Theorem nntopi 8104

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 2239	. . 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 2531	. 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 2239	. . 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 2531	. 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 2239	. . 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 2531	. 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 2239	. . 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 2531	. 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 7525	. . 3 ⊢ 1o ∈ N
11		eqid 2229	. . 3 ⊢ 1 = 1
12		opeq1 3860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 1o → ⟨𝑧, 1o⟩ = ⟨1o, 1o⟩)
13	12	eceq1d 6733	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
14		df-1nqqs 7561	. . . . . . . . . . . . . . . 16 ⊢ 1Q = [⟨1o, 1o⟩] ~Q
15	13, 14	eqtr4di 2280	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 1o → [⟨𝑧, 1o⟩] ~Q = 1Q)
16	15	breq2d 4098	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → (𝑙 <Q [⟨𝑧, 1o⟩] ~Q ↔ 𝑙 <Q 1Q))
17	16	abbidv 2347	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1o → {𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q 1Q})
18	15	breq1d 4096	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 1o → ([⟨𝑧, 1o⟩] ~Q <Q 𝑢 ↔ 1Q <Q 𝑢))
19	18	abbidv 2347	. . . . . . . . . . . . 13 ⊢ (𝑧 = 1o → {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
20	17, 19	opeq12d 3868	. . . . . . . . . . . 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 7677	. . . . . . . . . . . 12 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
22	20, 21	eqtr4di 2280	. . . . . . . . . . 11 ⊢ (𝑧 = 1o → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ = 1P)
23	22	oveq1d 6028	. . . . . . . . . 10 ⊢ (𝑧 = 1o → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (1P +P 1P))
24	23	opeq1d 3866	. . . . . . . . 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 6733	. . . . . . . 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 7942	. . . . . . . 8 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
27	25, 26	eqtr4di 2280	. . . . . . 7 ⊢ (𝑧 = 1o → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = 1R)
28	27	opeq1d 3866	. . . . . 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 8030	. . . . . 6 ⊢ 1 = ⟨1R, 0R⟩
30	28, 29	eqtr4di 2280	. . . . 5 ⊢ (𝑧 = 1o → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1)
31	30	eqeq1d 2238	. . . 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 2908	. . 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 7537	. . . . . . 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 8057	. . . . . . . 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 6028	. . . . . . 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 2264	. . . . . 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 3860	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑧 +N 1o) → ⟨𝑣, 1o⟩ = ⟨(𝑧 +N 1o), 1o⟩)
43	42	eceq1d 6733	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑧 +N 1o) → [⟨𝑣, 1o⟩] ~Q = [⟨(𝑧 +N 1o), 1o⟩] ~Q )
44	43	breq2d 4098	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 +N 1o) → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q ↔ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q ))
45	44	abbidv 2347	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 +N 1o) → {𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨(𝑧 +N 1o), 1o⟩] ~Q })
46	43	breq1d 4096	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑧 +N 1o) → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢))
47	46	abbidv 2347	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑧 +N 1o) → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑧 +N 1o), 1o⟩] ~Q <Q 𝑢})
48	45, 47	opeq12d 3868	. . . . . . . . . . . 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 6028	. . . . . . . . . . 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 3866	. . . . . . . . . 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 6733	. . . . . . . . 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 3866	. . . . . . . 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 2238	. . . . . . 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 2908	. . . . . 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 2648	. . 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 3860	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑧 → ⟨𝑣, 1o⟩ = ⟨𝑧, 1o⟩)
59	58	eceq1d 6733	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑧 → [⟨𝑣, 1o⟩] ~Q = [⟨𝑧, 1o⟩] ~Q )
60	59	breq2d 4098	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → (𝑙 <Q [⟨𝑣, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q ))
61	60	abbidv 2347	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → {𝑙 ∣ 𝑙 <Q [⟨𝑣, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑧, 1o⟩] ~Q })
62	59	breq1d 4096	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑧 → ([⟨𝑣, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑧, 1o⟩] ~Q <Q 𝑢))
63	62	abbidv 2347	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → {𝑢 ∣ [⟨𝑣, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑧, 1o⟩] ~Q <Q 𝑢})
64	61, 63	opeq12d 3868	. . . . . . . . 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 6028	. . . . . . . 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 3866	. . . . . . 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 6733	. . . . . 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 3866	. . . . 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 2238	. . . 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 2766	. . 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 8103	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 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  ⟨cop 3670  ∩ cint 3926   class class class wbr 4086  (class class class)co 6013  1_oc1o 6570  [cec 6695  Ncnpi 7482   +_N cpli 7483   ~_Q ceq 7489  1_Qc1q 7491   <_Q cltq 7495  1_Pc1p 7502   +_P cpp 7503   ~_R cer 7506  0_Rc0r 7508  1_Rc1r 7509  1c1 8023   + caddc 8025

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-i1p 7677  df-iplp 7678  df-enr 7936  df-nr 7937  df-plr 7938  df-0r 7941  df-1r 7942  df-c 8028  df-1 8030  df-r 8032  df-add 8033

This theorem is referenced by:  axcaucvglemres  8109