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Theorem ballotlemfrceq 30899
Description: Value of 𝐹 for a reverse counting (𝑅𝐶). (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 30883 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
11 1zzd 11600 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 1 ∈ ℤ)
121, 2, 3, 4, 5, 6, 7, 8ballotlemiex 30872 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1312adantr 472 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1413simpld 477 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
15 elfzelz 12535 . . . . . . . . . 10 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
1614, 15syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ ℤ)
17 elfzuz3 12532 . . . . . . . . . . . . 13 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
18 fzss2 12574 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝐼𝐶)))
2119, 20sseldd 3745 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
221, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsdom 30882 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
2321, 22syldan 488 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
24 elfzelz 12535 . . . . . . . . . 10 (((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
2523, 24syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
26 fzsubel 12570 . . . . . . . . 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 1478 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)) ↔ (((𝑆𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼𝐶) − 1))))
2810, 27mpbid 222 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼𝐶) − 1)))
29 1m1e0 11281 . . . . . . . 8 (1 − 1) = 0
3029oveq1i 6823 . . . . . . 7 ((1 − 1)...((𝐼𝐶) − 1)) = (0...((𝐼𝐶) − 1))
3128, 30syl6eleq 2849 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1)))
3212simpld 477 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
3332, 15syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
34 1zzd 11600 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → 1 ∈ ℤ)
3533, 34zsubcld 11679 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℤ)
36 nnaddcl 11234 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
371, 2, 36mp2an 710 . . . . . . . . . . 11 (𝑀 + 𝑁) ∈ ℕ
3837nnzi 11593 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℤ
3938a1i 11 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℤ)
40 elfzle2 12538 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
4132, 40syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
42 zlem1lt 11621 . . . . . . . . . . . 12 (((𝐼𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4333, 39, 42syl2anc 696 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4435zred 11674 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℝ)
4539zred 11674 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℝ)
46 ltle 10318 . . . . . . . . . . . 12 ((((𝐼𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4744, 45, 46syl2anc 696 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (((𝐼𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4843, 47sylbid 230 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4941, 48mpd 15 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁))
50 eluz2 11885 . . . . . . . . 9 ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) ↔ (((𝐼𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
5135, 39, 49, 50syl3anbrc 1429 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)))
52 fzss2 12574 . . . . . . . 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 3744 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
5531, 54syldan 488 . . . . 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 30896 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = (𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))))
5955, 58syldan 488 . . . 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 30897 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
6159, 60oveq12d 6831 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
62 fzsplit3 29862 . . . . . 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 6829 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = (𝐶 ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
65 fz1ssfz0 12629 . . . . . . . 8 (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
6665sseli 3740 . . . . . . 7 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ (0...(𝑀 + 𝑁)))
671, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57ballotlemfg 30896 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
6866, 67sylan2 492 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
6914, 68syldan 488 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
7013simprd 482 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = 0)
7169, 70eqtr3d 2796 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = 0)
72 fzfi 12965 . . . . . . 7 (1...(𝑀 + 𝑁)) ∈ Fin
73 eldifi 3875 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
741, 2, 3ballotlemelo 30858 . . . . . . . . 9 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
7574simplbi 478 . . . . . . . 8 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
7673, 75syl 17 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
77 ssfi 8345 . . . . . . 7 (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
7872, 76, 77sylancr 698 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ∈ Fin)
7978adantr 472 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ∈ Fin)
80 fzfid 12966 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...(((𝑆𝐶)‘𝐽) − 1)) ∈ Fin)
81 fzfid 12966 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin)
8225zred 11674 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℝ)
83 ltm1 11055 . . . . . 6 (((𝑆𝐶)‘𝐽) ∈ ℝ → (((𝑆𝐶)‘𝐽) − 1) < ((𝑆𝐶)‘𝐽))
84 fzdisj 12561 . . . . . 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 30895 . . . 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 472 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶𝑂)
9025, 11zsubcld 11679 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℤ)
911, 2, 3, 4, 5, 89, 90ballotlemfelz 30861 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℤ)
9291zcnd 11675 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℂ)
931, 2, 3, 4, 5, 6, 7, 8, 9, 56ballotlemro 30893 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
9493adantr 472 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
95 elfzelz 12535 . . . . . 6 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
9620, 95syl 17 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
971, 2, 3, 4, 5, 94, 96ballotlemfelz 30861 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℤ)
9897zcnd 11675 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℂ)
99 addeq0 29819 . . 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 696 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽)))
10188, 100mpbid 222 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058  ifcif 4230  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881  cima 5269  cfv 6049  (class class class)co 6813  cmpt2 6815  Fincfn 8121  infcinf 8512  cc 10126  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   < clt 10266  cle 10267  cmin 10458  -cneg 10459   / cdiv 10876  cn 11212  cz 11569  cuz 11879  ...cfz 12519  chash 13311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-hash 13312
This theorem is referenced by:  ballotlemfrcn0  30900
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