ballotlemfrceq - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ballotlemfrceq		Structured version Visualization version GIF version

Theorem ballotlemfrceq 29917

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 29901	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
11		1zzd 11285	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 1 ∈ ℤ)
12	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 29890	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
13	12	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
14	13	simpld 474	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
15		elfzelz 12213	. . . . . . . . . 10 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ ℤ)
17		elfzuz3 12210	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ_≥‘(𝐼‘𝐶)))
18		fzss2 12252	. . . . . . . . . . . . 13 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘(𝐼‘𝐶)) → (1...(𝐼‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
19	14, 17, 18	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...(𝐼‘𝐶)) ⊆ (1...(𝑀 + 𝑁)))
20		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝐼‘𝐶)))
21	19, 20	sseldd 3569	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
22	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsdom 29900	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
23	21, 22	syldan 486	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
24		elfzelz 12213	. . . . . . . . . 10 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
25	23, 24	syl 17	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
26		fzsubel 12248	. . . . . . . . 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))))
27	11, 16, 25, 11, 26	syl22anc 1319	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) ↔ (((𝑆‘𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼‘𝐶) − 1))))
28	10, 27	mpbid 221	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼‘𝐶) − 1)))
29		1m1e0 10966	. . . . . . . 8 ⊢ (1 − 1) = 0
30	29	oveq1i 6559	. . . . . . 7 ⊢ ((1 − 1)...((𝐼‘𝐶) − 1)) = (0...((𝐼‘𝐶) − 1))
31	28, 30	syl6eleq 2698	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...((𝐼‘𝐶) − 1)))
32	12	simpld 474	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
33	32, 15	syl 17	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
34		1zzd 11285	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℤ)
35	33, 34	zsubcld 11363	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℤ)
36		nnaddcl 10919	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
37	1, 2, 36	mp2an 704	. . . . . . . . . . 11 ⊢ (𝑀 + 𝑁) ∈ ℕ
38	37	nnzi 11278	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℤ
39	38	a1i 11	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℤ)
40		elfzle2 12216	. . . . . . . . . . 11 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
41	32, 40	syl 17	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
42		zlem1lt 11306	. . . . . . . . . . . 12 ⊢ (((𝐼‘𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
43	33, 39, 42	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
44	35	zred 11358	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℝ)
45	39	zred 11358	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℝ)
46		ltle 10005	. . . . . . . . . . . 12 ⊢ ((((𝐼‘𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼‘𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
47	44, 45, 46	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (((𝐼‘𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
48	43, 47	sylbid 229	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
49	41, 48	mpd 15	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁))
50		eluz2 11569	. . . . . . . . 9 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘((𝐼‘𝐶) − 1)) ↔ (((𝐼‘𝐶) − 1) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
51	35, 39, 49, 50	syl3anbrc 1239	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ (ℤ_≥‘((𝐼‘𝐶) − 1)))
52		fzss2 12252	. . . . . . . 8 ⊢ ((𝑀 + 𝑁) ∈ (ℤ_≥‘((𝐼‘𝐶) − 1)) → (0...((𝐼‘𝐶) − 1)) ⊆ (0...(𝑀 + 𝑁)))
53	51, 52	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (0...((𝐼‘𝐶) − 1)) ⊆ (0...(𝑀 + 𝑁)))
54	53	sselda 3568	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...((𝐼‘𝐶) − 1))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
55	31, 54	syldan 486	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
56		ballotth.r	. . . . . 6 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
57		ballotlemg	. . . . . 6 ⊢ ↑ = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((#‘(𝑣 ∩ 𝑢)) − (#‘(𝑣 ∖ 𝑢))))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57	ballotlemfg 29914	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = (𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))))
59	55, 58	syldan 486	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = (𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))))
60	1, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57	ballotlemfrc 29915	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
61	59, 60	oveq12d 6567	. . 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)))))
62		fzsplit3 28940	. . . . . 6 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) → (1...(𝐼‘𝐶)) = ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∪ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
63	10, 62	syl 17	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...(𝐼‘𝐶)) = ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∪ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
64	63	oveq2d 6565	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (1...(𝐼‘𝐶))) = (𝐶 ↑ ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∪ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))))
65		1eluzge0 11608	. . . . . . . . 9 ⊢ 1 ∈ (ℤ_≥‘0)
66		fzss1 12251	. . . . . . . . 9 ⊢ (1 ∈ (ℤ_≥‘0) → (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁)))
67	65, 66	ax-mp 5	. . . . . . . 8 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
68	67	sseli 3564	. . . . . . 7 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ (0...(𝑀 + 𝑁)))
69	1, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57	ballotlemfg 29914	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
70	68, 69	sylan2 490	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
71	14, 70	syldan 486	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
72	13	simprd 478	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
73	71, 72	eqtr3d 2646	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (1...(𝐼‘𝐶))) = 0)
74		fzfi 12633	. . . . . . 7 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
75		eldifi 3694	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
76	1, 2, 3	ballotlemelo 29876	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))
77	76	simplbi 475	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
78	75, 77	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
79		ssfi 8065	. . . . . . 7 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
80	74, 78, 79	sylancr 694	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ Fin)
81	80	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ Fin)
82		fzfid 12634	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...(((𝑆‘𝐶)‘𝐽) − 1)) ∈ Fin)
83		fzfid 12634	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∈ Fin)
84	25	zred 11358	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ ℝ)
85		ltm1 10742	. . . . . 6 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ ℝ → (((𝑆‘𝐶)‘𝐽) − 1) < ((𝑆‘𝐶)‘𝐽))
86		fzdisj 12239	. . . . . 6 ⊢ ((((𝑆‘𝐶)‘𝐽) − 1) < ((𝑆‘𝐶)‘𝐽) → ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∩ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) = ∅)
87	84, 85, 86	3syl 18	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((1...(((𝑆‘𝐶)‘𝐽) − 1)) ∩ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))) = ∅)
88	1, 2, 3, 4, 5, 6, 7, 8, 9, 56, 57, 81, 82, 83, 87	ballotlemgun 29913	. . . 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)))))
89	64, 73, 88	3eqtr3rd 2653	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐶 ↑ (1...(((𝑆‘𝐶)‘𝐽) − 1))) + (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)))) = 0)
90	61, 89	eqtrd 2644	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅‘𝐶))‘𝐽)) = 0)
91	75	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ 𝑂)
92	25, 11	zsubcld 11363	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℤ)
93	1, 2, 3, 4, 5, 91, 92	ballotlemfelz 29879	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ∈ ℤ)
94	93	zcnd 11359	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ∈ ℂ)
95	1, 2, 3, 4, 5, 6, 7, 8, 9, 56	ballotlemro 29911	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
96	95	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
97		elfzelz 12213	. . . . . 6 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
98	20, 97	syl 17	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
99	1, 2, 3, 4, 5, 96, 98	ballotlemfelz 29879	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℤ)
100	99	zcnd 11359	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ)
101		addeq0 28898	. . 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)))
102	94, 100, 101	syl2anc 691	. 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)))
103	90, 102	mpbid 221	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 class class class wbr 4583 ↦ cmpt 4643 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Fincfn 7841 infcinf 8230 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 #chash 12979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-hash 12980

This theorem is referenced by: ballotlemfrcn0 29918

W3C validator