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Theorem ballotlemfrceq 32015
 Description: Value of 𝐹 for a reverse counting (𝑅‘𝐶). (Contributed by Thierry Arnoux, 27-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)))
ballotth.r 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
ballotlemg = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
Assertion
Ref Expression
ballotlemfrceq ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽))
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   𝑣,𝑢,𝐶   𝑢,𝐼,𝑣   𝑢,𝐽,𝑣   𝑢,𝑅,𝑣   𝑢,𝑆,𝑣   𝑖,𝐽
Allowed substitution hints:   𝐶(𝑥,𝑐)   𝑃(𝑥,𝑣,𝑢,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑘,𝑐)   𝑆(𝑥)   𝐸(𝑥,𝑣,𝑢)   (𝑥,𝑣,𝑢,𝑖,𝑘,𝑐)   𝐹(𝑥,𝑣,𝑢)   𝐼(𝑥)   𝐽(𝑥,𝑐)   𝑀(𝑥,𝑣,𝑢)   𝑁(𝑥,𝑣,𝑢)   𝑂(𝑥,𝑣,𝑢)

Proof of Theorem ballotlemfrceq
StepHypRef Expression
1 ballotth.m . . . . . . . . 9 𝑀 ∈ ℕ
2 ballotth.n . . . . . . . . 9 𝑁 ∈ ℕ
3 ballotth.o . . . . . . . . 9 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
4 ballotth.p . . . . . . . . 9 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
5 ballotth.f . . . . . . . . 9 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6 ballotth.e . . . . . . . . 9 𝐸 = {𝑐𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹𝑐)‘𝑖)}
7 ballotth.mgtn . . . . . . . . 9 𝑁 < 𝑀
8 ballotth.i . . . . . . . . 9 𝐼 = (𝑐 ∈ (𝑂𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹𝑐)‘𝑘) = 0}, ℝ, < ))
9 ballotth.s . . . . . . . . 9 𝑆 = (𝑐 ∈ (𝑂𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼𝑐), (((𝐼𝑐) + 1) − 𝑖), 𝑖)))
101, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsel1i 31999 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
11 1zzd 12053 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 1 ∈ ℤ)
121, 2, 3, 4, 5, 6, 7, 8ballotlemiex 31988 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1312adantr 485 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1413simpld 499 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
15 elfzelz 12957 . . . . . . . . . 10 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
1614, 15syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ ℤ)
17 elfzuz3 12954 . . . . . . . . . . . . 13 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
18 fzss2 12997 . . . . . . . . . . . . 13 ((𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)) → (1...(𝐼𝐶)) ⊆ (1...(𝑀 + 𝑁)))
1914, 17, 183syl 18 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...(𝐼𝐶)) ⊆ (1...(𝑀 + 𝑁)))
20 simpr 489 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝐼𝐶)))
2119, 20sseldd 3894 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
221, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsdom 31998 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
2321, 22syldan 595 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
24 elfzelz 12957 . . . . . . . . . 10 (((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
2523, 24syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
26 fzsubel 12993 . . . . . . . . 9 (((1 ∈ ℤ ∧ (𝐼𝐶) ∈ ℤ) ∧ (((𝑆𝐶)‘𝐽) ∈ ℤ ∧ 1 ∈ ℤ)) → (((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)) ↔ (((𝑆𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼𝐶) − 1))))
2711, 16, 25, 11, 26syl22anc 838 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)) ↔ (((𝑆𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼𝐶) − 1))))
2810, 27mpbid 235 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼𝐶) − 1)))
29 1m1e0 11747 . . . . . . . 8 (1 − 1) = 0
3029oveq1i 7161 . . . . . . 7 ((1 − 1)...((𝐼𝐶) − 1)) = (0...((𝐼𝐶) − 1))
3128, 30eleqtrdi 2863 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1)))
3212simpld 499 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
3332, 15syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
34 1zzd 12053 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → 1 ∈ ℤ)
3533, 34zsubcld 12132 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℤ)
36 nnaddcl 11698 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
371, 2, 36mp2an 692 . . . . . . . . . . 11 (𝑀 + 𝑁) ∈ ℕ
3837nnzi 12046 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℤ
3938a1i 11 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℤ)
40 elfzle2 12961 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
4132, 40syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
42 zlem1lt 12074 . . . . . . . . . . . 12 (((𝐼𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4333, 39, 42syl2anc 588 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4435zred 12127 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℝ)
4539zred 12127 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℝ)
46 ltle 10768 . . . . . . . . . . . 12 ((((𝐼𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4744, 45, 46syl2anc 588 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (((𝐼𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4843, 47sylbid 243 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4941, 48mpd 15 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁))
50 eluz2 12289 . . . . . . . . 9 ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) ↔ (((𝐼𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
5135, 39, 49, 50syl3anbrc 1341 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)))
52 fzss2 12997 . . . . . . . 8 ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) → (0...((𝐼𝐶) − 1)) ⊆ (0...(𝑀 + 𝑁)))
5351, 52syl 17 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (0...((𝐼𝐶) − 1)) ⊆ (0...(𝑀 + 𝑁)))
5453sselda 3893 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
5531, 54syldan 595 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
56 ballotth.r . . . . . 6 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
