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Theorem ballotlemsf1o 30884
 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 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
ballotth.mgtn 𝑁 < 𝑀
ballotth.i 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
Assertion
Ref Expression
ballotlemsf1o (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
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.
StepHypRef 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 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6 ballotth.e . . . . 5 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
7 ballotth.mgtn . . . . 5 𝑁 < 𝑀
8 ballotth.i . . . . 5 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
9 ballotth.s . . . . 5 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
101, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsval 30879 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
111, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsv 30880 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑖) = if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖))
121, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsdom 30882 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑖) ∈ (1...(𝑀 + 𝑁)))
1311, 12eqeltrrd 2840 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖) ∈ (1...(𝑀 + 𝑁)))
141, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsv 30880 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑗) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
151, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsdom 30882 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑗) ∈ (1...(𝑀 + 𝑁)))
1614, 15eqeltrrd 2840 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) ∈ (1...(𝑀 + 𝑁)))
17 oveq2 6821 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)))
18 id 22 . . . . . 6 (𝑖 = 𝑗𝑖 = 𝑗)
19 breq1 4807 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (𝑖 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶)))
20 breq1 4807 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ≤ (𝐼𝐶) ↔ 𝑗 ≤ (𝐼𝐶)))
211, 2, 3, 4, 5, 6, 7, 8ballotlemiex 30872 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
2221simpld 477 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
23 elfzelz 12535 . . . . . . . . . . . 12 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
2423peano2zd 11677 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → ((𝐼𝐶) + 1) ∈ ℤ)
2522, 24syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℤ)
2625zcnd 11675 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℂ)
2726adantr 472 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → ((𝐼𝐶) + 1) ∈ ℂ)
28 elfzelz 12535 . . . . . . . . . 10 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
2928zcnd 11675 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℂ)
3029ad2antll 767 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℂ)
3127, 30nncand 10589 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)) = 𝑗)
3231eqcomd 2766 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)))
3322, 23syl 17 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
3433adantr 472 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝐼𝐶) ∈ ℤ)
35 elfznn 12563 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ)
3635ad2antll 767 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℕ)
3734, 36ltesubnnd 29877 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶))
3837adantr 472 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶))
39 vex 3343 . . . . . . 7 𝑗 ∈ V
4039a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ V)
41 ovexd 6843 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑗) ∈ V)
4217, 18, 19, 20, 32, 38, 40, 41ifeqeqx 29668 . . . . 5 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗)) → 𝑗 = if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖))
43 oveq2 6821 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (((𝐼𝐶) + 1) − 𝑗) = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)))
44 id 22 . . . . . 6 (𝑗 = 𝑖𝑗 = 𝑖)
45 breq1 4807 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (𝑗 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶)))
46 breq1 4807 . . . . . 6 (𝑗 = 𝑖 → (𝑗 ≤ (𝐼𝐶) ↔ 𝑖 ≤ (𝐼𝐶)))
47 elfzelz 12535 . . . . . . . . . 10 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℤ)
4847zcnd 11675 . . . . . . . . 9 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℂ)
4948ad2antrl 766 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ ℂ)
5027, 49nncand 10589 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)) = 𝑖)
5150eqcomd 2766 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)))
5234adantr 472 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (𝐼𝐶) ∈ ℤ)
53 simplrl 819 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → 𝑖 ∈ (1...(𝑀 + 𝑁)))
54 elfznn 12563 . . . . . . . 8 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℕ)
5553, 54syl 17 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → 𝑖 ∈ ℕ)
5652, 55ltesubnnd 29877 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶))
57 vex 3343 . . . . . . 7 𝑖 ∈ V
5857a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ V)
59 ovexd 6843 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑖) ∈ V)
6043, 44, 45, 46, 51, 56, 58, 59ifeqeqx 29668 . . . . 5 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 = if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)) → 𝑖 = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
6142, 60impbida 913 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝑖 = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) ↔ 𝑗 = if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
6210, 13, 16, 61f1o3d 29740 . . 3 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))))
6362simpld 477 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
64 oveq2 6821 . . . . . 6 (𝑖 = 𝑗 → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − 𝑗))
6520, 64, 18ifbieq12d 4257 . . . . 5 (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
6665cbvmptv 4902 . . . 4 (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
6766a1i 11 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗)))
6862simprd 482 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗)))
6967, 10, 683eqtr4rd 2805 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑆𝐶))
7063, 69jca 555 1 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  Vcvv 3340   ∖ cdif 3712   ∩ cin 3714  ifcif 4230  𝒫 cpw 4302   class class class wbr 4804   ↦ cmpt 4881  ◡ccnv 5265  –1-1-onto→wf1o 6048  ‘cfv 6049  (class class class)co 6813  infcinf 8512  ℂcc 10126  ℝcr 10127  0cc0 10128  1c1 10129   + caddc 10131   < clt 10266   ≤ cle 10267   − cmin 10458   / cdiv 10876  ℕcn 11212  ℤcz 11569  ...cfz 12519  ♯chash 13311 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-hash 13312 This theorem is referenced by:  ballotlemsima  30886  ballotlemscr  30889  ballotlemrv  30890  ballotlemro  30893  ballotlemfrc  30897  ballotlemrinv0  30903
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