Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlemfrcn0 Structured version   Visualization version   GIF version

Theorem ballotlemfrcn0 30719
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 11446 . . . . 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 11080 . . . . . . . 8 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
52, 3, 4mp2an 708 . . . . . . 7 (𝑀 + 𝑁) ∈ ℕ
65nnzi 11439 . . . . . 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 30701 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
16 elfzelz 12380 . . . . . . . 8 (((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
1715, 16syl 17 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
18173adant3 1101 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
1918, 1zsubcld 11525 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℤ)
202, 3, 8, 9, 10, 11, 12, 13, 14ballotlemsgt1 30700 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 < ((𝑆𝐶)‘𝐽))
21 zltlem1 11468 . . . . . . 7 ((1 ∈ ℤ ∧ ((𝑆𝐶)‘𝐽) ∈ ℤ) → (1 < ((𝑆𝐶)‘𝐽) ↔ 1 ≤ (((𝑆𝐶)‘𝐽) − 1)))
2221biimpa 500 . . . . . 6 (((1 ∈ ℤ ∧ ((𝑆𝐶)‘𝐽) ∈ ℤ) ∧ 1 < ((𝑆𝐶)‘𝐽)) → 1 ≤ (((𝑆𝐶)‘𝐽) − 1))
231, 18, 20, 22syl21anc 1365 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 ≤ (((𝑆𝐶)‘𝐽) − 1))
2418zred 11520 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ∈ ℝ)
25 1red 10093 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 ∈ ℝ)
2624, 25resubcld 10496 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℝ)
27 simp1 1081 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐶 ∈ (𝑂𝐸))
282, 3, 8, 9, 10, 11, 12, 13ballotlemiex 30691 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
2928simpld 474 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
30 elfzelz 12380 . . . . . . . 8 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
3127, 29, 303syl 18 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝐼𝐶) ∈ ℤ)
3231zred 11520 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝐼𝐶) ∈ ℝ)
337zred 11520 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝑀 + 𝑁) ∈ ℝ)
34 elfzelz 12380 . . . . . . . . . . . 12 (𝐽 ∈ (1...(𝑀 + 𝑁)) → 𝐽 ∈ ℤ)
35343ad2ant2 1103 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 ∈ ℤ)
36 elfzle1 12382 . . . . . . . . . . . 12 (𝐽 ∈ (1...(𝑀 + 𝑁)) → 1 ≤ 𝐽)
37363ad2ant2 1103 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 1 ≤ 𝐽)
3835zred 11520 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 ∈ ℝ)
39 simp3 1083 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 < (𝐼𝐶))
4038, 32, 39ltled 10223 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 ≤ (𝐼𝐶))
41 elfz4 12373 . . . . . . . . . . 11 (((1 ∈ ℤ ∧ (𝐼𝐶) ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽𝐽 ≤ (𝐼𝐶))) → 𝐽 ∈ (1...(𝐼𝐶)))
421, 31, 35, 37, 40, 41syl32anc 1374 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → 𝐽 ∈ (1...(𝐼𝐶)))
432, 3, 8, 9, 10, 11, 12, 13, 14ballotlemsel1i 30702 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
4427, 42, 43syl2anc 694 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
45 elfzle2 12383 . . . . . . . . 9 (((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ≤ (𝐼𝐶))
4644, 45syl 17 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝑆𝐶)‘𝐽) ≤ (𝐼𝐶))
47 zlem1lt 11467 . . . . . . . . 9 ((((𝑆𝐶)‘𝐽) ∈ ℤ ∧ (𝐼𝐶) ∈ ℤ) → (((𝑆𝐶)‘𝐽) ≤ (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶)))
4818, 31, 47syl2anc 694 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) ≤ (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶)))
4946, 48mpbid 222 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶))
5026, 32, 49ltled 10223 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ≤ (𝐼𝐶))
51 elfzle2 12383 . . . . . . 7 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
5227, 29, 513syl 18 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
5326, 32, 33, 50, 52letrd 10232 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁))
54 elfz4 12373 . . . . 5 (((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ ℤ) ∧ (1 ≤ (((𝑆𝐶)‘𝐽) − 1) ∧ (((𝑆𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)))
551, 7, 19, 23, 53, 54syl32anc 1374 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)))
56 biid 251 . . . . . . . . 9 ((((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶))
