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

Theorem ballotlemfrceq 30930
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 30914 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
11 1zzd 11610 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 1 ∈ ℤ)
121, 2, 3, 4, 5, 6, 7, 8ballotlemiex 30903 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1312adantr 466 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1413simpld 482 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
15 elfzelz 12549 . . . . . . . . . 10 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
1614, 15syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ ℤ)
17 elfzuz3 12546 . . . . . . . . . . . . 13 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
18 fzss2 12588 . . . . . . . . . . . . 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 471 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝐼𝐶)))
2119, 20sseldd 3753 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
221, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsdom 30913 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
2321, 22syldan 579 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
24 elfzelz 12549 . . . . . . . . . 10 (((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
2523, 24syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
26 fzsubel 12584 . . . . . . . . 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 1477 . . . . . . . 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 11291 . . . . . . . 8 (1 − 1) = 0
3029oveq1i 6803 . . . . . . 7 ((1 − 1)...((𝐼𝐶) − 1)) = (0...((𝐼𝐶) − 1))
3128, 30syl6eleq 2860 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1)))
3212simpld 482 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
3332, 15syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
34 1zzd 11610 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → 1 ∈ ℤ)
3533, 34zsubcld 11689 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℤ)
36 nnaddcl 11244 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
371, 2, 36mp2an 672 . . . . . . . . . . 11 (𝑀 + 𝑁) ∈ ℕ
3837nnzi 11603 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℤ
3938a1i 11 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℤ)
40 elfzle2 12552 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
4132, 40syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
42 zlem1lt 11631 . . . . . . . . . . . 12 (((𝐼𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4333, 39, 42syl2anc 573 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4435zred 11684 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℝ)
4539zred 11684 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℝ)
46 ltle 10328 . . . . . . . . . . . 12 ((((𝐼𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4744, 45, 46syl2anc 573 . . . . . . . . . . 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 11894 . . . . . . . . 9 ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) ↔ (((𝐼𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
5135, 39, 49, 50syl3anbrc 1428 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)))
52 fzss2 12588 . . . . . . . 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 3752 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
5531, 54syldan 579 . . . . 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 30927 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = (𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))))
5955, 58syldan 579 . . . 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 30928 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
6159, 60oveq12d 6811 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
62 fzsplit3 29893 . . . . . 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 6809 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = (𝐶 ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
65 fz1ssfz0 12643 . . . . . . . 8 (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
6665sseli 3748 . . . . . . 7 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ (0...(𝑀 + 𝑁)))
671, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57ballotlemfg 30927 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
6866, 67sylan2 580 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
6914, 68syldan 579 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
7013simprd 483 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = 0)
7169, 70eqtr3d 2807 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = 0)
72 fzfi 12979 . . . . . . 7 (1...(𝑀 + 𝑁)) ∈ Fin
73 eldifi 3883 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
741, 2, 3ballotlemelo 30889 . . . . . . . . 9 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
7574simplbi 485 . . . . . . . 8 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
7673, 75syl 17 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
77 ssfi 8336 . . . . . . 7 (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
7872, 76, 77sylancr 575 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ∈ Fin)
7978adantr 466 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ∈ Fin)
80 fzfid 12980 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...(((𝑆𝐶)‘𝐽) − 1)) ∈ Fin)
81 fzfid 12980 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin)
8225zred 11684 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℝ)
83 ltm1 11065 . . . . . 6 (((𝑆𝐶)‘𝐽) ∈ ℝ → (((𝑆𝐶)‘𝐽) − 1) < ((𝑆𝐶)‘𝐽))
84 fzdisj 12575 . . . . . 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 30926 . . . 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 2814 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))) = 0)
8861, 87eqtrd 2805 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0)
8973adantr 466 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶𝑂)
9025, 11zsubcld 11689 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℤ)
911, 2, 3, 4, 5, 89, 90ballotlemfelz 30892 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℤ)
9291zcnd 11685 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℂ)
931, 2, 3, 4, 5, 6, 7, 8, 9, 56ballotlemro 30924 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
9493adantr 466 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
95 elfzelz 12549 . . . . . 6 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
9620, 95syl 17 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
971, 2, 3, 4, 5, 94, 96ballotlemfelz 30892 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℤ)
9897zcnd 11685 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℂ)
99 addeq0 29850 . . 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 573 . 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 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  ifcif 4225  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  cima 5252  cfv 6031  (class class class)co 6793  cmpt2 6795  Fincfn 8109  infcinf 8503  cc 10136  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   < clt 10276  cle 10277  cmin 10468  -cneg 10469   / cdiv 10886  cn 11222  cz 11579  cuz 11888  ...cfz 12533  chash 13321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-z 11580  df-uz 11889  df-rp 12036  df-fz 12534  df-hash 13322
This theorem is referenced by:  ballotlemfrcn0  30931
  Copyright terms: Public domain W3C validator