] ~Q = [⟨𝑧, 1o⟩] ~Q )
60	59	breq2d 4046	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q ))
61	60	abbidv 2314	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → {𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q })
62	59	breq1d 4044	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑧, 1o⟩] ~Q <Q 𝑢))
63	62	abbidv 2314	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢})
64	61, 63	opeq12d 3817	. . . . . . . . 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 5940	. . . . . . . 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 3815	. . . . . . 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 6637	. . . . . 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 3815	. . . . 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 2205	. . . 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 2730	. . 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 7977	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 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  ⟨cop 3626  ∩ cint 3875   class class class wbr 4034  (class class class)co 5925  1_oc1o 6476  [cec 6599  Ncnpi 7356   +_N cpli 7357   ~_Q ceq 7363  1_Qc1q 7365   <_Q cltq 7369  1_Pc1p 7376   +_P cpp 7377   ~_R cer 7380  0_Rc0r 7382  1_Rc1r 7383  1c1 7897   + caddc 7899

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-enr 7810  df-nr 7811  df-plr 7812  df-0r 7815  df-1r 7816  df-c 7902  df-1 7904  df-r 7906  df-add 7907

This theorem is referenced by:  axcaucvglemres  7983