Theorem ballotfi 13226

Description: Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-Dec-2016.) (Revised by Jim Kingdon, 17-Jun-2026.)

Hypotheses

Ref	Expression
ballotfi.m	⊢ 𝑀 ∈ ℕ
ballotfi.n	⊢ 𝑁 ∈ ℕ
ballotfi.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfi.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotfi.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotfi.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotfi.mgtn	⊢ 𝑁 < 𝑀

Assertion

Ref	Expression
ballotfi	⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

Distinct variable groups:   𝐸,𝑐,𝑖,𝑥   𝐹,𝑐,𝑖,𝑥   𝑀,𝑐,𝑖,𝑥   𝑁,𝑐,𝑖,𝑥   𝑂,𝑐,𝑖,𝑥

Allowed substitution hints:   𝑃(𝑥,𝑖,𝑐)

Proof of Theorem ballotfi

Dummy variables 𝑘 𝑞 𝑟 𝑠 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotfi.m	. 2 ⊢ 𝑀 ∈ ℕ
2		ballotfi.n	. 2 ⊢ 𝑁 ∈ ℕ
3		ballotfi.o	. 2 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
4		ballotfi.p	. 2 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotfi.f	. 2 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotfi.e	. 2 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotfi.mgtn	. 2 ⊢ 𝑁 < 𝑀
8		fveq2 5675	. . . . . . . 8 ⊢ (𝑞 = 𝑐 → (𝐹‘𝑞) = (𝐹‘𝑐))
9	8	fveq1d 5677	. . . . . . 7 ⊢ (𝑞 = 𝑐 → ((𝐹‘𝑞)‘𝑝) = ((𝐹‘𝑐)‘𝑝))
10	9	eqeq1d 2243	. . . . . 6 ⊢ (𝑞 = 𝑐 → (((𝐹‘𝑞)‘𝑝) = 0 ↔ ((𝐹‘𝑐)‘𝑝) = 0))
11	10	rabbidv 2804	. . . . 5 ⊢ (𝑞 = 𝑐 → {𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0} = {𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0})
12	11	infeq1d 7316	. . . 4 ⊢ (𝑞 = 𝑐 → inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ) = inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0}, ℝ, < ))
13	12	cbvmptv 4211	. . 3 ⊢ (𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < )) = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0}, ℝ, < ))
14		fveqeq2 5684	. . . . . 6 ⊢ (𝑝 = 𝑘 → (((𝐹‘𝑐)‘𝑝) = 0 ↔ ((𝐹‘𝑐)‘𝑘) = 0))
15	14	cbvrabv 2814	. . . . 5 ⊢ {𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}
16	15	infeq1i 7317	. . . 4 ⊢ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0}, ℝ, < ) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )
17	16	mpteq2i 4202	. . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0}, ℝ, < )) = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
18	13, 17	eqtri 2255	. 2 ⊢ (𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < )) = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
19		fveq2 5675	. . . . . . 7 ⊢ (𝑟 = 𝑐 → ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) = ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐))
20	19	breq2d 4126	. . . . . 6 ⊢ (𝑟 = 𝑐 → (𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) ↔ 𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐)))
21	19	oveq1d 6073	. . . . . . 7 ⊢ (𝑟 = 𝑐 → (((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) + 1) = (((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1))
22	21	oveq1d 6073	. . . . . 6 ⊢ (𝑟 = 𝑐 → ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) + 1) − 𝑠) = ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑠))
23	20, 22	ifbieq1d 3649	. . . . 5 ⊢ (𝑟 = 𝑐 → if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) + 1) − 𝑠), 𝑠) = if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑠), 𝑠))
24	23	mpteq2dv 4206	. . . 4 ⊢ (𝑟 = 𝑐 → (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) + 1) − 𝑠), 𝑠)) = (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑠), 𝑠)))
25	24	cbvmptv 4211	. . 3 ⊢ (𝑟 ∈ (𝑂 ∖ 𝐸) ↦ (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) + 1) − 𝑠), 𝑠))) = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑠), 𝑠)))
26		breq1 4117	. . . . . 6 ⊢ (𝑠 = 𝑖 → (𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) ↔ 𝑖 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐)))
27		oveq2 6066	. . . . . 6 ⊢ (𝑠 = 𝑖 → ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑠) = ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑖))
28		id 19	. . . . . 6 ⊢ (𝑠 = 𝑖 → 𝑠 = 𝑖)
29	26, 27, 28	ifbieq12d 3653	. . . . 5 ⊢ (𝑠 = 𝑖 → if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑠), 𝑠) = if(𝑖 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑖), 𝑖))
30	29	cbvmptv 4211	. . . 4 ⊢ (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑠), 𝑠)) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑖), 𝑖))
31	30	mpteq2i 4202	. . 3 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑠), 𝑠))) = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑖), 𝑖)))
32	25, 31	eqtri 2255	. 2 ⊢ (𝑟 ∈ (𝑂 ∖ 𝐸) ↦ (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) + 1) − 𝑠), 𝑠))) = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑐) + 1) − 𝑖), 𝑖)))
33		eqid 2234	. 2 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (((𝑟 ∈ (𝑂 ∖ 𝐸) ↦ (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) + 1) − 𝑠), 𝑠)))‘𝑐) “ 𝑐)) = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (((𝑟 ∈ (𝑂 ∖ 𝐸) ↦ (𝑠 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑠 ≤ ((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟), ((((𝑞 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0}, ℝ, < ))‘𝑟) + 1) − 𝑠), 𝑠)))‘𝑐) “ 𝑐))
34	1, 2, 3, 4, 5, 6, 7, 18, 32, 33	ballotfilemth 13225	1 ⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

Syntax hints:   = wceq 1398   ∈ wcel 2205  ∀wral 2522  {crab 2526   ∖ cdif 3211   ∩ cin 3213  ifcif 3624  𝒫 cpw 3674   class class class wbr 4114   ↦ cmpt 4176   “ cima 4757  ‘cfv 5357  (class class class)co 6058  Fincfn 6988  infcinf 7287  ℝcr 8142  0cc0 8143  1c1 8144   + caddc 8146   < clt 8324   ≤ cle 8325   − cmin 8460   / cdiv 8963  ℕcn 9254  ℤcz 9594  ...cfz 10361  ♯chash 11163

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-fac 11113  df-bc 11135  df-ihash 11164

This theorem is referenced by: (None)