57 ballotlemg . . . . . 6 = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
581, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57ballotlemfg 32012 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = (𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))))
5955, 58syldan 595 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = (𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))))
601, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57ballotlemfrc 32013 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
6159, 60oveq12d 7169 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
62 fzsplit3 30640 . . . . . 6 (((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)) → (1...(𝐼𝐶)) = ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
6310, 62syl 17 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...(𝐼𝐶)) = ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
6463oveq2d 7167 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = (𝐶 ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
65 fz1ssfz0 13053 . . . . . . . 8 (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
6665sseli 3889 . . . . . . 7 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ (0...(𝑀 + 𝑁)))
671, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57ballotlemfg 32012 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
6866, 67sylan2 596 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
6914, 68syldan 595 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
7013simprd 500 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = 0)
7169, 70eqtr3d 2796 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = 0)
72 fzfi 13390 . . . . . . 7 (1...(𝑀 + 𝑁)) ∈ Fin
73 eldifi 4033 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
741, 2, 3ballotlemelo 31974 . . . . . . . . 9 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
7574simplbi 502 . . . . . . . 8 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
7673, 75syl 17 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
77 ssfi 8743 . . . . . . 7 (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
7872, 76, 77sylancr 591 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ∈ Fin)
7978adantr 485 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ∈ Fin)
80 fzfid 13391 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...(((𝑆𝐶)‘𝐽) − 1)) ∈ Fin)
81 fzfid 13391 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin)
8225zred 12127 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℝ)
83 ltm1 11521 . . . . . 6 (((𝑆𝐶)‘𝐽) ∈ ℝ → (((𝑆𝐶)‘𝐽) − 1) < ((𝑆𝐶)‘𝐽))
84 fzdisj 12984 . . . . . 6 ((((𝑆𝐶)‘𝐽) − 1) < ((𝑆𝐶)‘𝐽) → ((1...(((𝑆𝐶)‘𝐽) − 1)) ∩ (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ∅)
8582, 83, 843syl 18 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((1...(((𝑆𝐶)‘𝐽) − 1)) ∩ (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ∅)
861, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57, 79, 80, 81, 85ballotlemgun 32011 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))) = ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
8764, 71, 863eqtr3rd 2803 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))) = 0)
8861, 87eqtrd 2794 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0)
8973adantr 485 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶𝑂)
9025, 11zsubcld 12132 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℤ)
911, 2, 3, 4, 5, 89, 90ballotlemfelz 31977 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℤ)
9291zcnd 12128 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℂ)
931, 2, 3, 4, 5, 6, 7, 8, 9, 56ballotlemro 32009 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
9493adantr 485 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
95 elfzelz 12957 . . . . . 6 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
9620, 95syl 17 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
971, 2, 3, 4, 5, 94, 96ballotlemfelz 31977 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℤ)
9897zcnd 12128 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℂ)
99 addeq0 11102 . . 3 ((((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℂ ∧ ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℂ) → ((((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽)))
10092, 98, 99syl2anc 588 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽)))
10188, 100mpbid 235 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075   ∖ cdif 3856   ∪ cun 3857   ∩ cin 3858   ⊆ wss 3859  ∅c0 4226  ifcif 4421  𝒫 cpw 4495   class class class wbr 5033   ↦ cmpt 5113   “ cima 5528  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153  Fincfn 8528  infcinf 8939  ℂcc 10574  ℝcr 10575  0cc0 10576  1c1 10577   + caddc 10579   < clt 10714   ≤ cle 10715   − cmin 10909  -cneg 10910   / cdiv 11336  ℕcn 11675  ℤcz 12021  ℤ≥cuz 12283  ...cfz 12940  ♯chash 13741 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10632  ax-resscn 10633  ax-1cn 10634  ax-icn 10635  ax-addcl 10636  ax-addrcl 10637  ax-mulcl 10638  ax-mulrcl 10639  ax-mulcom 10640  ax-addass 10641  ax-mulass 10642  ax-distr 10643  ax-i2m1 10644  ax-1ne0 10645  ax-1rid 10646  ax-rnegex 10647  ax-rrecex 10648  ax-cnre 10649  ax-pre-lttri 10650  ax-pre-lttrn 10651  ax-pre-ltadd 10652  ax-pre-mulgt0 10653 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-oadd 8117  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-sup 8940  df-inf 8941  df-dju 9364  df-card 9402  df-pnf 10716  df-mnf 10717  df-xr 10718  df-ltxr 10719  df-le 10720  df-sub 10911  df-neg 10912  df-nn 11676  df-2 11738  df-n0 11936  df-z 12022  df-uz 12284  df-rp 12432  df-fz 12941  df-hash 13742 This theorem is referenced by:  ballotlemfrcn0  32016
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