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Theorem ballotlemfrc 31683
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 31670 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
1110simpld 495 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
12 f1of1 6607 . . . . . . 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 481 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
151, 2, 3, 4, 5, 6, 7, 8ballotlemiex 31658 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1615simpld 495 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
1716adantr 481 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
18 elfzuz3 12893 . . . . . . . . 9 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
1917, 18syl 17 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
20 elfzuz3 12893 . . . . . . . . 9 (𝐽 ∈ (1...(𝐼𝐶)) → (𝐼𝐶) ∈ (ℤ𝐽))
2120adantl 482 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (ℤ𝐽))
22 uztrn 12249 . . . . . . . 8 (((𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)) ∧ (𝐼𝐶) ∈ (ℤ𝐽)) → (𝑀 + 𝑁) ∈ (ℤ𝐽))
2319, 21, 22syl2anc 584 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ𝐽))
24 fzss2 12935 . . . . . . 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 4211 . . . . . 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 6622 . . . . 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 584 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
30 ovex 7178 . . . . . 6 (1...𝐽) ∈ V
3130inex1 5212 . . . . 5 ((1...𝐽) ∩ (𝑅𝐶)) ∈ V
3231f1oen 8518 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) → ((1...𝐽) ∩ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
33 hasheni 13696 . . . 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 4116 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36 f1ores 6622 . . . . 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 584 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
38 difexg 5222 . . . . . 6 ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅𝐶)) ∈ V)
3930, 38ax-mp 5 . . . . 5 ((1...𝐽) ∖ (𝑅𝐶)) ∈ V
4039f1oen 8518 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
41 hasheni 13696 . . . 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 7163 . 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 31679 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
4645adantr 481 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
47 elfzelz 12896 . . . 4 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
4847adantl 482 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
491, 2, 3, 4, 5, 46, 48ballotlemfval 31646 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅𝐶))) − (♯‘((1...𝐽) ∖ (𝑅𝐶)))))
50 fzfi 13328 . . . . 5 (1...(𝑀 + 𝑁)) ∈ Fin
51 eldifi 4100 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
521, 2, 3ballotlemelo 31644 . . . . . . . 8 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
5352simplbi 498 . . . . . . 7 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
5451, 53syl 17 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
5554adantr 481 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
56 ssfi 8726 . . . . 5 (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
5750, 55, 56sylancr 587 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ∈ Fin)
58 fzfid 13329 . . . 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 31680 . . . 4 ((𝐶 ∈ Fin ∧ (((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∈ Fin) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
6157, 58, 60syl2anc 584 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
62 dff1o3 6614 . . . . . . . . 9 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ↔ ((𝑆𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) ∧ Fun (𝑆𝐶)))
6362simprbi 497 . . . . . . . 8 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun (𝑆𝐶))
64 imain 6432 . . . . . . . 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 481 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))))
671, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsima 31672 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) = (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))
681, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemscr 31675 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
6968adantr 481 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
7067, 69ineq12d 4187 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7166, 70eqtrd 2853 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7271fveq2d 6667 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)))
73 imadif 6431 . . . . . . . 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 481 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))))
7667, 69difeq12d 4097 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7775, 76eqtrd 2853 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7877fveq2d 6667 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶)))
7972, 78oveq12d 7163 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
8061, 79eqtr4d 2856 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))))
8143, 49, 803eqtr4d 2863 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cdif 3930  cin 3932  wss 3933  ifcif 4463  𝒫 cpw 4535   class class class wbr 5057  cmpt 5137  ccnv 5547  cres 5550  cima 5551  Fun wfun 6342  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  cen 8494  Fincfn 8497  infcinf 8893  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  cz 11969  cuz 12231  ...cfz 12880  chash 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-hash 13679
This theorem is referenced by:  ballotlemfrci  31684  ballotlemfrceq  31685
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