Theorem ballotlemfrc 35212

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 35199	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) ∧ ◡(𝑆‘𝐶) = (𝑆‘𝐶)))
11	10	simpld 500	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)))
12		f1of1 6831	. . . . . . 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 486	. . . . 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 35187	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
16	15	simpld 500	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
17	16	adantr 486	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
18		elfzuz3 13663	. . . . . . . . 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 13663	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
21	20	adantl 487	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (ℤ≥‘𝐽))
22		uztrn 12993	. . . . . . . 8 ⊢ (((𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)) ∧ (𝐼‘𝐶) ∈ (ℤ≥‘𝐽)) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
23	19, 21, 22	syl2anc 596	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑀 + 𝑁) ∈ (ℤ≥‘𝐽))
24		fzss2 13708	. . . . . . 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 4144	. . . . . 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 6847	. . . . 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 596	. . . 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 7460	. . . . . 6 ⊢ (1...𝐽) ∈ V
31	30	inex1 5280	. . . . 5 ⊢ ((1...𝐽) ∩ (𝑅‘𝐶)) ∈ V
32	31	f1oen 9010	. . . 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 14502	. . . 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 4085	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...𝐽) ∖ (𝑅‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
36		f1ores 6847	. . . . 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 596	. . . 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 5294	. . . . . 6 ⊢ ((1...𝐽) ∈ V → ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ V)
39	30, 38	ax-mp 5	. . . . 5 ⊢ ((1...𝐽) ∖ (𝑅‘𝐶)) ∈ V
40	39	f1oen 9010	. . . 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 14502	. . . 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 7445	. 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 35208	. . . 4 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
46	45	adantr 486	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
47		elfzelz 13666	. . . 4 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
48	47	adantl 487	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
49	1, 2, 3, 4, 5, 46, 48	ballotlemfval 35175	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = ((♯‘((1...𝐽) ∩ (𝑅‘𝐶))) − (♯‘((1...𝐽) ∖ (𝑅‘𝐶)))))
50		fzfi 14125	. . . . 5 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
51		eldifi 4069	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
52	1, 2, 3	ballotlemelo 35173	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
53	52	simplbi 502	. . . . . . 7 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
54	51, 53	syl 18	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
55	54	adantr 486	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
56		ssfi 9199	. . . . 5 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
57	50, 55, 56	sylancr 599	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ Fin)
58		fzfid 14126	. . . 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 35209	. . . 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 596	. . 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 6839	. . . . . . . . 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 503	. . . . . . . 8 ⊢ ((𝑆‘𝐶):(1...(𝑀 + 𝑁))–1-1-onto→(1...(𝑀 + 𝑁)) → Fun ◡(𝑆‘𝐶))
64		imain 6633	. . . . . . . 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 486	. . . . . 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 35201	. . . . . . 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 35204	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
69	68	adantr 486	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ (𝑅‘𝐶)) = 𝐶)
70	67, 69	ineq12d 4120	. . . . . 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 2798	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶))
72	71	fveq2d 6897	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∩ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∩ 𝐶)))
73		imadif 6632	. . . . . . . 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 486	. . . . . 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 4066	. . . . . 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 2798	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶))) = ((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶))
78	77	fveq2d 6897	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (♯‘((𝑆‘𝐶) “ ((1...𝐽) ∖ (𝑅‘𝐶)))) = (♯‘((((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∖ 𝐶)))
79	72, 78	oveq12d 7445	. . 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 2801	. 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 2808	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3079  {crab 3415  Vcvv 3453   ⊆ wss 3899   ∖ cdif 4047   ∩ cin 4091  ifcif 4484  𝒫 cpw 4559   class class class wbr 5105   ↦ cmpt 5188  ^◡ccnv 5653   ↾ cres 5656   “ cima 5657  Fun wfun 6540  –1-1→wf1 6543  –onto→wfo 6544  –1-1-onto→wf1o 6545  ‘cfv 6546  (class class class)co 7427   ∈ cmpo 7429   ≈ cen 8981  Fincfn 8984  infcinf 9443  ℝcr 11209  0cc0 11210  1c1 11211   + caddc 11213   < clt 11353   ≤ cle 11354   − cmin 11551   / cdiv 11983  ℕcn 12345  ℤcz 12703  ℤ_≥cuz 12975  ...cfz 13649  ♯chash 14484

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-oadd 8491  df-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9444  df-inf 9445  df-dju 9992  df-card 10030  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-n0 12617  df-z 12704  df-uz 12976  df-rp 13131  df-fz 13650  df-hash 14485

This theorem is used by:  ballotlemfrci  35213  ballotlemfrceq  35214