Theorem ballotfilemfrceq 13274

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
ballotfilem.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfilem.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘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 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))

Assertion

Ref	Expression
ballotfilemfrceq	⊢ ((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   𝑢,𝑂,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)

Proof of Theorem ballotfilemfrceq

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . . . . 9 ⊢ 𝑀 ∈ ℕ
2		ballotth.n	. . . . . . . . 9 ⊢ 𝑁 ∈ ℕ
3		ballotfilem.o	. . . . . . . . 9 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
4		ballotfilem.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘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	ballotfilemsel1i 13258	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
11		1zzd 9673	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 1 ∈ ℤ)
12	1, 2, 3, 4, 5, 6, 7, 8	ballotfilemiex 13246	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
13	12	adantr 276	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
14	13	simpld 112	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
15	14	elfzelzd 10431	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ ℤ)
16		elfzuz3 10427	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)))
17		fzss2 10472	. . . . . . . . . . . . 13 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘(𝐼‘𝐶)) → (1...(𝐼‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
18	14, 16, 17	3syl 17	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...(𝐼‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
19		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝐼‘𝐶)))
20	18, 19	sseldd 3249	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
21	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotfilemsdom 13257	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
22	20, 21	syldan 282	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
23	22	elfzelzd 10431	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
24		fzsubel 10468	. . . . . . . . 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 1279	. . . . . . . 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 147	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼‘𝐶) − 1)))
27		1m1e0 9374	. . . . . . . 8 ⊢ (1 − 1) = 0
28	27	oveq1i 6095	. . . . . . 7 ⊢ ((1 − 1)...((𝐼‘𝐶) − 1)) = (0...((𝐼‘𝐶) − 1))
29	26, 28	eleqtrdi 2331	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...((𝐼‘𝐶) − 1)))
30	12	simpld 112	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
31	30	elfzelzd 10431	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
32		1zzd 9673	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℤ)
33	31, 32	zsubcld 9775	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℤ)
34		nnaddcl 9325	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
35	1, 2, 34	mp2an 430	. . . . . . . . . . 11 ⊢ (𝑀 + 𝑁) ∈ ℕ
36	35	nnzi 9667	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℤ
37	36	a1i 9	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℤ)
38		elfzle2 10434	. . . . . . . . . . 11 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
39	30, 38	syl 14	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
40		zlem1lt 9703	. . . . . . . . . . . 12 ⊢ (((𝐼‘𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
41	31, 37, 40	syl2anc 415	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
42	33	zred 9770	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℝ)
43	37	zred 9770	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℝ)
44		ltle 8413	. . . . . . . . . . . 12 ⊢ ((((𝐼‘𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼‘𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
45	42, 43, 44	syl2anc 415	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (((𝐼‘𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
46	41, 45	sylbid 150	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
47	39, 46	mpd 13	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁))
48		eluz2 9929	. . . . . . . . 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 1212	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)))
50		fzss2 10472	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) ∈ (ℤ≥‘((𝐼‘𝐶) − 1)) → (0...((𝐼‘𝐶) − 1)) ⊆ (0...(𝑀 + 𝑁)))
51	49, 50	syl 14	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (0...((𝐼‘𝐶) − 1)) ⊆ (0...(𝑀 + 𝑁)))
52	51	sselda 3248	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...((𝐼‘𝐶) − 1))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
53	29, 52	syldan 282	. . . . 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 ∈ 𝑂, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 55	ballotfilemfg 13271	. . . . 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 282	. . . 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	ballotfilemfrc 13272	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
59	57, 58	oveq12d 6103	. . 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		eldifi 3351	. . . . 5 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
61	60	adantr 276	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ 𝑂)
62	1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 55, 61, 10	ballotfilemgun 13270	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (1...(𝐼‘𝐶))) = ((𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))) + (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))))
63		fz1ssfz0 10526	. . . . . . 7 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
64	63	sseli 3244	. . . . . 6 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ (0...(𝑀 + 𝑁)))
65	1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 55	ballotfilemfg 13271	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
66	64, 65	sylan2 286	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
67	14, 66	syldan 282	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
68	13	simprd 114	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
69	67, 68	eqtr3d 2273	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (1...(𝐼‘𝐶))) = 0)
70	59, 62, 69	3eqtr2d 2277	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅‘𝐶))‘𝐽)) = 0)
71	23, 11	zsubcld 9775	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℤ)
72	1, 2, 3, 4, 5, 61, 71	ballotfilemfelz 13232	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ∈ ℤ)
73	72	zcnd 9771	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ∈ ℂ)
74	1, 2, 3, 4, 5, 6, 7, 8, 9, 54	ballotfilemro 13268	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
75	74	adantr 276	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
76	19	elfzelzd 10431	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
77	1, 2, 3, 4, 5, 75, 76	ballotfilemfelz 13232	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℤ)
78	77	zcnd 9771	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ)
79		addeq0 8703	. . 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)))
80	73, 78, 79	syl2anc 415	. 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)))
81	70, 80	mpbid 147	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ∖ cdif 3217   ∩ cin 3219   ⊆ wss 3220  ifcif 3638  𝒫 cpw 3688   class class class wbr 4130   ↦ cmpt 4192   “ cima 4777  ‘cfv 5377  (class class class)co 6085   ∈ cmpo 6087  Fincfn 7022  infcinf 7323  ℂcc 8177  ℝcr 8178  0cc0 8179  1c1 8180   + caddc 8182   < clt 8360   ≤ cle 8361   − cmin 8497  -cneg 8498   / cdiv 9003  ℕcn 9305  ℤcz 9646  ℤ_≥cuz 9923  ...cfz 10413  ♯chash 11216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-ihash 11217

This theorem is used by:  ballotfilemfrcn0  13275