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Theorem ballotlemfrc 33594
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 33581 . . . . . . . 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 6832 . . . . . . 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 33569 . . . . . . . . . . 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 13500 . . . . . . . . 9 ((𝐼𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
1917, 18syl 17 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑀 + 𝑁) ∈ (ℤ‘(𝐼𝐶)))
20 elfzuz3 13500 . . . . . . . . 9 (𝐽 ∈ (1...(𝐼𝐶)) → (𝐼𝐶) ∈ (ℤ𝐽))
2120adantl 482 . . . . . . . 8 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐼𝐶) ∈ (ℤ𝐽))
22 uztrn 12842 . . . . . . . 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 13543 . . . . . . 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 4237 . . . . . 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 6847 . . . . 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 7444 . . . . . 6 (1...𝐽) ∈ V
3130inex1 5317 . . . . 5 ((1...𝐽) ∩ (𝑅𝐶)) ∈ V
3231f1oen 8971 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∩ (𝑅𝐶))):((1...𝐽) ∩ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) → ((1...𝐽) ∩ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))))
33 hasheni 14310 . . . 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 4142 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36 f1ores 6847 . . . . 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 5327 . . . . . 6 ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅𝐶)) ∈ V)
3930, 38ax-mp 5 . . . . 5 ((1...𝐽) ∖ (𝑅𝐶)) ∈ V
4039f1oen 8971 . . . 4 (((𝑆𝐶) ↾ ((1...𝐽) ∖ (𝑅𝐶))):((1...𝐽) ∖ (𝑅𝐶))–1-1-onto→((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) → ((1...𝐽) ∖ (𝑅𝐶)) ≈ ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))
41 hasheni 14310 . . . 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 7429 . 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 33590 . . . 4 (𝐶 ∈ (𝑂𝐸) → (𝑅𝐶) ∈ 𝑂)
4645adantr 481 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝑅𝐶) ∈ 𝑂)
47 elfzelz 13503 . . . 4 (𝐽 ∈ (1...(𝐼𝐶)) → 𝐽 ∈ ℤ)
4847adantl 482 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → 𝐽 ∈ ℤ)
491, 2, 3, 4, 5, 46, 48ballotlemfval 33557 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝐹‘(𝑅𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅𝐶))) − (♯‘((1...𝐽) ∖ (𝑅𝐶)))))
50 fzfi 13939 . . . . 5 (1...(𝑀 + 𝑁)) ∈ Fin
51 eldifi 4126 . . . . . . 7 (𝐶 ∈ (𝑂𝐸) → 𝐶𝑂)
521, 2, 3ballotlemelo 33555 . . . . . . . 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 9175 . . . . 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 13940 . . . 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 33591 . . . 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 6839 . . . . . . . . 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 6633 . . . . . . . 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 33583 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (1...𝐽)) = (((𝑆𝐶)‘𝐽)...(𝐼𝐶)))
681, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemscr 33586 . . . . . . . 8 (𝐶 ∈ (𝑂𝐸) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
6968adantr 481 . . . . . . 7 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ (𝑅𝐶)) = 𝐶)
7067, 69ineq12d 4213 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∩ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7166, 70eqtrd 2772 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶))
7271fveq2d 6895 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)))
73 imadif 6632 . . . . . . . 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 4123 . . . . . 6 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (((𝑆𝐶) “ (1...𝐽)) ∖ ((𝑆𝐶) “ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7775, 76eqtrd 2772 . . . . 5 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))) = ((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))
7877fveq2d 6895 . . . 4 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶)))) = (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶)))
7972, 78oveq12d 7429 . . 3 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))) = ((♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∩ 𝐶)) − (♯‘((((𝑆𝐶)‘𝐽)...(𝐼𝐶)) ∖ 𝐶))))
8061, 79eqtr4d 2775 . 2 ((𝐶 ∈ (𝑂𝐸) ∧ 𝐽 ∈ (1...(𝐼𝐶))) → (𝐶 (((𝑆𝐶)‘𝐽)...(𝐼𝐶))) = ((♯‘((𝑆𝐶) “ ((1...𝐽) ∩ (𝑅𝐶)))) − (♯‘((𝑆𝐶) “ ((1...𝐽) ∖ (𝑅𝐶))))))
8143, 49, 803eqtr4d 2782 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 1541  wcel 2106  wral 3061  {crab 3432  Vcvv 3474  cdif 3945  cin 3947  wss 3948  ifcif 4528  𝒫 cpw 4602   class class class wbr 5148  cmpt 5231  ccnv 5675  cres 5678  cima 5679  Fun wfun 6537  1-1wf1 6540  ontowfo 6541  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7411  cmpo 7413  cen 8938  Fincfn 8941  infcinf 9438  cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11250  cle 11251  cmin 11446   / cdiv 11873  cn 12214  cz 12560  cuz 12824  ...cfz 13486  chash 14292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-fz 13487  df-hash 14293
This theorem is referenced by:  ballotlemfrci  33595  ballotlemfrceq  33596
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