Theorem nntopi 8006

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 2214	. . 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 2506	. 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 2214	. . 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 2506	. 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 2214	. . 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 2506	. 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 2214	. . 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 2506	. 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 7427	. . 3 ⊢ 1o ∈ N
11		eqid 2204	. . 3 ⊢ 1 = 1
12		opeq1 3818	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 1o → ⟨𝑧, 1o⟩ = ⟨1o, 1o⟩)
13	12	eceq1d 6655	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
14		df-1nqqs 7463	. . . . . . . . . . . . . . . 16 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
15	13, 14	eqtr4di 2255	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = 1Q)
16	15	breq2d 4055	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → (𝑙 <Q [⟨𝑧, 1o⟩] ~Q ↔ 𝑙 <Q 1Q))
17	16	abbidv 2322	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1o → {𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q 1Q})
18	15	breq1d 4053	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → ([⟨𝑧, 1o⟩] ~Q <Q 𝑢 ↔ 1Q <Q 𝑢))
19	18	abbidv 2322	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1o → {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
20	17, 19	opeq12d 3826	. . . . . . . . . . . 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 7579	. . . . . . . . . . . 12 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
22	20, 21	eqtr4di 2255	. . . . . . . . . . 11 ⊢ (𝑧 = 1o → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ = 1P)
23	22	oveq1d 5958	. . . . . . . . . 10 ⊢ (𝑧 = 1o → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (1P +P 1P))
24	23	opeq1d 3824	. . . . . . . . 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 6655	. . . . . . . 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 7844	. . . . . . . 8 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
27	25, 26	eqtr4di 2255	. . . . . . 7 ⊢ (𝑧 = 1o → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R)
28	27	opeq1d 3824	. . . . . 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 7932	. . . . . 6 ⊢ 1 = ⟨1R, 0R⟩
30	28, 29	eqtr4di 2255	. . . . 5 ⊢ (𝑧 = 1o → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
31	30	eqeq1d 2213	. . . 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 2876	. . 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 7439	. . . . . . 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 7959	. . . . . . . 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 5958	. . . . . . 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 2239	. . . . . 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 3818	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑧 +N 1o) → ⟨𝑣, 1o⟩ = ⟨(𝑧 +N 1o), 1o⟩)
43	42	eceq1d 6655	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 +N 1o) → [⟨𝑣, 1o⟩] ~Q = [⟨(𝑧 +N 1o), 1o⟩] ~Q )
44	43	breq2d 4055	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 +N 1o) → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q ↔ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q ))
45	44	abbidv 2322	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 +N 1o) → {𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q })
46	43	breq1d 4053	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 +N 1o) → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢))
47	46	abbidv 2322	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 +N 1o) → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢})
48	45, 47	opeq12d 3826	. . . . . . . . . . . 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 5958	. . . . . . . . . . 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 3824	. . . . . . . . . 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 6655	. . . . . . . . 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 3824	. . . . . . . 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 2213	. . . . . . 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 2876	. . . . . 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 2622	. . 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 3818	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ⟨𝑣, 1o⟩ = ⟨𝑧, 1o⟩)
59	58	eceq1d 6655	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → [⟨𝑣, 1o⟩] ~Q = [⟨𝑧, 1o⟩] ~Q )
60	59	breq2d 4055	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q ))
61	60	abbidv 2322	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → {𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q })
62	59	breq1d 4053	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑧, 1o⟩] ~Q <Q 𝑢))
63	62	abbidv 2322	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢})
64	61, 63	opeq12d 3826	. . . . . . . . 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 5958	. . . . . . . 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 3824	. . . . . . 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 6655	. . . . . 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 3824	. . . . 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 2213	. . . 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 2738	. . 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 8005	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 1372   ∈ wcel 2175  {cab 2190  ∀wral 2483  ∃wrex 2484  ⟨cop 3635  ∩ cint 3884   class class class wbr 4043  (class class class)co 5943  1_oc1o 6494  [cec 6617  Ncnpi 7384   +_N cpli 7385   ~_Q ceq 7391  1_Qc1q 7393   <_Q cltq 7397  1_Pc1p 7404   +_P cpp 7405   ~_R cer 7408  0_Rc0r 7410  1_Rc1r 7411  1c1 7925   + caddc 7927

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-i1p 7579  df-iplp 7580  df-enr 7838  df-nr 7839  df-plr 7840  df-0r 7843  df-1r 7844  df-c 7930  df-1 7932  df-r 7934  df-add 7935

This theorem is referenced by:  axcaucvglemres  8011