Theorem ballotfilemfrc 13253

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
ballotfilem.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfilem.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
ballotlemg	⊢ ↑ = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))

Assertion

Ref	Expression
ballotfilemfrc	⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))

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   𝑢,𝑂,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)

Proof of Theorem ballotfilemfrc

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . 5 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . 5 ⊢ 𝑁 ∈ ℕ
3		ballotfilem.o	. . . . 5 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
4		ballotfilem.p	. . . . . . 7 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
5		ballotth.f	. . . . . . 7 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . . . . 7 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . . . . 7 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . . . . 7 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9		ballotth.s	. . . . . . 7 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10		ballotth.r	. . . . . . 7 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemro 13249	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
12	11	adantr 276	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
13		elfzelz 10411	. . . . . 6 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
14	13	adantl 277	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
15	1, 2, 3, 12, 14	ballotfilemcinfi 13207	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∩ (𝑅‘𝐶)) ∈ Fin)
16	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotfilemsf1o 13240	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))
17	16	simpld 112	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
18		f1of1 5636	. . . . . . 7 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
19	17, 18	syl 14	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
20	19	adantr 276	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
21	1, 2, 3, 4, 5, 6, 7, 8	ballotfilemiex 13227	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
22	21	simpld 112	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
23	22	adantr 276	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
24		elfzuz3 10408	. . . . . . . . 9 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)))
25	23, 24	syl 14	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)))
26		elfzuz3 10408	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
27	26	adantl 277	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
28		uztrn 9922	. . . . . . . 8 ⊢ (((𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)) ∧ (𝐼‘𝐶) ∈ (ℤ≥‘𝐽)) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
29	25, 27, 28	syl2anc 415	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
30		fzss2 10453	. . . . . . 7 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝐽) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
31	29, 30	syl 14	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
32		ssinss1 3460	. . . . . 6 ⊢ ((1...𝐽) ⊆ (1...(𝑀 + 𝑁)) → ((1...𝐽) ∩ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
33	31, 32	syl 14	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∩ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
34		f1ores 5652	. . . . 5 ⊢ (((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ ((1...𝐽) ∩ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶) ↾ ((1...𝐽) ∩ (𝑅‘𝐶))):((1...𝐽) ∩ (𝑅‘𝐶))–1-1-onto→((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))))
35	20, 33, 34	syl2anc 415	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) ↾ ((1...𝐽) ∩ (𝑅‘𝐶))):((1...𝐽) ∩ (𝑅‘𝐶))–1-1-onto→((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))))
36	15, 35	fihasheqf1od 11211	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((1...𝐽) ∩ (𝑅‘𝐶))) = (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))))
37	1, 2, 3, 12, 14	ballotfilemdifcfi 13208	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ Fin)
38	31	ssdifssd 3367	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∖ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
39		f1ores 5652	. . . . 5 ⊢ (((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)) ∧ ((1...𝐽) ∖ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶) ↾ ((1...𝐽) ∖ (𝑅‘𝐶))):((1...𝐽) ∖ (𝑅‘𝐶))–1-1-onto→((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))))
40	20, 38, 39	syl2anc 415	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) ↾ ((1...𝐽) ∖ (𝑅‘𝐶))):((1...𝐽) ∖ (𝑅‘𝐶))–1-1-onto→((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))))
41	37, 40	fihasheqf1od 11211	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((1...𝐽) ∖ (𝑅‘𝐶))) = (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))))
42	36, 41	oveq12d 6097	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((♯‘((1...𝐽) ∩ (𝑅‘𝐶))) − (♯‘((1...𝐽) ∖ (𝑅‘𝐶)))) = ((♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))) − (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))))))
43	1, 2, 3, 4, 5, 12, 14	ballotfilemfval 13212	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅‘𝐶))) − (♯‘((1...𝐽) ∖ (𝑅‘𝐶)))))
44		ballotlemg	. . . 4 ⊢ ↑ = (𝑢 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
45		eldifi 3351	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
46	45	adantr 276	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ 𝑂)
47	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotfilemsel1i 13239	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
48	47	elfzelzd 10412	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
49	23	elfzelzd 10412	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ ℤ)
50		eqidd 2239	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) = (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))
51	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 44, 46, 48, 49, 50	ballotfilemgval 13250	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) = ((♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶)) − (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))))
52		dff1o3 5643	. . . . . . . . 9 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ↔ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) ∧ Fun ◡(𝑆‘𝐶)))
53	52	simprbi 275	. . . . . . . 8 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun ◡(𝑆‘𝐶))
54		imain 5461	. . . . . . . 8 ⊢ (Fun ◡(𝑆‘𝐶) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
55	17, 53, 54	3syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
56	55	adantr 276	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
57	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotfilemsima 13242	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (1...𝐽)) = (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	ballotfilemscr 13245	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
59	58	adantr 276	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
60	57, 59	ineq12d 3433	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶))
61	56, 60	eqtrd 2271	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶))
62	61	fveq2d 5697	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶)))
63		imadif 5459	. . . . . . . 8 ⊢ (Fun ◡(𝑆‘𝐶) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
64	17, 53, 63	3syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
65	64	adantr 276	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
66	57, 59	difeq12d 3348	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))
67	65, 66	eqtrd 2271	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))
68	67	fveq2d 5697	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶)))
69	62, 68	oveq12d 6097	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))) − (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))))) = ((♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶)) − (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))))
70	51, 69	eqtr4d 2274	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) = ((♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))) − (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))))))
71	42, 43, 70	3eqtr4d 2281	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ∖ cdif 3217   ∩ cin 3219   ⊆ wss 3220  ifcif 3638  𝒫 cpw 3688   class class class wbr 4128   ↦ cmpt 4190  ^◡ccnv 4771   ↾ cres 4774   “ cima 4775  Fun wfun 5369  –1-1→wf1 5372  –onto→wfo 5373  –1-1-onto→wf1o 5374  ‘cfv 5375  (class class class)co 6079   ∈ cmpo 6081  Fincfn 7016  infcinf 7317  ℝcr 8172  0cc0 8173  1c1 8174   + caddc 8176   < clt 8354   ≤ cle 8355   − cmin 8491   / cdiv 8996  ℕcn 9287  ℤcz 9627  ℤ_≥cuz 9904  ...cfz 10394  ♯chash 11197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-ihash 11198

This theorem is referenced by:  ballotfilemfrci  13254  ballotfilemfrceq  13255