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Theorem ballotlemsf1o 31670
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 31665 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
111, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsv 31666 . . . . 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 31668 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑖) ∈ (1...(𝑀 + 𝑁)))
1311, 12eqeltrrd 2911 . . . 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 31666 . . . . 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 31668 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑗) ∈ (1...(𝑀 + 𝑁)))
1614, 15eqeltrrd 2911 . . . 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 5060 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (𝑖 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶)))
20 breq1 5060 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ≤ (𝐼𝐶) ↔ 𝑗 ≤ (𝐼𝐶)))
211, 2, 3, 4, 5, 6, 7, 8ballotlemiex 31658 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
2221simpld 495 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
23 elfzelz 12896 . . . . . . . . . . . 12 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
2423peano2zd 12078 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → ((𝐼𝐶) + 1) ∈ ℤ)
2522, 24syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℤ)
2625zcnd 12076 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℂ)
2726adantr 481 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → ((𝐼𝐶) + 1) ∈ ℂ)
28 elfzelz 12896 . . . . . . . . . 10 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
2928zcnd 12076 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℂ)
3029ad2antll 725 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℂ)
3127, 30nncand 10990 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)) = 𝑗)
3231eqcomd 2824 . . . . . 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 12924 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ)
3635ad2antll 725 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℕ)
3734, 36ltesubnnd 30465 . . . . . . 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 3495 . . . . . . 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 30224 . . . . 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 5060 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (𝑗 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶)))
46 breq1 5060 . . . . . 6 (𝑗 = 𝑖 → (𝑗 ≤ (𝐼𝐶) ↔ 𝑖 ≤ (𝐼𝐶)))
47 elfzelz 12896 . . . . . . . . . 10 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℤ)
4847zcnd 12076 . . . . . . . . 9 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℂ)
4948ad2antrl 724 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ ℂ)
5027, 49nncand 10990 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)) = 𝑖)
5150eqcomd 2824 . . . . . 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 12924 . . . . . . . 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 30465 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶))
57 vex 3495 . . . . . . 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 30224 . . . . 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 30300 . . 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 4490 . . . . 5 (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
6665cbvmptv 5160 . . . 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 2864 . 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 3135  {crab 3139  Vcvv 3492  cdif 3930  cin 3932  ifcif 4463  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  ccnv 5547  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  infcinf 8893  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  cz 11969  ...cfz 12880  chash 13678
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 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
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 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-hash 13679
This theorem is referenced by:  ballotlemsima  31672  ballotlemscr  31675  ballotlemrv  31676  ballotlemro  31679  ballotlemfrc  31683  ballotlemrinv0  31689
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