Theorem ballotlemfrceq 34828

Description: Value of 𝐹 for a reverse counting (𝑅‘𝐶). (Contributed by Thierry Arnoux, 27-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
ballotlemfrceq	⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽))

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 ballotlemfrceq

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	ballotlemsel1i 34812	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
11		1zzd 12604	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 1 ∈ ℤ)
12	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 34801	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
13	12	adantr 484	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
14	13	simpld 498	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
15	14	elfzelzd 13532	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ ℤ)
16		elfzuz3 13528	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)))
17		fzss2 13571	. . . . . . . . . . . . 13 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)) → (1...(𝐼‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
18	14, 16, 17	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...(𝐼‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
19		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝐼‘𝐶)))
20	18, 19	sseldd 3939	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
21	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsdom 34811	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
22	20, 21	syldan 600	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
23	22	elfzelzd 13532	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
24		fzsubel 13567	. . . . . . . . 9 ⊢ (((1 ∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ) ∧ (((𝑆‘𝐶)‘𝐽) ∈ ℤ ∧ 1 ∈ ℤ)) → (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) ↔ (((𝑆‘𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼‘𝐶) − 1))))
25	11, 15, 23, 11, 24	syl22anc 849	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) ↔ (((𝑆‘𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼‘𝐶) − 1))))
26	10, 25	mpbid 234	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼‘𝐶) − 1)))
27		1m1e0 12292	. . . . . . . 8 ⊢ (1 − 1) = 0
28	27	oveq1i 7408	. . . . . . 7 ⊢ ((1 − 1)...((𝐼‘𝐶) − 1)) = (0...((𝐼‘𝐶) − 1))
29	26, 28	eleqtrdi 2874	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...((𝐼‘𝐶) − 1)))
30	12	simpld 498	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
31	30	elfzelzd 13532	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
32		1zzd 12604	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℤ)
33	31, 32	zsubcld 12684	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℤ)
34		nnaddcl 12235	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
35	1, 2, 34	mp2an 702	. . . . . . . . . . 11 ⊢ (𝑀 + 𝑁) ∈ ℕ
36	35	nnzi 12597	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℤ
37	36	a1i 11	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℤ)
38		elfzle2 13535	. . . . . . . . . . 11 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
39	30, 38	syl 17	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
40		zlem1lt 12625	. . . . . . . . . . . 12 ⊢ (((𝐼‘𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
41	31, 37, 40	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
42	33	zred 12679	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℝ)
43	37	zred 12679	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℝ)
44		ltle 11273	. . . . . . . . . . . 12 ⊢ ((((𝐼‘𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼‘𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
45	42, 43, 44	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (((𝐼‘𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
46	41, 45	sylbid 242	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
47	39, 46	mpd 15	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁))
48		eluz2 12847	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) ↔ (((𝐼‘𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
49	33, 37, 47, 48	syl3anbrc 1358	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)))
50		fzss2 13571	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) → (0...((𝐼‘𝐶) − 1)) ⊆ (0...(𝑀 + 𝑁)))
51	49, 50	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (0...((𝐼‘𝐶) − 1)) ⊆ (0...(𝑀 + 𝑁)))
52	51	sselda 3938	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...((𝐼‘𝐶) − 1))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
53	29, 52	syldan 600	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
54		ballotth.r	. . . . . 6 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
55		ballotlemg	. . . . . 6 ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 55	ballotlemfg 34825	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = (𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))))
57	53, 56	syldan 600	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = (𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 55	ballotlemfrc 34826	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
59	57, 58	oveq12d 7416	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅‘𝐶))‘𝐽)) = ((𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))) + (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))))
60		fzsplit3 32997	. . . . . 6 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) → (1...(𝐼‘𝐶)) = ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∪ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
61	10, 60	syl 17	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...(𝐼‘𝐶)) = ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∪ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
62	61	oveq2d 7414	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (1...(𝐼‘𝐶))) = (𝐶 ↑ ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∪ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))))
63		fz1ssfz0 13630	. . . . . . . 8 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
64	63	sseli 3934	. . . . . . 7 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ (0...(𝑀 + 𝑁)))
65	1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 55	ballotlemfg 34825	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
66	64, 65	sylan2 602	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
67	14, 66	syldan 600	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
68	13	simprd 499	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
69	67, 68	eqtr3d 2801	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (1...(𝐼‘𝐶))) = 0)
70		fzfi 13987	. . . . . . 7 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
71		eldifi 4086	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
72	1, 2, 3	ballotlemelo 34787	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
73	72	simplbi 500	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
74	71, 73	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
75		ssfi 9143	. . . . . . 7 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
76	70, 74, 75	sylancr 596	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ Fin)
77	76	adantr 484	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ Fin)
78		fzfid 13988	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...(((𝑆‘𝐶)‘𝐽) − 1)) ∈ Fin)
79		fzfid 13988	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∈ Fin)
80	23	zred 12679	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ ℝ)
81		ltm1 12035	. . . . . 6 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ ℝ → (((𝑆‘𝐶)‘𝐽) − 1) < ((𝑆‘𝐶)‘𝐽))
82		fzdisj 13558	. . . . . 6 ⊢ ((((𝑆‘𝐶)‘𝐽) − 1) < ((𝑆‘𝐶)‘𝐽) → ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∩ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) = ∅)
83	80, 81, 82	3syl 18	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∩ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) = ∅)
84	1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 55, 77, 78, 79, 83	ballotlemgun 34824	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∪ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))) = ((𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))) + (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))))
85	62, 69, 84	3eqtr3rd 2808	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))) + (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))) = 0)
86	59, 85	eqtrd 2799	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅‘𝐶))‘𝐽)) = 0)
87	71	adantr 484	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ 𝑂)
88	23, 11	zsubcld 12684	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℤ)
89	1, 2, 3, 4, 5, 87, 88	ballotlemfelz 34790	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ∈ ℤ)
90	89	zcnd 12680	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ∈ ℂ)
91	1, 2, 3, 4, 5, 6, 7, 8, 9, 54	ballotlemro 34822	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
92	91	adantr 484	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
93	19	elfzelzd 13532	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
94	1, 2, 3, 4, 5, 92, 93	ballotlemfelz 34790	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℤ)
95	94	zcnd 12680	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ)
96		addeq0 11612	. . 3 ⊢ ((((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ∈ ℂ ∧ ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ) → ((((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅‘𝐶))‘𝐽)) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽)))
97	90, 95, 96	syl2anc 593	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅‘𝐶))‘𝐽)) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽)))
98	86, 97	mpbid 234	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  ifcif 4482  𝒫 cpw 4557   class class class wbr 5102   ↦ cmpt 5183   “ cima 5652  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  Fincfn 8929  infcinf 9389  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078   < clt 11218   ≤ cle 11219   − cmin 11416  -cneg 11417   / cdiv 11846  ℕcn 12212  ℤcz 12570  ℤ_≥cuz 12841  ...cfz 13514  ♯chash 14345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-oadd 8443  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-sup 9390  df-inf 9391  df-dju 9861  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-n0 12484  df-z 12571  df-uz 12842  df-rp 12996  df-fz 13515  df-hash 14346

This theorem is referenced by:  ballotlemfrcn0  34829