Theorem ballotlemfrc 34684

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 34671	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))
11	10	simpld 494	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
12		f1of1 6773	. . . . . . 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 480	. . . . 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 34659	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
16	15	simpld 494	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
18		elfzuz3 13437	. . . . . . . . 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 13437	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
21	20	adantl 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
22		uztrn 12769	. . . . . . . 8 ⊢ (((𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)) ∧ (𝐼‘𝐶) ∈ (ℤ≥‘𝐽)) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
23	19, 21, 22	syl2anc 584	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
24		fzss2 13480	. . . . . . 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 4198	. . . . . 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 6788	. . . . 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 584	. . . 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 7391	. . . . . 6 ⊢ (1...𝐽) ∈ V
31	30	inex1 5262	. . . . 5 ⊢ ((1...𝐽) ∩ (𝑅‘𝐶)) ∈ V
32	31	f1oen 8909	. . . 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 14271	. . . 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 4099	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∖ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36		f1ores 6788	. . . . 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 584	. . . 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 5274	. . . . . 6 ⊢ ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ V)
39	30, 38	ax-mp 5	. . . . 5 ⊢ ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ V
40	39	f1oen 8909	. . . 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 14271	. . . 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 7376	. 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 34680	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
46	45	adantr 480	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
47		elfzelz 13440	. . . 4 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
48	47	adantl 481	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
49	1, 2, 3, 4, 5, 46, 48	ballotlemfval 34647	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅‘𝐶))) − (♯‘((1...𝐽) ∖ (𝑅‘𝐶)))))
50		fzfi 13895	. . . . 5 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
51		eldifi 4083	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
52	1, 2, 3	ballotlemelo 34645	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
53	52	simplbi 497	. . . . . . 7 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
54	51, 53	syl 17	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
55	54	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
56		ssfi 9097	. . . . 5 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
57	50, 55, 56	sylancr 587	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ Fin)
58		fzfid 13896	. . . 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 34681	. . . 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 584	. . 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 6780	. . . . . . . . 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 496	. . . . . . . 8 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun ◡(𝑆‘𝐶))
64		imain 6577	. . . . . . . 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 480	. . . . . 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 34673	. . . . . . 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 34676	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
69	68	adantr 480	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
70	67, 69	ineq12d 4173	. . . . . 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 2771	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶))
72	71	fveq2d 6838	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶)))
73		imadif 6576	. . . . . . . 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 480	. . . . . 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 4079	. . . . . 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 2771	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))
78	77	fveq2d 6838	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶)))
79	72, 78	oveq12d 7376	. . 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 2774	. 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 2781	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ifcif 4479  𝒫 cpw 4554   class class class wbr 5098   ↦ cmpt 5179  ^◡ccnv 5623   ↾ cres 5626   “ cima 5627  Fun wfun 6486  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   ≈ cen 8880  Fincfn 8883  infcinf 9344  ℝcr 11025  0cc0 11026  1c1 11027   + caddc 11029   < clt 11166   ≤ cle 11167   − cmin 11364   / cdiv 11794  ℕcn 12145  ℤcz 12488  ℤ_≥cuz 12751  ...cfz 13423  ♯chash 14253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-fz 13424  df-hash 14254

This theorem is referenced by:  ballotlemfrci  34685  ballotlemfrceq  34686