Theorem ballotlemsf1o 35216

Description: The defined 𝑆 is a bijection, and an involution. (Contributed by Thierry Arnoux, 14-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotth.o	⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p	⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))

Assertion

Ref	Expression
ballotlemsf1o	⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))⟶1-1onto(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑖,𝑐,𝐹,𝑘   𝐶,𝑖,𝑘   𝑖,𝐸,𝑘   𝐶,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐

Allowed substitution hints:   𝐶(𝑥, 𝑐)   𝑃(𝑥, 𝑖, 𝑘, 𝑐)   𝑆(𝑥, 𝑖, 𝑘, 𝑐)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemsf1o

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . 5 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . 5 ⊢ 𝑁 ∈ ℕ
3		ballotth.o	. . . . 5 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4		ballotth.p	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . . . 5 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . . 5 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . . 5 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . . 5 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9		ballotth.s	. . . . 5 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsval 35211	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))
11	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsv 35212	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑖) = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖))
12	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsdom 35214	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑖) ∈ (1...(𝑀 + 𝑁)))
13	11, 12	eqeltrrd 2865	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖) ∈ (1...(𝑀 + 𝑁)))
14	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsv 35212	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑗) = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
15	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsdom 35214	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝑗) ∈ (1...(𝑀 + 𝑁)))
16	14, 15	eqeltrrd 2865	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) ∈ (1...(𝑀 + 𝑁)))
17		oveq2 7451	. . . . . 6 ⊢ (𝑖 = (((𝐼‘𝐶) + 1) − 𝑗) → (((𝐼‘𝐶) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑗)))
18		id 23	. . . . . 6 ⊢ (𝑖 = 𝑗 → 𝑖 = 𝑗)
19		breq1 5109	. . . . . 6 ⊢ (𝑖 = (((𝐼‘𝐶) + 1) − 𝑗) → (𝑖 ≤ (𝐼‘𝐶) ↔ (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶)))
20		breq1 5109	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 ≤ (𝐼‘𝐶) ↔ 𝑗 ≤ (𝐼‘𝐶)))
21	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 35204	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
22	21	simpld 500	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
23		elfzelz 13683	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ)
24	23	peano2zd 12833	. . . . . . . . . . 11 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → ((𝐼‘𝐶) + 1) ∈ ℤ)
25	22, 24	syl 18	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) + 1) ∈ ℤ)
26	25	zcnd 12831	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) + 1) ∈ ℂ)
27	26	adantr 486	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → ((𝐼‘𝐶) + 1) ∈ ℂ)
28		elfzelz 13683	. . . . . . . . . 10 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
29	28	zcnd 12831	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℂ)
30	29	ad2antll 742	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℂ)
31	27, 30	nncand 11701	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑗)) = 𝑗)
32	31	eqcomd 2769	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑗)))
33	22, 23	syl 18	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
34	33	adantr 486	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝐼‘𝐶) ∈ ℤ)
35		elfznn 13714	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ)
36	35	ad2antll 742	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℕ)
37	34, 36	ltesubnnd 33484	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶))
38	37	adantr 486	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 ≤ (𝐼‘𝐶)) → (((𝐼‘𝐶) + 1) − 𝑗) ≤ (𝐼‘𝐶))
39		vex 3458	. . . . . . 7 ⊢ 𝑗 ∈ V
40	39	a1i 11	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ V)
41		ovexd 7478	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − 𝑗) ∈ V)
42	17, 18, 19, 20, 32, 38, 40, 41	ifeqeqx 33210	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)) → 𝑗 = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖))
43		oveq2 7451	. . . . . 6 ⊢ (𝑗 = (((𝐼‘𝐶) + 1) − 𝑖) → (((𝐼‘𝐶) + 1) − 𝑗) = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑖)))
44		id 23	. . . . . 6 ⊢ (𝑗 = 𝑖 → 𝑗 = 𝑖)
45		breq1 5109	. . . . . 6 ⊢ (𝑗 = (((𝐼‘𝐶) + 1) − 𝑖) → (𝑗 ≤ (𝐼‘𝐶) ↔ (((𝐼‘𝐶) + 1) − 𝑖) ≤ (𝐼‘𝐶)))
46		breq1 5109	. . . . . 6 ⊢ (𝑗 = 𝑖 → (𝑗 ≤ (𝐼‘𝐶) ↔ 𝑖 ≤ (𝐼‘𝐶)))
47		elfzelz 13683	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℤ)
48	47	zcnd 12831	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℂ)
49	48	ad2antrl 741	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ ℂ)
50	27, 49	nncand 11701	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑖)) = 𝑖)
51	50	eqcomd 2769	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 = (((𝐼‘𝐶) + 1) − (((𝐼‘𝐶) + 1) − 𝑖)))
52	34	adantr 486	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℤ)
53		simplrl 789	. . . . . . . 8 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → 𝑖 ∈ (1...(𝑀 + 𝑁)))
54		elfznn 13714	. . . . . . . 8 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℕ)
55	53, 54	syl 18	. . . . . . 7 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → 𝑖 ∈ ℕ)
56	52, 55	ltesubnnd 33484	. . . . . 6 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼‘𝐶)) → (((𝐼‘𝐶) + 1) − 𝑖) ≤ (𝐼‘𝐶))
57		vex 3458	. . . . . . 7 ⊢ 𝑖 ∈ V
58	57	a1i 11	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ V)
59		ovexd 7478	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼‘𝐶) + 1) − 𝑖) ∈ V)
60	43, 44, 45, 46, 51, 56, 58, 59	ifeqeqx 33210	. . . . 5 ⊢ (((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) → 𝑖 = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
61	42, 60	impbida 813	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝑖 = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗) ↔ 𝑗 = if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)))
62	10, 13, 16, 61	f1o3d 33292	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))⟶1-1onto(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))))
63	62	simpld 500	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))⟶1-1onto(1...(𝑀 + 𝑁)))
64		oveq2 7451	. . . . . 6 ⊢ (𝑖 = 𝑗 → (((𝐼‘𝐶) + 1) − 𝑖) = (((𝐼‘𝐶) + 1) − 𝑗))
65	20, 64, 18	ifbieq12d 4514	. . . . 5 ⊢ (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
66	65	cbvmptv 5212	. . . 4 ⊢ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗))
67	66	a1i 11	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)))
68	62	simprd 501	. . 3 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ◡(𝑆‘𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼‘𝐶), (((𝐼‘𝐶) + 1) − 𝑗), 𝑗)))
69	67, 10, 68	3eqtr4rd 2809	. 2 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ◡(𝑆‘𝐶) = (𝑆‘𝐶))
70	63, 69	jca 521	1 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))⟶1-1onto(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092  ifcif 4485  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5189  ^◡ccnv 5654  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443  infcinf 9460  ℂcc 11225  ℝcr 11226  0cc0 11227  1c1 11228   + caddc 11230   < clt 11370   ≤ cle 11371   − cmin 11568   / cdiv 12000  ℕcn 12362  ℤcz 12720  ...cfz 13666  ♯chash 14501

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-sup 9461  df-inf 9462  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-n0 12634  df-z 12721  df-uz 12993  df-rp 13148  df-fz 13667  df-hash 14502

This theorem is used by:  ballotlemsima  35218  ballotlemscr  35221  ballotlemrv  35222  ballotlemro  35225  ballotlemfrc  35229  ballotlemrinv0  35235