Theorem ballotfi 13265

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 5693	. . . . . . . 8 ⊢ (𝑞 = 𝑐 → (𝐹‘𝑞) = (𝐹‘𝑐))
9	8	fveq1d 5695	. . . . . . 7 ⊢ (𝑞 = 𝑐 → ((𝐹‘𝑞)‘𝑝) = ((𝐹‘𝑐)‘𝑝))
10	9	eqeq1d 2247	. . . . . 6 ⊢ (𝑞 = 𝑐 → (((𝐹‘𝑞)‘𝑝) = 0 ↔ ((𝐹‘𝑐)‘𝑝) = 0))
11	10	rabbidv 2810	. . . . 5 ⊢ (𝑞 = 𝑐 → {𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑞)‘𝑝) = 0} = {𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0})
12	11	infeq1d 7346	. . . 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 4225	. . 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 5702	. . . . . 6 ⊢ (𝑝 = 𝑘 → (((𝐹‘𝑐)‘𝑝) = 0 ↔ ((𝐹‘𝑐)‘𝑘) = 0))
15	14	cbvrabv 2820	. . . . 5 ⊢ {𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0} = {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}
16	15	infeq1i 7347	. . . 4 ⊢ inf({𝑝 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑝) = 0}, ℝ, < ) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < )
17	16	mpteq2i 4216	. . 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 2259	. 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 5693	. . . . . . 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 4140	. . . . . 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 6094	. . . . . . 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 6094	. . . . . 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 3663	. . . . 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 4220	. . . 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 4225	. . 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 4131	. . . . . 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 6087	. . . . . 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 3667	. . . . 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 4225	. . . 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 4216	. . 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 2259	. 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 2238	. 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 13264	1 ⊢ (𝑃‘𝐸) = ((𝑀 − 𝑁) / (𝑀 + 𝑁))

Syntax hints:   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ∖ cdif 3217   ∩ cin 3219  ifcif 3638  𝒫 cpw 3688   class class class wbr 4128   ↦ cmpt 4190   “ cima 4775  ‘cfv 5375  (class class class)co 6079  Fincfn 7016  infcinf 7317  ℝcr 8172  0cc0 8173  1c1 8174   + caddc 8176   < clt 8354   ≤ cle 8355   − cmin 8491   / cdiv 8996  ℕcn 9287  ℤcz 9627  ...cfz 10394  ♯chash 11197

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-fac 11147  df-bc 11169  df-ihash 11198

This theorem is referenced by: (None)