Metamath Proof Explorer

Theorem ballotlem8

Description: There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B . (Contributed by Thierry Arnoux, 7-Dec-2016)

Ref Expression
Hypotheses 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 = c ∈ O ⟼ i ∈ ℤ ⟼ 1 … i ∩ c − 1 … i ∖ c
ballotth.e ⊢ E = c ∈ O | ∀ i ∈ 1 … M + N 0 < F ⁡ c ⁡ i
ballotth.mgtn ⊢ N < M
ballotth.i ⊢ I = c ∈ O ∖ E ⟼ sup k ∈ 1 … M + N | F ⁡ c ⁡ k = 0 ℝ <
ballotth.s ⊢ S = c ∈ O ∖ E ⟼ i ∈ 1 … M + N ⟼ if i ≤ I ⁡ c I ⁡ c + 1 - i i
ballotth.r ⊢ R = c ∈ O ∖ E ⟼ S ⁡ c c
Assertion ballotlem8 ⊢ c ∈ O ∖ E | 1 ∈ c = c ∈ O ∖ E | ¬ 1 ∈ c

Proof

Step Hyp Ref Expression
1 ballotth.m ⊢ M ∈ ℕ
2 ballotth.n ⊢ N ∈ ℕ
3 ballotth.o ⊢ O = c ∈ 𝒫 1 … M + N | c = M
4 ballotth.p ⊢ P = x ∈ 𝒫 O ⟼ x O
5 ballotth.f ⊢ F = c ∈ O ⟼ i ∈ ℤ ⟼ 1 … i ∩ c − 1 … i ∖ c
6 ballotth.e ⊢ E = c ∈ O | ∀ i ∈ 1 … M + N 0 < F ⁡ c ⁡ i
7 ballotth.mgtn ⊢ N < M
8 ballotth.i ⊢ I = c ∈ O ∖ E ⟼ sup k ∈ 1 … M + N | F ⁡ c ⁡ k = 0 ℝ <
9 ballotth.s ⊢ S = c ∈ O ∖ E ⟼ i ∈ 1 … M + N ⟼ if i ≤ I ⁡ c I ⁡ c + 1 - i i
10 ballotth.r ⊢ R = c ∈ O ∖ E ⟼ S ⁡ c c
11 1 2 3 4 5 6 7 8 9 10 ballotlem7 ⊢ R ↾ c ∈ O ∖ E | 1 ∈ c : c ∈ O ∖ E | 1 ∈ c ⟶ 1-1 onto c ∈ O ∖ E | ¬ 1 ∈ c
12 1 2 3 ballotlemoex ⊢ O ∈ V
13 difexg ⊢ O ∈ V → O ∖ E ∈ V
14 12 13 ax-mp ⊢ O ∖ E ∈ V
15 14 rabex ⊢ c ∈ O ∖ E | 1 ∈ c ∈ V
16 15 f1oen ⊢ R ↾ c ∈ O ∖ E | 1 ∈ c : c ∈ O ∖ E | 1 ∈ c ⟶ 1-1 onto c ∈ O ∖ E | ¬ 1 ∈ c → c ∈ O ∖ E | 1 ∈ c ≈ c ∈ O ∖ E | ¬ 1 ∈ c
17 hasheni ⊢ c ∈ O ∖ E | 1 ∈ c ≈ c ∈ O ∖ E | ¬ 1 ∈ c → c ∈ O ∖ E | 1 ∈ c = c ∈ O ∖ E | ¬ 1 ∈ c
18 11 16 17 mp2b ⊢ c ∈ O ∖ E | 1 ∈ c = c ∈ O ∖ E | ¬ 1 ∈ c