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Theorem ballotlemfrcn0 34533
Description: Value of 𝐹 for a reversed counting (𝑅𝐶), before the first tie, cannot be zero. (Contributed by Thierry Arnoux, 25-Apr-2017.) (Revised by AV, 6-Oct-2020.)
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)))
ballotth.r 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
Assertion
Ref Expression
ballotlemfrcn0 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝐹‘(𝑅𝐶))‘𝐽) ≠ 0)
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   𝑘,𝐽   𝑆,𝑘,𝑖,𝑐   𝑅,𝑖   𝑖,𝐽
Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑘,𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝐽(𝑥,𝑐)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemfrcn0
Dummy variables 𝑣 𝑢 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1zzd 12650 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 ∈ ℤ)
2 ballotth.m . . . . . . . 8 𝑀 ∈ ℕ
3 ballotth.n . . . . . . . 8 𝑁 ∈ ℕ
4 nnaddcl 12290 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
52, 3, 4mp2an 692 . . . . . . 7 (𝑀 + 𝑁) ∈ ℕ
65nnzi 12643 . . . . . 6 (𝑀 + 𝑁) ∈ ℤ
76a1i 11 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝑀 + 𝑁) ∈ ℤ)
8 ballotth.o . . . . . . . . 9 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
9 ballotth.p . . . . . . . . 9 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
10 ballotth.f . . . . . . . . 9 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
11 ballotth.e . . . . . . . . 9 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
12 ballotth.mgtn . . . . . . . . 9 𝑁 < 𝑀
13 ballotth.i . . . . . . . . 9 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
14 ballotth.s . . . . . . . . 9 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
152, 3, 8, 9, 10, 11, 12, 13, 14ballotlemsdom 34515 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
1615elfzelzd 13566 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
17163adant3 1132 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
1817, 1zsubcld 12729 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℤ)
192, 3, 8, 9, 10, 11, 12, 13, 14ballotlemsgt1 34514 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 < ((𝑆𝐶)‘𝐽))
20 zltlem1 12672 . . . . . . 7 ((1 ∈ ℤ ∧ ((𝑆𝐶)‘𝐽) ∈ ℤ) → (1 < ((𝑆𝐶)‘𝐽) ↔ 1 ≤ (((𝑆𝐶)‘𝐽) − 1)))
2120biimpa 476 . . . . . 6 (((1 ∈ ℤ ∧ ((𝑆𝐶)‘𝐽) ∈ ℤ) ∧ 1 < ((𝑆𝐶)‘𝐽)) → 1 ≤ (((𝑆𝐶)‘𝐽) − 1))
221, 17, 19, 21syl21anc 837 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 ≤ (((𝑆𝐶)‘𝐽) − 1))
2317zred 12724 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ∈ ℝ)
24 1red 11263 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 ∈ ℝ)
2523, 24resubcld 11692 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℝ)
26 simp1 1136 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐶 ∈ (𝑂𝐸))
272, 3, 8, 9, 10, 11, 12, 13ballotlemiex 34505 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
2827simpld 494 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
29 elfzelz 13565 . . . . . . . 8 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
3026, 28, 293syl 18 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝐼𝐶) ∈ ℤ)
3130zred 12724 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝐼𝐶) ∈ ℝ)
327zred 12724 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝑀 + 𝑁) ∈ ℝ)
33 elfzelz 13565 . . . . . . . . . . . 12 (𝐽 ∈ (1...(𝑀 + 𝑁)) → 𝐽 ∈ ℤ)
34333ad2ant2 1134 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 ∈ ℤ)
35 elfzle1 13568 . . . . . . . . . . . 12 (𝐽 ∈ (1...(𝑀 + 𝑁)) → 1 ≤ 𝐽)
36353ad2ant2 1134 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 ≤ 𝐽)
3734zred 12724 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 ∈ ℝ)
38 simp3 1138 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 < (𝐼𝐶))
3937, 31, 38ltled 11410 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 ≤ (𝐼𝐶))
401, 30, 34, 36, 39elfzd 13556 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 ∈ (1...(𝐼𝐶)))
412, 3, 8, 9, 10, 11, 12, 13, 14ballotlemsel1i 34516 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
4226, 40, 41syl2anc 584 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
43 elfzle2 13569 . . . . . . . . 9 (((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ≤ (𝐼𝐶))
4442, 43syl 17 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ≤ (𝐼𝐶))
45 zlem1lt 12671 . . . . . . . . 9 ((((𝑆𝐶)‘𝐽) ∈ ℤ ∧ (𝐼𝐶) ∈ ℤ) → (((𝑆𝐶)‘𝐽) ≤ (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶)))
4617, 30, 45syl2anc 584 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) ≤ (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶)))
4744, 46mpbid 232 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶))
4825, 31, 47ltled 11410 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ≤ (𝐼𝐶))
