Theorem pitonnlem2 8110

Description: Lemma for pitonn 8111. 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 8083	. . . 4 ⊢ 1 = ⟨1R, 0R⟩
2	1	oveq2i 6039	. . 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 7816	. . . . . . . 8 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
4		1pr 7817	. . . . . . . 8 ⊢ 1P ∈ P
5		addclpr 7800	. . . . . . . 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 4763	. . . . . . 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 8000	. . . . . . 7 ⊢ ~R ∈ V
10	9	ecelqsi 6801	. . . . . 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 7990	. . . . 5 ⊢ R = ((P × P) / ~R )
13	11, 12	eleqtrrdi 2325	. . . 4 ⊢ (𝐾 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
14		1sr 8014	. . . 4 ⊢ 1R ∈ R
15		addresr 8100	. . . 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 2276	. 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 8109	. . . . 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 7995	. . . . . 6 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
21	20	oveq2i 6039	. . . . 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 7800	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
23	4, 4, 22	mp2an 426	. . . . . . 7 ⊢ (1P +P 1P) ∈ P
24		addsrpr 8008	. . . . . . . 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 2276	. . . 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 7727	. . . . . . . . . . 11 ⊢ (𝐾 ∈ N → [⟨(𝐾 +N 1o), 1o⟩] ~Q = ([⟨𝐾, 1o⟩] ~Q +Q 1Q))
30	29	breq2d 4105	. . . . . . . . . 10 ⊢ (𝐾 ∈ N → (𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q ↔ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)))
31	30	abbidv 2350	. . . . . . . . 9 ⊢ (𝐾 ∈ N → {𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)})
32	29	breq1d 4103	. . . . . . . . . 10 ⊢ (𝐾 ∈ N → ([⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢 ↔ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢))
33	32	abbidv 2350	. . . . . . . . 9 ⊢ (𝐾 ∈ N → {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢})
34	31, 33	opeq12d 3875	. . . . . . . 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 7685	. . . . . . . . 9 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
36		addnqpr1 7825	. . . . . . . . 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 2264	. . . . . . 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 6043	. . . . . 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 3873	. . . . 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 6781	. . . 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 2274	. . 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 3873	. 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 2264	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 1398   ∈ wcel 2202  {cab 2217  ⟨cop 3676   class class class wbr 4093   × cxp 4729  (class class class)co 6028  1_oc1o 6618  [cec 6743   / cqs 6744  Ncnpi 7535   +_N cpli 7536   ~_Q ceq 7542  Qcnq 7543  1_Qc1q 7544   +_Q cplq 7545   <_Q cltq 7548  Pcnp 7554  1_Pc1p 7555   +_P cpp 7556   ~_R cer 7559  Rcnr 7560  0_Rc0r 7561  1_Rc1r 7562   +_R cplr 7564  1c1 8076   + caddc 8078

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-i1p 7730  df-iplp 7731  df-enr 7989  df-nr 7990  df-plr 7991  df-0r 7994  df-1r 7995  df-c 8081  df-1 8083  df-add 8086

This theorem is referenced by:  pitonn  8111  nntopi  8157