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Theorem ballotlemfrceq 30389
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 30373 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝐼𝐶)))
11 1zzd 11359 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 1 ∈ ℤ)
121, 2, 3, 4, 5, 6, 7, 8ballotlemiex 30362 . . . . . . . . . . . 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 475 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
15 elfzelz 12291 . . . . . . . . . 10 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ ℤ)
1614, 15syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ ℤ)
17 elfzuz3 12288 . . . . . . . . . . . . 13 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
18 fzss2 12330 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 12 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝐼𝐶)))
2119, 20sseldd 3588 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
221, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsdom 30372 . . . . . . . . . . 11 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
2321, 22syldan 487 . . . . . . . . . 10 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
24 elfzelz 12291 . . . . . . . . . 10 (((𝑆𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
2523, 24syl 17 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℤ)
26 fzsubel 12326 . . . . . . . . 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 1324 . . . . . . . 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 11040 . . . . . . . 8 (1 − 1) = 0
3029oveq1i 6620 . . . . . . 7 ((1 − 1)...((𝐼𝐶) − 1)) = (0...((𝐼𝐶) − 1))
3128, 30syl6eleq 2708 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1)))
3212simpld 475 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
3332, 15syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ ℤ)
34 1zzd 11359 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → 1 ∈ ℤ)
3533, 34zsubcld 11438 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℤ)
36 nnaddcl 10993 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
371, 2, 36mp2an 707 . . . . . . . . . . 11 (𝑀 + 𝑁) ∈ ℕ
3837nnzi 11352 . . . . . . . . . 10 (𝑀 + 𝑁) ∈ ℤ
3938a1i 11 . . . . . . . . 9 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℤ)
40 elfzle2 12294 . . . . . . . . . . 11 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
4132, 40syl 17 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ≤ (𝑀 + 𝑁))
42 zlem1lt 11380 . . . . . . . . . . . 12 (((𝐼𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4333, 39, 42syl2anc 692 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼𝐶) − 1) < (𝑀 + 𝑁)))
4435zred 11433 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) − 1) ∈ ℝ)
4539zred 11433 . . . . . . . . . . . 12 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ ℝ)
46 ltle 10077 . . . . . . . . . . . 12 ((((𝐼𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
4744, 45, 46syl2anc 692 . . . . . . . . . . 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 11644 . . . . . . . . 9 ((𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)) ↔ (((𝐼𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ ((𝐼𝐶) − 1) ≤ (𝑀 + 𝑁)))
5135, 39, 49, 50syl3anbrc 1244 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → (𝑀 + 𝑁) ∈ (ℤ‘((𝐼𝐶) − 1)))
52 fzss2 12330 . . . . . . . 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 3587 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...((𝐼𝐶) − 1))) → (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
5531, 54syldan 487 . . . . 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 30386 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ (((𝑆𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = (𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))))
5955, 58syldan 487 . . . 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 30387 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
6159, 60oveq12d 6628 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
62 fzsplit3 29412 . . . . . 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 6626 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = (𝐶 ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
65 1eluzge0 11683 . . . . . . . . 9 1 ∈ (ℤ‘0)
66 fzss1 12329 . . . . . . . . 9 (1 ∈ (ℤ‘0) → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
6765, 66ax-mp 5 . . . . . . . 8 (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
6867sseli 3583 . . . . . . 7 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼𝐶) ∈ (0...(𝑀 + 𝑁)))
691, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57ballotlemfg 30386 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
7068, 69sylan2 491 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ (𝐼𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
7114, 70syldan 487 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = (𝐶 (1...(𝐼𝐶))))
7213simprd 479 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(𝐼𝐶)) = 0)
7371, 72eqtr3d 2657 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (1...(𝐼𝐶))) = 0)
74 fzfi 12718 . . . . . . 7 (1...(𝑀 + 𝑁)) ∈ Fin
75 eldifi 3715 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
761, 2, 3ballotlemelo 30348 . . . . . . . . 9 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))
7776simplbi 476 . . . . . . . 8 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
7875, 77syl 17 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
79 ssfi 8131 . . . . . . 7 (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
8074, 78, 79sylancr 694 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ∈ Fin)
8180adantr 481 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ∈ Fin)
82 fzfid 12719 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...(((𝑆𝐶)‘𝐽) − 1)) ∈ Fin)
83 fzfid 12719 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin)
8425zred 11433 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶)‘𝐽) ∈ ℝ)
85 ltm1 10814 . . . . . 6 (((𝑆𝐶)‘𝐽) ∈ ℝ → (((𝑆𝐶)‘𝐽) − 1) < ((𝑆𝐶)‘𝐽))
86 fzdisj 12317 . . . . . 6 ((((𝑆𝐶)‘𝐽) − 1) < ((𝑆𝐶)‘𝐽) → ((1...(((𝑆𝐶)‘𝐽) − 1)) ∩ (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ∅)
8784, 85, 863syl 18 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((1...(((𝑆𝐶)‘𝐽) − 1)) ∩ (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ∅)
881, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57, 81, 82, 83, 87ballotlemgun 30385 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 ((1...(((𝑆𝐶)‘𝐽) − 1)) ∪ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))) = ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))))
8964, 73, 883eqtr3rd 2664 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐶 (1...(((𝑆𝐶)‘𝐽) − 1))) + (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))) = 0)
9061, 89eqtrd 2655 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0)
9175adantr 481 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶𝑂)
9225, 11zsubcld 11438 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽) − 1) ∈ ℤ)
931, 2, 3, 4, 5, 91, 92ballotlemfelz 30351 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℤ)
9493zcnd 11434 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) ∈ ℂ)
951, 2, 3, 4, 5, 6, 7, 8, 9, 56ballotlemro 30383 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
9695adantr 481 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
97 elfzelz 12291 . . . . . 6 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
9820, 97syl 17 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
991, 2, 3, 4, 5, 96, 98ballotlemfelz 30351 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℤ)
10099zcnd 11434 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) ∈ ℂ)
101 addeq0 29371 . . 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)))
10294, 100, 101syl2anc 692 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅𝐶))‘𝐽)) = 0 ↔ ((𝐹𝐶)‘(((𝑆𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅𝐶))‘𝐽)))
10390, 102mpbid 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 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  ifcif 4063  𝒫 cpw 4135   class class class wbr 4618  cmpt 4678  cima 5082  cfv 5852  (class class class)co 6610  cmpt2 6612  Fincfn 7906  infcinf 8298  cc 9885  cr 9886  0cc0 9887  1c1 9888   + caddc 9890   < clt 10025  cle 10026  cmin 10217  -cneg 10218   / cdiv 10635  cn 10971  cz 11328  cuz 11638  ...cfz 12275  #chash 13064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-inf 8300  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-n0 11244  df-z 11329  df-uz 11639  df-rp 11784  df-fz 12276  df-hash 13065
This theorem is referenced by:  ballotlemfrcn0  30390
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