5749, 56sylibr 224 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶))
582, 3, 8, 9, 10, 11, 12, 13ballotlemi 30690 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < ))
5958breq2d 4697 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((((𝑆𝐶)‘𝐽) − 1) < (𝐼𝐶) ↔ (((𝑆𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < )))
60593ad2ant1 1102 . . . . . . . 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, < )))
6157, 60mpbid 222 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝑆𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0}, ℝ, < ))
62 ltso 10156 . . . . . . . . . 10 < Or ℝ
6362a1i 11 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → < Or ℝ)
642, 3, 8, 9, 10, 11, 12, 13ballotlemsup 30694 . . . . . . . . 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)))
6563, 64inflb 8436 . . . . . . . 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, < )))
6665con2d 129 . . . . . . 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}))
6727, 61, 66sylc 65 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ¬ (((𝑆𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0})
68 fveq2 6229 . . . . . . . 8 (𝑘 = (((𝑆𝐶)‘𝐽) − 1) → ((𝐹𝐶)‘𝑘) = ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)))
6968eqeq1d 2653 . . . . . . 7 (𝑘 = (((𝑆𝐶)‘𝐽) − 1) → (((𝐹𝐶)‘𝑘) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
7069elrab 3396 . . . . . 6 ((((𝑆𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝐶)‘𝑘) = 0} ↔ ((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
7167, 70sylnib 317 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ¬ ((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
72 imnan 437 . . . . 5 (((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) → ¬ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0) ↔ ¬ ((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
7371, 72sylibr 224 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((((𝑆𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) → ¬ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
7455, 73mpd 15 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ¬ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0)
7574neqned 2830 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ≠ 0)
76 ballotth.r . . . . . . . . . 10 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
772, 3, 8, 9, 10, 11, 12, 13, 14, 76ballotlemro 30712 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
7877adantr 480 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
79 elfzelz 12380 . . . . . . . . 9 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
8079adantl 481 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
812, 3, 8, 9, 10, 78, 80ballotlemfelz 30680 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℤ)
8281zcnd 11521 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℂ)
8382negeq0d 10422 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹‘(𝑅𝐶))‘𝐽) = 0 ↔ -((𝐹‘(𝑅𝐶))‘𝐽) = 0))
84 eqid 2651 . . . . . . 7 (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣𝑢)) − (#‘(𝑣𝑢))))
852, 3, 8, 9, 10, 11, 12, 13, 14, 76, 84ballotlemfrceq 30718 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽))
8685eqeq1d 2653 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0 ↔ -((𝐹‘(𝑅𝐶))‘𝐽) = 0))
8783, 86bitr4d 271 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹‘(𝑅𝐶))‘𝐽) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = 0))
8887necon3bid 2867 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹‘(𝑅𝐶))‘𝐽) ≠ 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ≠ 0))
8927, 42, 88syl2anc 694 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼𝐶)) → (((𝐹‘(𝑅𝐶))‘𝐽) ≠ 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ≠ 0))
9075, 89mpbird 247 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  cdif 3604  cin 3606  ifcif 4119  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762   Or wor 5063  cima 5146  cfv 5926  (class class class)co 6690  cmpt2 6692  Fincfn 7997  infcinf 8388  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  -cneg 10305   / cdiv 10722  cn 11058  cz 11415  ...cfz 12364  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-hash 13158
This theorem is referenced by:  ballotlemirc  30721
  Copyright terms: Public domain W3C validator