Theorem pitonnlem2 7846

Description: Lemma for pitonn 7847. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)

Assertion

Ref	Expression
pitonnlem2	⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

Distinct variable group:   𝐾,𝑙,𝑢

Proof of Theorem pitonnlem2

Step	Hyp	Ref	Expression
1		df-1 7819	. . . 4 ⊢ 1 = ⟨1R, 0R⟩
2	1	oveq2i 5886	. . 3 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩)
3		nnprlu 7552	. . . . . . . 8 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
4		1pr 7553	. . . . . . . 8 ⊢ 1P ∈ P
5		addclpr 7536	. . . . . . . 8 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1P ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
6	3, 4, 5	sylancl 413	. . . . . . 7 ⊢ (𝐾 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
7		opelxpi 4659	. . . . . . 7 ⊢ (((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1P ∈ P) → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
8	6, 4, 7	sylancl 413	. . . . . 6 ⊢ (𝐾 ∈ N → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
9		enrex 7736	. . . . . . 7 ⊢ ~R ∈ V
10	9	ecelqsi 6589	. . . . . 6 ⊢ (⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P) → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
11	8, 10	syl 14	. . . . 5 ⊢ (𝐾 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
12		df-nr 7726	. . . . 5 ⊢ R = ((P × P) / ~R )
13	11, 12	eleqtrrdi 2271	. . . 4 ⊢ (𝐾 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
14		1sr 7750	. . . 4 ⊢ 1R ∈ R
15		addresr 7836	. . . 4 ⊢ (([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R ∧ 1R ∈ R) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩) = ⟨([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
16	13, 14, 15	sylancl 413	. . 3 ⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩) = ⟨([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
17	2, 16	eqtrid 2222	. 2 ⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
18		pitonnlem1p1 7845	. . . . 5 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P → [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
19	6, 18	syl 14	. . . 4 ⊢ (𝐾 ∈ N → [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
20		df-1r 7731	. . . . . 6 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
21	20	oveq2i 5886	. . . . 5 ⊢ ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
22		addclpr 7536	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
23	4, 4, 22	mp2an 426	. . . . . . 7 ⊢ (1P +P 1P) ∈ P
24		addsrpr 7744	. . . . . . . 8 ⊢ ((((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1P ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
25	4, 24	mpanl2 435	. . . . . . 7 ⊢ (((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
26	23, 4, 25	mpanr12 439	. . . . . 6 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
27	6, 26	syl 14	. . . . 5 ⊢ (𝐾 ∈ N → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
28	21, 27	eqtrid 2222	. . . 4 ⊢ (𝐾 ∈ N → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
29		addpinq1 7463	. . . . . . . . . . 11 ⊢ (𝐾 ∈ N → [⟨(𝐾 +N 1o), 1o⟩] ~Q = ([⟨𝐾, 1o⟩] ~Q +Q 1Q))
30	29	breq2d 4016	. . . . . . . . . 10 ⊢ (𝐾 ∈ N → (𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q ↔ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)))
31	30	abbidv 2295	. . . . . . . . 9 ⊢ (𝐾 ∈ N → {𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)})
32	29	breq1d 4014	. . . . . . . . . 10 ⊢ (𝐾 ∈ N → ([⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢 ↔ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢))
33	32	abbidv 2295	. . . . . . . . 9 ⊢ (𝐾 ∈ N → {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢})
34	31, 33	opeq12d 3787	. . . . . . . 8 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢}⟩)
35		nnnq 7421	. . . . . . . . 9 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
36		addnqpr1 7561	. . . . . . . . 9 ⊢ ([⟨𝐾, 1o⟩] ~Q ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
37	35, 36	syl 14	. . . . . . . 8 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
38	34, 37	eqtrd 2210	. . . . . . 7 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
39	38	oveq1d 5890	. . . . . 6 ⊢ (𝐾 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P))
40	39	opeq1d 3785	. . . . 5 ⊢ (𝐾 ∈ N → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩)
41	40	eceq1d 6571	. . . 4 ⊢ (𝐾 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
42	19, 28, 41	3eqtr4d 2220	. . 3 ⊢ (𝐾 ∈ N → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
43	42	opeq1d 3785	. 2 ⊢ (𝐾 ∈ N → ⟨([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
44	17, 43	eqtrd 2210	1 ⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  {cab 2163  ⟨cop 3596   class class class wbr 4004   × cxp 4625  (class class class)co 5875  1_oc1o 6410  [cec 6533   / cqs 6534  Ncnpi 7271   +_N cpli 7272   ~_Q ceq 7278  Qcnq 7279  1_Qc1q 7280   +_Q cplq 7281   <_Q cltq 7284  Pcnp 7290  1_Pc1p 7291   +_P cpp 7292   ~_R cer 7295  Rcnr 7296  0_Rc0r 7297  1_Rc1r 7298   +_R cplr 7300  1c1 7812   + caddc 7814

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-i1p 7466  df-iplp 7467  df-enr 7725  df-nr 7726  df-plr 7727  df-0r 7730  df-1r 7731  df-c 7817  df-1 7819  df-add 7822

This theorem is referenced by:  pitonn  7847  nntopi  7893