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Theorem ballotlemfrc 31792
 Description: Express the value of (𝐹‘(𝑅‘𝐶)) 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 31779 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
1110simpld 497 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
12 f1of1 6590 . . . . . . 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 483 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
151, 2, 3, 4, 5, 6, 7, 8ballotlemiex 31767 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1615simpld 497 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
1716adantr 483 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
18 elfzuz3 12889 . . . . . . . . 9 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
1917, 18syl 17 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
20 elfzuz3 12889 . . . . . . . . 9 (𝐽 ∈ (1...(𝐼𝐶)) → (𝐼𝐶) ∈ (ℤ𝐽))
2120adantl 484 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (ℤ𝐽))
22 uztrn 12240 . . . . . . . 8 (((𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)) ∧ (𝐼𝐶) ∈ (ℤ𝐽)) → (𝑀 + 𝑁) ∈ (ℤ𝐽))
2319, 21, 22syl2anc 586 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ𝐽))
24 fzss2 12931 . . . . . . 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 4192 . . . . . 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 6605 . . . . 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 586 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
30 ovex 7166 . . . . . 6 (1...𝐽) ∈ V
3130inex1 5197 . . . . 5 ((1...𝐽) ∩ (𝑅𝐶)) ∈ V
3231f1oen 8508 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) → ((1...𝐽) ∩ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
33 hasheni 13693 . . . 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 4098 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36 f1ores 6605 . . . . 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 586 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
38 difexg 5207 . . . . . 6 ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅𝐶)) ∈ V)
3930, 38ax-mp 5 . . . . 5 ((1...𝐽) ∖ (𝑅𝐶)) ∈ V
4039f1oen 8508 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
41 hasheni 13693 . . . 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 7151 . 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 31788 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
4645adantr 483 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
47 elfzelz 12892 . . . 4 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
4847adantl 484 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
491, 2, 3, 4, 5, 46, 48ballotlemfval 31755 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅𝐶))) − (♯‘((1...𝐽) ∖ (𝑅𝐶)))))
50 fzfi 13324 . . . . 5 (1...(𝑀 + 𝑁)) ∈ Fin
51 eldifi 4082 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
521, 2, 3ballotlemelo 31753 . . . . . . . 8 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
5352simplbi 500 . . . . . . 7 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
5451, 53syl 17 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
5554adantr 483 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
56 ssfi 8716 . . . . 5 (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
5750, 55, 56sylancr 589 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ∈ Fin)
58 fzfid 13325 . . . 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 31789 . . . 4 ((𝐶 ∈ Fin ∧ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
6157, 58, 60syl2anc 586 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
62 dff1o3 6597 . . . . . . . . 9 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ↔ ((𝑆𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) ∧ Fun (𝑆𝐶)))
6362simprbi 499 . . . . . . . 8 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun (𝑆𝐶))
64 imain 6415 . . . . . . . 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 483 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))))
671, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsima 31781 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) = (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))
681, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemscr 31784 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
6968adantr 483 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
7067, 69ineq12d 4168 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7166, 70eqtrd 2855 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7271fveq2d 6650 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)))
73 imadif 6414 . . . . . . . 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 483 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))))
7667, 69difeq12d 4079 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7775, 76eqtrd 2855 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7877fveq2d 6650 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶)))
7972, 78oveq12d 7151 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
8061, 79eqtr4d 2858 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))))
8143, 49, 803eqtr4d 2865 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3125  {crab 3129  Vcvv 3473   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ifcif 4443  𝒫 cpw 4515   class class class wbr 5042   ↦ cmpt 5122  ◡ccnv 5530   ↾ cres 5533   “ cima 5534  Fun wfun 6325  –1-1→wf1 6328  –onto→wfo 6329  –1-1-onto→wf1o 6330  ‘cfv 6331  (class class class)co 7133   ∈ cmpo 7135   ≈ cen 8484  Fincfn 8487  infcinf 8883  ℝcr 10514  0cc0 10515  1c1 10516   + caddc 10518   < clt 10653   ≤ cle 10654   − cmin 10848   / cdiv 11275  ℕcn 11616  ℤcz 11960  ℤ≥cuz 12222  ...cfz 12876  ♯chash 13675 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-sup 8884  df-inf 8885  df-dju 9308  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-2 11679  df-n0 11877  df-z 11961  df-uz 12223  df-rp 12369  df-fz 12877  df-hash 13676 This theorem is referenced by:  ballotlemfrci  31793  ballotlemfrceq  31794
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