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

Theorem ballotlemfrc 34530
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 34517 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ (𝑆𝐶) = (𝑆𝐶)))
1110simpld 494 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
12 f1of1 6846 . . . . . . 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 480 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑆𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
151, 2, 3, 4, 5, 6, 7, 8ballotlemiex 34505 . . . . . . . . . . 11 (𝐶 ∈ (𝑂𝐸) → ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹𝐶)‘(𝐼𝐶)) = 0))
1615simpld 494 . . . . . . . . . 10 (𝐶 ∈ (𝑂𝐸) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
1716adantr 480 . . . . . . . . 9 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (1...(𝑀 + 𝑁)))
18 elfzuz3 13562 . . . . . . . . 9 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
1917, 18syl 17 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
20 elfzuz3 13562 . . . . . . . . 9 (𝐽 ∈ (1...(𝐼𝐶)) → (𝐼𝐶) ∈ (ℤ𝐽))
2120adantl 481 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (ℤ𝐽))
22 uztrn 12897 . . . . . . . 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 13605 . . . . . . 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 4245 . . . . . 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 6861 . . . . 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 7465 . . . . . 6 (1...𝐽) ∈ V
3130inex1 5316 . . . . 5 ((1...𝐽) ∩ (𝑅𝐶)) ∈ V
3231f1oen 9014 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) → ((1...𝐽) ∩ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
33 hasheni 14388 . . . 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 4146 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36 f1ores 6861 . . . . 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 5328 . . . . . 6 ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅𝐶)) ∈ V)
3930, 38ax-mp 5 . . . . 5 ((1...𝐽) ∖ (𝑅𝐶)) ∈ V
4039f1oen 9014 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
41 hasheni 14388 . . . 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 7450 . 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 34526 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
4645adantr 480 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
47 elfzelz 13565 . . . 4 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
4847adantl 481 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
491, 2, 3, 4, 5, 46, 48ballotlemfval 34493 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅𝐶))) − (♯‘((1...𝐽) ∖ (𝑅𝐶)))))
50 fzfi 14014 . . . . 5 (1...(𝑀 + 𝑁)) ∈ Fin
51 eldifi 4130 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
521, 2, 3ballotlemelo 34491 . . . . . . . 8 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
5352simplbi 497 . . . . . . 7 (𝐶𝑂𝐶 ⊆ (1...(𝑀 + 𝑁)))
5451, 53syl 17 . . . . . 6 (𝐶 ∈ (𝑂𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
5554adantr 480 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
56 ssfi 9214 . . . . 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 14015 . . . 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 34527 . . . 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 6853 . . . . . . . . 9 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ↔ ((𝑆𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) ∧ Fun (𝑆𝐶)))
6362simprbi 496 . . . . . . . 8 ((𝑆𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun (𝑆𝐶))
64 imain 6650 . . . . . . . 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 480 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))))
671, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsima 34519 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) = (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))
681, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemscr 34522 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
6968adantr 480 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
7067, 69ineq12d 4220 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7166, 70eqtrd 2776 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7271fveq2d 6909 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)))
73 imadif 6649 . . . . . . . 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 480 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))))
7667, 69difeq12d 4126 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7775, 76eqtrd 2776 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7877fveq2d 6909 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶)))
7972, 78oveq12d 7450 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
8061, 79eqtr4d 2779 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))))
8143, 49, 803eqtr4d 2786 1 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  cdif 3947  cin 3949  wss 3950  ifcif 4524  𝒫 cpw 4599   class class class wbr 5142  cmpt 5224  ccnv 5683  cres 5686  cima 5687  Fun wfun 6554  1-1wf1 6557  ontowfo 6558  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  cmpo 7434  cen 8983  Fincfn 8986  infcinf 9482  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   < clt 11296  cle 11297  cmin 11493   / cdiv 11921  cn 12267  cz 12615  cuz 12879  ...cfz 13548  chash 14370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-hash 14371
This theorem is referenced by:  ballotlemfrci  34531  ballotlemfrceq  34532
  Copyright terms: Public domain W3C validator