Theorem ballotlemfrc 31134

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 = (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 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))

Assertion

Ref	Expression
ballotlemfrc	⊢ ((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   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

Step	Hyp	Ref	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 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
6		ballotth.e	. . . . . . . . 9 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
7		ballotth.mgtn	. . . . . . . . 9 ⊢ 𝑁 < 𝑀
8		ballotth.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
9		ballotth.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsf1o 31121	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))
11	10	simpld 490	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
12		f1of1 6377	. . . . . . 7 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
13	11, 12	syl 17	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
14	13	adantr 474	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
15	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 31109	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
16	15	simpld 490	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
17	16	adantr 474	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
18		elfzuz3 12632	. . . . . . . . 9 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)))
19	17, 18	syl 17	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)))
20		elfzuz3 12632	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
21	20	adantl 475	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
22		uztrn 11985	. . . . . . . 8 ⊢ (((𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)) ∧ (𝐼‘𝐶) ∈ (ℤ≥‘𝐽)) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
23	19, 21, 22	syl2anc 581	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
24		fzss2 12674	. . . . . . 7 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝐽) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
25	23, 24	syl 17	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
26		ssinss1 4066	. . . . . 6 ⊢ ((1...𝐽) ⊆ (1...(𝑀 + 𝑁)) → ((1...𝐽) ∩ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
27	25, 26	syl 17	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∩ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
28		f1ores 6392	. . . . 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))))
29	14, 27, 28	syl2anc 581	. . . 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))))
30		ovex 6937	. . . . . 6 ⊢ (1...𝐽) ∈ V
31	30	inex1 5024	. . . . 5 ⊢ ((1...𝐽) ∩ (𝑅‘𝐶)) ∈ V
32	31	f1oen 8243	. . . 4 ⊢ (((𝑆‘𝐶) ↾ ((1...𝐽) ∩ (𝑅‘𝐶))):((1...𝐽) ∩ (𝑅‘𝐶))–1-1-onto→((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) → ((1...𝐽) ∩ (𝑅‘𝐶)) ≈ ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))))
33		hasheni 13428	. . . 4 ⊢ (((1...𝐽) ∩ (𝑅‘𝐶)) ≈ ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) → (♯‘((1...𝐽) ∩ (𝑅‘𝐶))) = (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))))
34	29, 32, 33	3syl 18	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((1...𝐽) ∩ (𝑅‘𝐶))) = (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))))
35	25	ssdifssd 3975	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∖ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36		f1ores 6392	. . . . 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))))
37	14, 35, 36	syl2anc 581	. . . 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))))
38		difexg 5033	. . . . . 6 ⊢ ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ V)
39	30, 38	ax-mp 5	. . . . 5 ⊢ ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ V
40	39	f1oen 8243	. . . 4 ⊢ (((𝑆‘𝐶) ↾ ((1...𝐽) ∖ (𝑅‘𝐶))):((1...𝐽) ∖ (𝑅‘𝐶))–1-1-onto→((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) → ((1...𝐽) ∖ (𝑅‘𝐶)) ≈ ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))))
41		hasheni 13428	. . . 4 ⊢ (((1...𝐽) ∖ (𝑅‘𝐶)) ≈ ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) → (♯‘((1...𝐽) ∖ (𝑅‘𝐶))) = (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))))
42	37, 40, 41	3syl 18	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((1...𝐽) ∖ (𝑅‘𝐶))) = (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))))
43	34, 42	oveq12d 6923	. 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))))))
44		ballotth.r	. . . . 5 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
45	1, 2, 3, 4, 5, 6, 7, 8, 9, 44	ballotlemro 31130	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
46	45	adantr 474	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
47		elfzelz 12635	. . . 4 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
48	47	adantl 475	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
49	1, 2, 3, 4, 5, 46, 48	ballotlemfval 31097	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅‘𝐶))) − (♯‘((1...𝐽) ∖ (𝑅‘𝐶)))))
50		fzfi 13066	. . . . 5 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
51		eldifi 3959	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
52	1, 2, 3	ballotlemelo 31095	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
53	52	simplbi 493	. . . . . . 7 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
54	51, 53	syl 17	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
55	54	adantr 474	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
56		ssfi 8449	. . . . 5 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
57	50, 55, 56	sylancr 583	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ Fin)
58		fzfid 13067	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∈ Fin)
59		ballotlemg	. . . . 5 ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
60	1, 2, 3, 4, 5, 6, 7, 8, 9, 44, 59	ballotlemgval 31131	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∈ Fin) → (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) = ((♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶)) − (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))))
61	57, 58, 60	syl2anc 581	. . 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))))
62		dff1o3 6384	. . . . . . . . 9 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ↔ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–onto→(1...(𝑀 + 𝑁)) ∧ Fun ◡(𝑆‘𝐶)))
63	62	simprbi 492	. . . . . . . 8 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun ◡(𝑆‘𝐶))
64		imain 6207	. . . . . . . 8 ⊢ (Fun ◡(𝑆‘𝐶) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
65	11, 63, 64	3syl 18	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
66	65	adantr 474	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
67	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsima 31123	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (1...𝐽)) = (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))
68	1, 2, 3, 4, 5, 6, 7, 8, 9, 44	ballotlemscr 31126	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
69	68	adantr 474	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
70	67, 69	ineq12d 4042	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶))
71	66, 70	eqtrd 2861	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶))
72	71	fveq2d 6437	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶)))
73		imadif 6206	. . . . . . . 8 ⊢ (Fun ◡(𝑆‘𝐶) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
74	11, 63, 73	3syl 18	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
75	74	adantr 474	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
76	67, 69	difeq12d 3956	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))
77	75, 76	eqtrd 2861	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))
78	77	fveq2d 6437	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶)))
79	72, 78	oveq12d 6923	. . 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))))
80	61, 79	eqtr4d 2864	. 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))))))
81	43, 49, 80	3eqtr4d 2871	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117  {crab 3121  Vcvv 3414   ∖ cdif 3795   ∩ cin 3797   ⊆ wss 3798  ifcif 4306  𝒫 cpw 4378   class class class wbr 4873   ↦ cmpt 4952  ^◡ccnv 5341   ↾ cres 5344   “ cima 5345  Fun wfun 6117  –1-1→wf1 6120  –onto→wfo 6121  –1-1-onto→wf1o 6122  ‘cfv 6123  (class class class)co 6905   ↦ cmpt2 6907   ≈ cen 8219  Fincfn 8222  infcinf 8616  ℝcr 10251  0cc0 10252  1c1 10253   + caddc 10255   < clt 10391   ≤ cle 10392   − cmin 10585   / cdiv 11009  ℕcn 11350  ℤcz 11704  ℤ_≥cuz 11968  ...cfz 12619  ♯chash 13410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-n0 11619  df-z 11705  df-uz 11969  df-rp 12113  df-fz 12620  df-hash 13411

This theorem is referenced by:  ballotlemfrci  31135  ballotlemfrceq  31136