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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlemfrceq Structured version   Visualization version   GIF version

Theorem ballotlemfrceq 31399
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 31383 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
11 1zzd 11867 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 1 ∈ ℤ)
121, 2, 3, 4, 5, 6, 7, 8ballotlemiex 31372 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1312adantr 481 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1413simpld 495 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
15 elfzelz 12762 . . . . . . . . . 10 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
1614, 15syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ ℤ)
17 elfzuz3 12759 . . . . . . . . . . . . 13 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
18 fzss2 12801 . . . . . . . . . . . . 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 485 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝐼𝐶)))
2119, 20sseldd 3896 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
221, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsdom 31382 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
2321, 22syldan 591 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
24 elfzelz 12762 . . . . . . . . . 10 (((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
2523, 24syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
26 fzsubel 12797 . . . . . . . . 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 835 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)) ↔ (((𝑆𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼𝐶) − 1))))
2810, 27mpbid 233 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼𝐶) − 1)))
29 1m1e0 11563 . . . . . . . 8 (1 − 1) = 0
3029oveq1i 7033 . . . . . . 7 ((1 − 1)...((𝐼𝐶) − 1)) = (0...((𝐼𝐶) − 1))
3128, 30syl6eleq 2895 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1)))
3212simpld 495 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
3332, 15syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
34 1zzd 11867 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → 1 ∈ ℤ)
3533, 34zsubcld 11946 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℤ)
36 nnaddcl 11514 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
371, 2, 36mp2an 688 . . . . . . . . . . 11 (𝑀 + 𝑁) ∈ ℕ
3837nnzi 11860 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℤ
3938a1i 11 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℤ)
40 elfzle2 12765 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
4132, 40syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
42 zlem1lt 11888 . . . . . . . . . . . 12 (((𝐼𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4333, 39, 42syl2anc 584 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4435zred 11941 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℝ)
4539zred 11941 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℝ)
46 ltle 10582 . . . . . . . . . . . 12 ((((𝐼𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4744, 45, 46syl2anc 584 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (((𝐼𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4843, 47sylbid 241 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4941, 48mpd 15 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁))
50 eluz2 12103 . . . . . . . . 9 ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) ↔ (((𝐼𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
5135, 39, 49, 50syl3anbrc 1336 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)))
52 fzss2 12801 . . . . . . . 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 3895 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
5531, 54syldan 591 . . . . 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 31396 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = (𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))))
5955, 58syldan 591 . . . 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 31397 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
6159, 60oveq12d 7041 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
62 fzsplit3 30199 . . . . . 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 7039 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = (𝐶 ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
65 fz1ssfz0 12857 . . . . . . . 8 (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
6665sseli 3891 . . . . . . 7 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ (0...(𝑀 + 𝑁)))
671, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57ballotlemfg 31396 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
6866, 67sylan2 592 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
6914, 68syldan 591 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
7013simprd 496 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = 0)
7169, 70eqtr3d 2835 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = 0)
72 fzfi 13194 . . . . . . 7 (1...(𝑀 + 𝑁)) ∈ Fin
73 eldifi 4030 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
741, 2, 3ballotlemelo 31358 . . . . . . . . 9 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
7574simplbi 498 . . . . . . . 8 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
7673, 75syl 17 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
77 ssfi 8591 . . . . . . 7 (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
7872, 76, 77sylancr 587 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ∈ Fin)
7978adantr 481 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ∈ Fin)
80 fzfid 13195 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...(((𝑆𝐶)‘𝐽) − 1)) ∈ Fin)
81 fzfid 13195 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin)
8225zred 11941 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℝ)
83 ltm1 11336 . . . . . 6 (((𝑆𝐶)‘𝐽) ∈ ℝ → (((𝑆𝐶)‘𝐽) − 1) < ((𝑆𝐶)‘𝐽))
84 fzdisj 12788 . . . . . 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 31395 . . . 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 2842 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))) = 0)
8861, 87eqtrd 2833 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0)
8973adantr 481 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶𝑂)
9025, 11zsubcld 11946 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℤ)
911, 2, 3, 4, 5, 89, 90ballotlemfelz 31361 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℤ)
9291zcnd 11942 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℂ)
931, 2, 3, 4, 5, 6, 7, 8, 9, 56ballotlemro 31393 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
9493adantr 481 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
95 elfzelz 12762 . . . . . 6 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
9620, 95syl 17 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
971, 2, 3, 4, 5, 94, 96ballotlemfelz 31361 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℤ)
9897zcnd 11942 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℂ)
99 addeq0 10917 . . 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 584 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽)))
10188, 100mpbid 233 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  wral 3107  {crab 3111  cdif 3862  cun 3863  cin 3864  wss 3865  c0 4217  ifcif 4387  𝒫 cpw 4459   class class class wbr 4968  cmpt 5047  cima 5453  cfv 6232  (class class class)co 7023  cmpo 7025  Fincfn 8364  infcinf 8758  cc 10388  cr 10389  0cc0 10390  1c1 10391   + caddc 10393   < clt 10528  cle 10529  cmin 10723  -cneg 10724   / cdiv 11151  cn 11492  cz 11835  cuz 12097  ...cfz 12746  chash 13544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-inf 8760  df-dju 9183  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-n0 11752  df-z 11836  df-uz 12098  df-rp 12244  df-fz 12747  df-hash 13545
This theorem is referenced by:  ballotlemfrcn0  31400
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