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Theorem ballotlemsf1o 31671
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 31666 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
111, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsv 31667 . . . . 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 31669 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑖) ∈ (1...(𝑀 + 𝑁)))
1311, 12eqeltrrd 2914 . . . 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 31667 . . . . 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 31669 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑗) ∈ (1...(𝑀 + 𝑁)))
1614, 15eqeltrrd 2914 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) ∈ (1...(𝑀 + 𝑁)))
17 oveq2 7153 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)))
18 id 22 . . . . . 6 (𝑖 = 𝑗𝑖 = 𝑗)
19 breq1 5061 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (𝑖 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶)))
20 breq1 5061 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ≤ (𝐼𝐶) ↔ 𝑗 ≤ (𝐼𝐶)))
211, 2, 3, 4, 5, 6, 7, 8ballotlemiex 31659 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
2221simpld 495 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
23 elfzelz 12898 . . . . . . . . . . . 12 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
2423peano2zd 12079 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → ((𝐼𝐶) + 1) ∈ ℤ)
2522, 24syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℤ)
2625zcnd 12077 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℂ)
2726adantr 481 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → ((𝐼𝐶) + 1) ∈ ℂ)
28 elfzelz 12898 . . . . . . . . . 10 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
2928zcnd 12077 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℂ)
3029ad2antll 725 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℂ)
3127, 30nncand 10991 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)) = 𝑗)
3231eqcomd 2827 . . . . . 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 481 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝐼𝐶) ∈ ℤ)
35 elfznn 12926 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ)
3635ad2antll 725 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℕ)
3734, 36ltesubnnd 30466 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶))
3837adantr 481 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶))
39 vex 3498 . . . . . . 7 𝑗 ∈ V
4039a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ V)
41 ovexd 7180 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑗) ∈ V)
4217, 18, 19, 20, 32, 38, 40, 41ifeqeqx 30225 . . . . 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 7153 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (((𝐼𝐶) + 1) − 𝑗) = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)))
44 id 22 . . . . . 6 (𝑗 = 𝑖𝑗 = 𝑖)
45 breq1 5061 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (𝑗 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶)))
46 breq1 5061 . . . . . 6 (𝑗 = 𝑖 → (𝑗 ≤ (𝐼𝐶) ↔ 𝑖 ≤ (𝐼𝐶)))
47 elfzelz 12898 . . . . . . . . . 10 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℤ)
4847zcnd 12077 . . . . . . . . 9 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℂ)
4948ad2antrl 724 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ ℂ)
5027, 49nncand 10991 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)) = 𝑖)
5150eqcomd 2827 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)))
5234adantr 481 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (𝐼𝐶) ∈ ℤ)
53 simplrl 773 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → 𝑖 ∈ (1...(𝑀 + 𝑁)))
54 elfznn 12926 . . . . . . . 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 30466 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶))
57 vex 3498 . . . . . . 7 𝑖 ∈ V
5857a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ V)
59 ovexd 7180 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑖) ∈ V)
6043, 44, 45, 46, 51, 56, 58, 59ifeqeqx 30225 . . . . 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 797 . . . 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 30301 . . 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 495 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
64 oveq2 7153 . . . . . 6 (𝑖 = 𝑗 → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − 𝑗))
6520, 64, 18ifbieq12d 4492 . . . . 5 (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
6665cbvmptv 5161 . . . 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 496 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗)))
6967, 10, 683eqtr4rd 2867 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑆𝐶))
7063, 69jca 512 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 396   = wceq 1528  wcel 2105  wral 3138  {crab 3142  Vcvv 3495  cdif 3932  cin 3934  ifcif 4465  𝒫 cpw 4537   class class class wbr 5058  cmpt 5138  ccnv 5548  1-1-ontowf1o 6348  cfv 6349  (class class class)co 7145  infcinf 8894  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664  cle 10665  cmin 10859   / cdiv 11286  cn 11627  cz 11970  ...cfz 12882  chash 13680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-hash 13681
This theorem is referenced by:  ballotlemsima  31673  ballotlemscr  31676  ballotlemrv  31677  ballotlemro  31680  ballotlemfrc  31684  ballotlemrinv0  31690
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