Theorem ballotlemfrc 34862

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 34849	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))
11	10	simpld 499	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
12		f1of1 6820	. . . . . . 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 18	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1→(1...(𝑀 + 𝑁)))
14	13	adantr 485	. . . . 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 34837	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
16	15	simpld 499	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
17	16	adantr 485	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
18		elfzuz3 13549	. . . . . . . . 9 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)))
19	17, 18	syl 18	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)))
20		elfzuz3 13549	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
21	20	adantl 486	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
22		uztrn 12880	. . . . . . . 8 ⊢ (((𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)) ∧ (𝐼‘𝐶) ∈ (ℤ≥‘𝐽)) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
23	19, 21, 22	syl2anc 595	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
24		fzss2 13592	. . . . . . 7 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘𝐽) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
25	23, 24	syl 18	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...𝐽) ⊆ (1...(𝑀 + 𝑁)))
26		ssinss1 4206	. . . . . 6 ⊢ ((1...𝐽) ⊆ (1...(𝑀 + 𝑁)) → ((1...𝐽) ∩ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
27	25, 26	syl 18	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∩ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
28		f1ores 6836	. . . . 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 595	. . . 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 7444	. . . . . 6 ⊢ (1...𝐽) ∈ V
31	30	inex1 5288	. . . . 5 ⊢ ((1...𝐽) ∩ (𝑅‘𝐶)) ∈ V
32	31	f1oen 8969	. . . 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 14384	. . . 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 19	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((1...𝐽) ∩ (𝑅‘𝐶))) = (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))))
35	25	ssdifssd 4109	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∖ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36		f1ores 6836	. . . . 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 595	. . . 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 5300	. . . . . 6 ⊢ ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ V)
39	30, 38	ax-mp 5	. . . . 5 ⊢ ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ V
40	39	f1oen 8969	. . . 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 14384	. . . 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 19	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((1...𝐽) ∖ (𝑅‘𝐶))) = (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))))
43	34, 42	oveq12d 7429	. 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 34858	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
46	45	adantr 485	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
47		elfzelz 13552	. . . 4 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
48	47	adantl 486	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
49	1, 2, 3, 4, 5, 46, 48	ballotlemfval 34825	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅‘𝐶))) − (♯‘((1...𝐽) ∖ (𝑅‘𝐶)))))
50		fzfi 14008	. . . . 5 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
51		eldifi 4093	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
52	1, 2, 3	ballotlemelo 34823	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
53	52	simplbi 501	. . . . . . 7 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
54	51, 53	syl 18	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
55	54	adantr 485	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
56		ssfi 9157	. . . . 5 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
57	50, 55, 56	sylancr 598	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ Fin)
58		fzfid 14009	. . . 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 34859	. . . 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 595	. . 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 6828	. . . . . . . . 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 502	. . . . . . . 8 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun ◡(𝑆‘𝐶))
64		imain 6622	. . . . . . . 8 ⊢ (Fun ◡(𝑆‘𝐶) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
65	11, 63, 64	3syl 19	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∩ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
66	65	adantr 485	. . . . . 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 34851	. . . . . . 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 34854	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
69	68	adantr 485	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
70	67, 69	ineq12d 4182	. . . . . 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 2804	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶))
72	71	fveq2d 6886	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶)))
73		imadif 6621	. . . . . . . 8 ⊢ (Fun ◡(𝑆‘𝐶) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
74	11, 63, 73	3syl 19	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = (((𝑆‘𝐶) “ (1...𝐽)) ∖ ((𝑆‘𝐶) “ (𝑅‘𝐶))))
75	74	adantr 485	. . . . . 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 4090	. . . . . 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 2804	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))
78	77	fveq2d 6886	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶)))
79	72, 78	oveq12d 7429	. . 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 2807	. 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 2814	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  ifcif 4492  𝒫 cpw 4567   class class class wbr 5113   ↦ cmpt 5196  ^◡ccnv 5661   ↾ cres 5664   “ cima 5665  Fun wfun 6531  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   ≈ cen 8940  Fincfn 8943  infcinf 9401  ℝcr 11099  0cc0 11100  1c1 11101   + caddc 11103   < clt 11243   ≤ cle 11244   − cmin 11441   / cdiv 11871  ℕcn 12233  ℤcz 12591  ℤ_≥cuz 12862  ...cfz 13535  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9402  df-inf 9403  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-rp 13017  df-fz 13536  df-hash 14367

This theorem is referenced by:  ballotlemfrci  34863  ballotlemfrceq  34864