Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlemsf1o Structured version   Visualization version   GIF version

Theorem ballotlemsf1o 30348
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 30343 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)))
111, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsv 30344 . . . . 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 30346 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑖 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑖) ∈ (1...(𝑀 + 𝑁)))
1311, 12eqeltrrd 2705 . . . 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 30344 . . . . 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 30346 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝑗) ∈ (1...(𝑀 + 𝑁)))
1614, 15eqeltrrd 2705 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁))) → if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗) ∈ (1...(𝑀 + 𝑁)))
17 oveq2 6613 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)))
18 id 22 . . . . . 6 (𝑖 = 𝑗𝑖 = 𝑗)
19 breq1 4621 . . . . . 6 (𝑖 = (((𝐼𝐶) + 1) − 𝑗) → (𝑖 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑗) ≤ (𝐼𝐶)))
20 breq1 4621 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ≤ (𝐼𝐶) ↔ 𝑗 ≤ (𝐼𝐶)))
211, 2, 3, 4, 5, 6, 7, 8ballotlemiex 30336 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
2221simpld 475 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
23 elfzelz 12281 . . . . . . . . . . . 12 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
2423peano2zd 11429 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → ((𝐼𝐶) + 1) ∈ ℤ)
2522, 24syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℤ)
2625zcnd 11427 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) + 1) ∈ ℂ)
2726adantr 481 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → ((𝐼𝐶) + 1) ∈ ℂ)
28 elfzelz 12281 . . . . . . . . . 10 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℤ)
2928zcnd 11427 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℂ)
3029ad2antll 764 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℂ)
3127, 30nncand 10342 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑗)) = 𝑗)
3231eqcomd 2632 . . . . . 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 12309 . . . . . . . . 9 (𝑗 ∈ (1...(𝑀 + 𝑁)) → 𝑗 ∈ ℕ)
3635ad2antll 764 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ ℕ)
3734, 36ltesubnnd 29401 . . . . . . 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 3194 . . . . . . 7 𝑗 ∈ V
4039a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑗 ∈ V)
41 ovex 6633 . . . . . . 7 (((𝐼𝐶) + 1) − 𝑗) ∈ V
4241a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑗) ∈ V)
4317, 18, 19, 20, 32, 38, 40, 42ifeqeqx 29199 . . . . 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))
44 oveq2 6613 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (((𝐼𝐶) + 1) − 𝑗) = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)))
45 id 22 . . . . . 6 (𝑗 = 𝑖𝑗 = 𝑖)
46 breq1 4621 . . . . . 6 (𝑗 = (((𝐼𝐶) + 1) − 𝑖) → (𝑗 ≤ (𝐼𝐶) ↔ (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶)))
47 breq1 4621 . . . . . 6 (𝑗 = 𝑖 → (𝑗 ≤ (𝐼𝐶) ↔ 𝑖 ≤ (𝐼𝐶)))
48 elfzelz 12281 . . . . . . . . . 10 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℤ)
4948zcnd 11427 . . . . . . . . 9 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℂ)
5049ad2antrl 763 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ ℂ)
5127, 50nncand 10342 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)) = 𝑖)
5251eqcomd 2632 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 = (((𝐼𝐶) + 1) − (((𝐼𝐶) + 1) − 𝑖)))
5334adantr 481 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (𝐼𝐶) ∈ ℤ)
54 simplrl 799 . . . . . . . 8 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → 𝑖 ∈ (1...(𝑀 + 𝑁)))
55 elfznn 12309 . . . . . . . 8 (𝑖 ∈ (1...(𝑀 + 𝑁)) → 𝑖 ∈ ℕ)
5654, 55syl 17 . . . . . . 7 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → 𝑖 ∈ ℕ)
5753, 56ltesubnnd 29401 . . . . . 6 (((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) ∧ 𝑖 ≤ (𝐼𝐶)) → (((𝐼𝐶) + 1) − 𝑖) ≤ (𝐼𝐶))
58 vex 3194 . . . . . . 7 𝑖 ∈ V
5958a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → 𝑖 ∈ V)
60 ovex 6633 . . . . . . 7 (((𝐼𝐶) + 1) − 𝑖) ∈ V
6160a1i 11 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝑖 ∈ (1...(𝑀 + 𝑁)) ∧ 𝑗 ∈ (1...(𝑀 + 𝑁)))) → (((𝐼𝐶) + 1) − 𝑖) ∈ V)
6244, 45, 46, 47, 52, 57, 59, 61ifeqeqx 29199 . . . . 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))
6343, 62impbida 876 . . . 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)))
6410, 13, 16, 63f1o3d 29266 . . 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))))
6564simpld 475 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
66 oveq2 6613 . . . . . 6 (𝑖 = 𝑗 → (((𝐼𝐶) + 1) − 𝑖) = (((𝐼𝐶) + 1) − 𝑗))
6720, 66, 18ifbieq12d 4090 . . . . 5 (𝑖 = 𝑗 → if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖) = if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
6867cbvmptv 4715 . . . 4 (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑖), 𝑖)) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗))
6968a1i 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)))
7064simprd 479 . . 3 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑗 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑗 ≤ (𝐼𝐶), (((𝐼𝐶) + 1) − 𝑗), 𝑗)))
7169, 10, 703eqtr4rd 2671 . 2 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶) = (𝑆𝐶))
7265, 71jca 554 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 384   = wceq 1480  wcel 1992  wral 2912  {crab 2916  Vcvv 3191  cdif 3557  cin 3559  ifcif 4063  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678  ccnv 5078  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  infcinf 8292  cc 9879  cr 9880  0cc0 9881  1c1 9882   + caddc 9884   < clt 10019  cle 10020  cmin 10211   / cdiv 10629  cn 10965  cz 11322  ...cfz 12265  #chash 13054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-inf 8294  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-fz 12266  df-hash 13055
This theorem is referenced by:  ballotlemsima  30350  ballotlemscr  30353  ballotlemrv  30354  ballotlemro  30357  ballotlemfrc  30361  ballotlemrinv0  30367
  Copyright terms: Public domain W3C validator