49 elfzle2 13569 . . . . . . 7 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
5026, 28, 493syl 18 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
5125, 31, 32, 48, 50letrd 11419 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁))
521, 7, 18, 22, 51elfzd 13556 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)))
53 biid 261 . . . . . . . . 9 ((((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶))
5447, 53sylibr 234 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶))
552, 3, 8, 9, 10, 11, 12, 13ballotlemi 34504 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < ))
5655breq2d 5154 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < )))
57563ad2ant1 1133 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < )))
5854, 57mpbid 232 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < ))
59 ltso 11342 . . . . . . . . . 10 < Or ℝ
6059a1i 11 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → < Or ℝ)
612, 3, 8, 9, 10, 11, 12, 13ballotlemsup 34508 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}𝑦 < 𝑤)))
6260, 61inflb 9530 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((((𝑆𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} → ¬ (((𝑆𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < )))
6362con2d 134 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → ((((𝑆𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < ) → ¬ (((𝑆𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}))
6426, 58, 63sylc 65 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ¬ (((𝑆𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0})
65 fveqeq2 6914 . . . . . . 7 (𝑘 = (((𝑆𝐶)‘𝐽) − 1) → (((𝐹𝐶)‘𝑘) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
6665elrab 3691 . . . . . 6 ((((𝑆𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ↔ ((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
6764, 66sylnib 328 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ¬ ((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
68 imnan 399 . . . . 5 (((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) → ¬ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0) ↔ ¬ ((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
6967, 68sylibr 234 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) → ¬ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
7052, 69mpd 15 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ¬ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0)
7170neqned 2946 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ≠ 0)
72 ballotth.r . . . . . . . . . 10 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
732, 3, 8, 9, 10, 11, 12, 13, 14, 72ballotlemro 34526 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
7473adantr 480 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
75 elfzelz 13565 . . . . . . . . 9 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
7675adantl 481 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
772, 3, 8, 9, 10, 74, 76ballotlemfelz 34494 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℤ)
7877zcnd 12725 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℂ)
7978negeq0d 11613 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹‘(𝑅𝐶))‘𝐽) = 0 ↔ -((𝐹‘(𝑅𝐶))‘𝐽) = 0))
80 eqid 2736 . . . . . . 7 (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
812, 3, 8, 9, 10, 11, 12, 13, 14, 72, 80ballotlemfrceq 34532 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽))
8281eqeq1d 2738 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0 ↔ -((𝐹‘(𝑅𝐶))‘𝐽) = 0))
8379, 82bitr4d 282 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹‘(𝑅𝐶))‘𝐽) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
8483necon3bid 2984 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹‘(𝑅𝐶))‘𝐽) ≠ 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ≠ 0))
8526, 40, 84syl2anc 584 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝐹‘(𝑅𝐶))‘𝐽) ≠ 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ≠ 0))
8671, 85mpbird 257 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝐹‘(𝑅𝐶))‘𝐽) ≠ 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  {crab 3435  cdif 3947  cin 3949  ifcif 4524  𝒫 cpw 4599   class class class wbr 5142  cmpt 5224   Or wor 5590  cima 5687  cfv 6560  (class class class)co 7432  cmpo 7434  Fincfn 8986  infcinf 9482  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   < clt 11296  cle 11297  cmin 11493  -cneg 11494   / cdiv 11921  cn 12267  cz 12615  ...cfz 13548  chash 14370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-hash 14371
This theorem is referenced by:  ballotlemirc  34535
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