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Theorem ballotlemsf1o 31999
 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 31994 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
111, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsv 31995 . . . . 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 31997 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑖) ∈ (1...(𝑀 + 𝑁)))
1311, 12eqeltrrd 2853 . . . 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 31995 . . . . 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 31997 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑗) ∈ (1...(𝑀 + 𝑁)))
1614, 15eqeltrrd 2853 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) ∈ (1...(𝑀 + 𝑁)))
17 oveq2 7158 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)))
18 id 22 . . . . . 6 (𝑖 = 𝑗𝑖 = 𝑗)
19 breq1 5035 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (𝑖 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶)))
20 breq1 5035 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ≤ (𝐼𝐶) ↔ 𝑗 ≤ (𝐼𝐶)))
211, 2, 3, 4, 5, 6, 7, 8ballotlemiex 31987 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
2221simpld 498 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
23 elfzelz 12956 . . . . . . . . . . . 12 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
2423peano2zd 12129 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → ((𝐼𝐶) + 1) ∈ ℤ)
2522, 24syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℤ)
2625zcnd 12127 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℂ)
2726adantr 484 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → ((𝐼𝐶) + 1) ∈ ℂ)
28 elfzelz 12956 . . . . . . . . . 10 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
2928zcnd 12127 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℂ)
3029ad2antll 728 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℂ)
3127, 30nncand 11040 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)) = 𝑗)
3231eqcomd 2764 . . . . . 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 484 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (𝐼𝐶) ∈ ℤ)
35 elfznn 12985 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ)
3635ad2antll 728 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℕ)
3734, 36ltesubnnd 30660 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶))
3837adantr 484 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑗 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶))
39 vex 3413 . . . . . . 7 𝑗 ∈ V
4039a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ V)
41 ovexd 7185 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑗) ∈ V)
4217, 18, 19, 20, 32, 38, 40, 41ifeqeqx 30407 . . . . 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 7158 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (((𝐼𝐶) + 1) − 𝑗) = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)))
44 id 22 . . . . . 6 (𝑗 = 𝑖𝑗 = 𝑖)
45 breq1 5035 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (𝑗 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶)))
46 breq1 5035 . . . . . 6 (𝑗 = 𝑖 → (𝑗 ≤ (𝐼𝐶) ↔ 𝑖 ≤ (𝐼𝐶)))
47 elfzelz 12956 . . . . . . . . . 10 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℤ)
4847zcnd 12127 . . . . . . . . 9 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℂ)
4948ad2antrl 727 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ ℂ)
5027, 49nncand 11040 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)) = 𝑖)
5150eqcomd 2764 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)))
5234adantr 484 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (𝐼𝐶) ∈ ℤ)
53 simplrl 776 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → 𝑖 ∈ (1...(𝑀 + 𝑁)))
54 elfznn 12985 . . . . . . . 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 30660 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶))
57 vex 3413 . . . . . . 7 𝑖 ∈ V
5857a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ V)
59 ovexd 7185 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑖) ∈ V)
6043, 44, 45, 46, 51, 56, 58, 59ifeqeqx 30407 . . . . 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 800 . . . 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 30484 . . 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 498 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
64 oveq2 7158 . . . . . 6 (𝑖 = 𝑗 → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − 𝑗))
6520, 64, 18ifbieq12d 4448 . . . . 5 (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
6665cbvmptv 5135 . . . 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 499 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗)))
6967, 10, 683eqtr4rd 2804 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑆𝐶))
7063, 69jca 515 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  Vcvv 3409   ∖ cdif 3855   ∩ cin 3857  ifcif 4420  𝒫 cpw 4494   class class class wbr 5032   ↦ cmpt 5112  ◡ccnv 5523  –1-1-onto→wf1o 6334  ‘cfv 6335  (class class class)co 7150  infcinf 8938  ℂcc 10573  ℝcr 10574  0cc0 10575  1c1 10576   + caddc 10578   < clt 10713   ≤ cle 10714   − cmin 10908   / cdiv 11335  ℕcn 11674  ℤcz 12020  ...cfz 12939  ♯chash 13740 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-oadd 8116  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-n0 11935  df-z 12021  df-uz 12283  df-rp 12431  df-fz 12940  df-hash 13741 This theorem is referenced by:  ballotlemsima  32001  ballotlemscr  32004  ballotlemrv  32005  ballotlemro  32008  ballotlemfrc  32012  ballotlemrinv0  32018
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