Theorem pitonnlem2 7975

Description: Lemma for pitonn 7976. 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 7948	. . . 4 ⊢ 1 = ⟨1R, 0R⟩
2	1	oveq2i 5967	. . 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 7681	. . . . . . . 8 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
4		1pr 7682	. . . . . . . 8 ⊢ 1P ∈ P
5		addclpr 7665	. . . . . . . 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 4714	. . . . . . 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 7865	. . . . . . 7 ⊢ ~R ∈ V
10	9	ecelqsi 6688	. . . . . 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 7855	. . . . 5 ⊢ R = ((P × P) / ~R )
13	11, 12	eleqtrrdi 2300	. . . 4 ⊢ (𝐾 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
14		1sr 7879	. . . 4 ⊢ 1R ∈ R
15		addresr 7965	. . . 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 2251	. 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 7974	. . . . 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 7860	. . . . . 6 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
21	20	oveq2i 5967	. . . . 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 7665	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
23	4, 4, 22	mp2an 426	. . . . . . 7 ⊢ (1P +P 1P) ∈ P
24		addsrpr 7873	. . . . . . . 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 2251	. . . 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 7592	. . . . . . . . . . 11 ⊢ (𝐾 ∈ N → [⟨(𝐾 +N 1o), 1o⟩] ~Q = ([⟨𝐾, 1o⟩] ~Q +Q 1Q))
30	29	breq2d 4062	. . . . . . . . . 10 ⊢ (𝐾 ∈ N → (𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q ↔ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)))
31	30	abbidv 2324	. . . . . . . . 9 ⊢ (𝐾 ∈ N → {𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)})
32	29	breq1d 4060	. . . . . . . . . 10 ⊢ (𝐾 ∈ N → ([⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢 ↔ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢))
33	32	abbidv 2324	. . . . . . . . 9 ⊢ (𝐾 ∈ N → {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢})
34	31, 33	opeq12d 3832	. . . . . . . 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 7550	. . . . . . . . 9 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
36		addnqpr1 7690	. . . . . . . . 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 2239	. . . . . . 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 5971	. . . . . 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 3830	. . . . 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 6668	. . . 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 2249	. . 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 3830	. 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 2239	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 1373   ∈ wcel 2177  {cab 2192  ⟨cop 3640   class class class wbr 4050   × cxp 4680  (class class class)co 5956  1_oc1o 6507  [cec 6630   / cqs 6631  Ncnpi 7400   +_N cpli 7401   ~_Q ceq 7407  Qcnq 7408  1_Qc1q 7409   +_Q cplq 7410   <_Q cltq 7413  Pcnp 7419  1_Pc1p 7420   +_P cpp 7421   ~_R cer 7424  Rcnr 7425  0_Rc0r 7426  1_Rc1r 7427   +_R cplr 7429  1c1 7941   + caddc 7943

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-eprel 4343  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-irdg 6468  df-1o 6514  df-2o 6515  df-oadd 6518  df-omul 6519  df-er 6632  df-ec 6634  df-qs 6638  df-ni 7432  df-pli 7433  df-mi 7434  df-lti 7435  df-plpq 7472  df-mpq 7473  df-enq 7475  df-nqqs 7476  df-plqqs 7477  df-mqqs 7478  df-1nqqs 7479  df-rq 7480  df-ltnqqs 7481  df-enq0 7552  df-nq0 7553  df-0nq0 7554  df-plq0 7555  df-mq0 7556  df-inp 7594  df-i1p 7595  df-iplp 7596  df-enr 7854  df-nr 7855  df-plr 7856  df-0r 7859  df-1r 7860  df-c 7946  df-1 7948  df-add 7951

This theorem is referenced by:  pitonn  7976  nntopi  8022