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Theorem ballotlemfrc 31134
Description: Express the value of (𝐹‘(𝑅𝐶)) in terms of the newly defined . (Contributed by Thierry Arnoux, 21-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
ballotlemfrc ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
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 ballotlemfrc
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, 9ballotlemsf1o 31121 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
1110simpld 490 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
12 f1of1 6377 . . . . . . 7 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
1311, 12syl 17 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
1413adantr 474 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
151, 2, 3, 4, 5, 6, 7, 8ballotlemiex 31109 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1615simpld 490 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
1716adantr 474 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
18 elfzuz3 12632 . . . . . . . . 9 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
1917, 18syl 17 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
20 elfzuz3 12632 . . . . . . . . 9 (𝐽 ∈ (1...(𝐼𝐶)) → (𝐼𝐶) ∈ (ℤ𝐽))
2120adantl 475 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (ℤ𝐽))
22 uztrn 11985 . . . . . . . 8 (((𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)) ∧ (𝐼𝐶) ∈ (ℤ𝐽)) → (𝑀 + 𝑁) ∈ (ℤ𝐽))
2319, 21, 22syl2anc 581 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ𝐽))
24 fzss2 12674 . . . . . . 7 ((𝑀 + 𝑁) ∈ (ℤ𝐽) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
2523, 24syl 17 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
26 ssinss1 4066 . . . . . 6 ((1...𝐽) ⊆ (1...(𝑀 + 𝑁)) → ((1...𝐽) ∩ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁)))
2725, 26syl 17 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((1...𝐽) ∩ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁)))
28 f1ores 6392 . . . . 5 (((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ ((1...𝐽) ∩ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁))) → ((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
2914, 27, 28syl2anc 581 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
30 ovex 6937 . . . . . 6 (1...𝐽) ∈ V
3130inex1 5024 . . . . 5 ((1...𝐽) ∩ (𝑅𝐶)) ∈ V
3231f1oen 8243 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) → ((1...𝐽) ∩ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
33 hasheni 13428 . . . 4 (((1...𝐽) ∩ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) → (♯‘((1...𝐽) ∩ (𝑅𝐶))) = (♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))))
3429, 32, 333syl 18 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((1...𝐽) ∩ (𝑅𝐶))) = (♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))))
3525ssdifssd 3975 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36 f1ores 6392 . . . . 5 (((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ ((1...𝐽) ∖ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁))) → ((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
3714, 35, 36syl2anc 581 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
38 difexg 5033 . . . . . 6 ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅𝐶)) ∈ V)
3930, 38ax-mp 5 . . . . 5 ((1...𝐽) ∖ (𝑅𝐶)) ∈ V
4039f1oen 8243 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
41 hasheni 13428 . . . 4 (((1...𝐽) ∖ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) → (♯‘((1...𝐽) ∖ (𝑅𝐶))) = (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶)))))
4237, 40, 413syl 18 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((1...𝐽) ∖ (𝑅𝐶))) = (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶)))))
4334, 42oveq12d 6923 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((♯‘((1...𝐽) ∩ (𝑅𝐶))) − (♯‘((1...𝐽) ∖ (𝑅𝐶)))) = ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))))
44 ballotth.r . . . . 5 𝑅 = (𝑐 ∈ (𝑂𝐸) ↦ ((𝑆𝑐) “ 𝑐))
451, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemro 31130 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
4645adantr 474 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
47 elfzelz 12635 . . . 4 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
4847adantl 475 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
491, 2, 3, 4, 5, 46, 48ballotlemfval 31097 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅𝐶))) − (♯‘((1...𝐽) ∖ (𝑅𝐶)))))
50 fzfi 13066 . . . . 5 (1...(𝑀 + 𝑁)) ∈ Fin
51 eldifi 3959 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
521, 2, 3ballotlemelo 31095 . . . . . . . 8 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
5352simplbi 493 . . . . . . 7 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
5451, 53syl 17 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
5554adantr 474 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
56 ssfi 8449 . . . . 5 (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
5750, 55, 56sylancr 583 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ∈ Fin)
58 fzfid 13067 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin)
59 ballotlemg . . . . 5 = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣𝑢)) − (♯‘(𝑣𝑢))))
601, 2, 3, 4, 5, 6, 7, 8, 9, 44, 59ballotlemgval 31131 . . . 4 ((𝐶 ∈ Fin ∧ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
6157, 58, 60syl2anc 581 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
62 dff1o3 6384 . . . . . . . . 9 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ↔ ((𝑆𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) ∧ Fun (𝑆𝐶)))
6362simprbi 492 . . . . . . . 8 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun (𝑆𝐶))
64 imain 6207 . . . . . . . 8 (Fun (𝑆𝐶) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))))
6511, 63, 643syl 18 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))))
6665adantr 474 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))))
671, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsima 31123 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) = (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))
681, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemscr 31126 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
6968adantr 474 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
7067, 69ineq12d 4042 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7166, 70eqtrd 2861 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7271fveq2d 6437 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)))
73 imadif 6206 . . . . . . . 8 (Fun (𝑆𝐶) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))))
7411, 63, 733syl 18 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))))
7574adantr 474 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))))
7667, 69difeq12d 3956 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7775, 76eqtrd 2861 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7877fveq2d 6437 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶)))
7972, 78oveq12d 6923 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
8061, 79eqtr4d 2864 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))))
8143, 49, 803eqtr4d 2871 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  wral 3117  {crab 3121  Vcvv 3414  cdif 3795  cin 3797  wss 3798  ifcif 4306  𝒫 cpw 4378   class class class wbr 4873  cmpt 4952  ccnv 5341  cres 5344  cima 5345  Fun wfun 6117  1-1wf1 6120  ontowfo 6121  1-1-ontowf1o 6122  cfv 6123  (class class class)co 6905  cmpt2 6907  cen 8219  Fincfn 8222  infcinf 8616  cr 10251  0cc0 10252  1c1 10253   + caddc 10255   < clt 10391  cle 10392  cmin 10585   / cdiv 11009  cn 11350  cz 11704  cuz 11968  ...cfz 12619  chash 13410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-n0 11619  df-z 11705  df-uz 11969  df-rp 12113  df-fz 12620  df-hash 13411
This theorem is referenced by:  ballotlemfrci  31135  ballotlemfrceq  31136
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