Metamath Proof Explorer

Theorem ballotlemrv1

Description: Value of R before the tie. (Contributed by Thierry Arnoux, 11-Apr-2017)

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 ballotlemrv1 ⊢ C ∈ O ∖ E ∧ J ∈ 1 … M + N ∧ J ≤ I ⁡ C → J ∈ R ⁡ C ↔ I ⁡ C + 1 - J ∈ 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 ballotlemrv ⊢ C ∈ O ∖ E ∧ J ∈ 1 … M + N → J ∈ R ⁡ C ↔ if J ≤ I ⁡ C I ⁡ C + 1 - J J ∈ C
12 11 3adant3 ⊢ C ∈ O ∖ E ∧ J ∈ 1 … M + N ∧ J ≤ I ⁡ C → J ∈ R ⁡ C ↔ if J ≤ I ⁡ C I ⁡ C + 1 - J J ∈ C
13 iftrue ⊢ J ≤ I ⁡ C → if J ≤ I ⁡ C I ⁡ C + 1 - J J = I ⁡ C + 1 - J
14 13 eleq1d ⊢ J ≤ I ⁡ C → if J ≤ I ⁡ C I ⁡ C + 1 - J J ∈ C ↔ I ⁡ C + 1 - J ∈ C
15 14 3ad2ant3 ⊢ C ∈ O ∖ E ∧ J ∈ 1 … M + N ∧ J ≤ I ⁡ C → if J ≤ I ⁡ C I ⁡ C + 1 - J J ∈ C ↔ I ⁡ C + 1 - J ∈ C
16 12 15 bitrd ⊢ C ∈ O ∖ E ∧ J ∈ 1 … M + N ∧ J ≤ I ⁡ C → J ∈ R ⁡ C ↔ I ⁡ C + 1 - J ∈ C