ballotlemfrceq - Mathbox for Thierry Arnoux

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Theorem ballotlemfrceq 33826

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 33810	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
11		1zzd 12598	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 1 ∈ ℤ)
12	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 33799	. . . . . . . . . . . 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 494	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
15	14	elfzelzd 13507	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐼‘𝐶) ∈ ℤ)
16		elfzuz3 13503	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝑀 + 𝑁) ∈ (ℤ_≥‘(𝐼‘𝐶)))
17		fzss2 13546	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝐼‘𝐶)))
20	18, 19	sseldd 3983	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ (1...(𝑀 + 𝑁)))
21	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsdom 33809	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
22	20, 21	syldan 590	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
23	22	elfzelzd 13507	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
24		fzsubel 13542	. . . . . . . . 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 836	. . . . . . . 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 231	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ((1 − 1)...((𝐼‘𝐶) − 1)))
27		1m1e0 12289	. . . . . . . 8 ⊢ (1 − 1) = 0
28	27	oveq1i 7422	. . . . . . 7 ⊢ ((1 − 1)...((𝐼‘𝐶) − 1)) = (0...((𝐼‘𝐶) − 1))
29	26, 28	eleqtrdi 2842	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...((𝐼‘𝐶) − 1)))
30	12	simpld 494	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
31	30	elfzelzd 13507	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ ℤ)
32		1zzd 12598	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 1 ∈ ℤ)
33	31, 32	zsubcld 12676	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℤ)
34		nnaddcl 12240	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
35	1, 2, 34	mp2an 689	. . . . . . . . . . 11 ⊢ (𝑀 + 𝑁) ∈ ℕ
36	35	nnzi 12591	. . . . . . . . . 10 ⊢ (𝑀 + 𝑁) ∈ ℤ
37	36	a1i 11	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℤ)
38		elfzle2 13510	. . . . . . . . . . 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 12619	. . . . . . . . . . . 12 ⊢ (((𝐼‘𝐶) ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
41	31, 37, 40	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ≤ (𝑀 + 𝑁) ↔ ((𝐼‘𝐶) − 1) < (𝑀 + 𝑁)))
42	33	zred 12671	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) − 1) ∈ ℝ)
43	37	zred 12671	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ ℝ)
44		ltle 11307	. . . . . . . . . . . 12 ⊢ ((((𝐼‘𝐶) − 1) ∈ ℝ ∧ (𝑀 + 𝑁) ∈ ℝ) → (((𝐼‘𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
45	42, 43, 44	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (((𝐼‘𝐶) − 1) < (𝑀 + 𝑁) → ((𝐼‘𝐶) − 1) ≤ (𝑀 + 𝑁)))
46	41, 45	sylbid 239	. . . . . . . . . 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 12833	. . . . . . . . 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 1342	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑀 + 𝑁) ∈ (ℤ_≥‘((𝐼‘𝐶) − 1)))
50		fzss2 13546	. . . . . . . 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 3982	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...((𝐼‘𝐶) − 1))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (0...(𝑀 + 𝑁)))
53	29, 52	syldan 590	. . . . 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 33823	. . . . 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 590	. . . 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 33824	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) = (𝐶 ↑ (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶))))
59	57, 58	oveq12d 7430	. . 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 32273	. . . . . 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 7428	. . . 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 13602	. . . . . . . 8 ⊢ (1...(𝑀 + 𝑁)) ⊆ (0...(𝑀 + 𝑁))
64	63	sseli 3978	. . . . . . 7 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ (0...(𝑀 + 𝑁)))
65	1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 55	ballotlemfg 33823	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (0...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
66	64, 65	sylan2 592	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
67	14, 66	syldan 590	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = (𝐶 ↑ (1...(𝐼‘𝐶))))
68	13	simprd 495	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0)
69	67, 68	eqtr3d 2773	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝐶 ↑ (1...(𝐼‘𝐶))) = 0)
70		fzfi 13942	. . . . . . 7 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
71		eldifi 4126	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ 𝑂)
72	1, 2, 3	ballotlemelo 33785	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
73	72	simplbi 497	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑂 → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
74	71, 73	syl 17	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ⊆ (1...(𝑀 + 𝑁)))
75		ssfi 9177	. . . . . . 7 ⊢ (((1...(𝑀 + 𝑁)) ∈ Fin ∧ 𝐶 ⊆ (1...(𝑀 + 𝑁))) → 𝐶 ∈ Fin)
76	70, 74, 75	sylancr 586	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → 𝐶 ∈ Fin)
77	76	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ Fin)
78		fzfid 13943	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (1...(((𝑆‘𝐶)‘𝐽) − 1)) ∈ Fin)
79		fzfid 13943	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽)...(𝐼‘𝐶)) ∈ Fin)
80	23	zred 12671	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ ℝ)
81		ltm1 12061	. . . . . 6 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ ℝ → (((𝑆‘𝐶)‘𝐽) − 1) < ((𝑆‘𝐶)‘𝐽))
82		fzdisj 13533	. . . . . 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 33822	. . . 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 2780	. . 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 2771	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) + ((𝐹‘(𝑅‘𝐶))‘𝐽)) = 0)
87	71	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐶 ∈ 𝑂)
88	23, 11	zsubcld 12676	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℤ)
89	1, 2, 3, 4, 5, 87, 88	ballotlemfelz 33788	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ∈ ℤ)
90	89	zcnd 12672	. . 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 33820	. . . . . 6 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
92	91	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
93	19	elfzelzd 13507	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
94	1, 2, 3, 4, 5, 92, 93	ballotlemfelz 33788	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℤ)
95	94	zcnd 12672	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ)
96		addeq0 11642	. . 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 583	. 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 231	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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ifcif 4528 𝒫 cpw 4602 class class class wbr 5148 ↦ cmpt 5231 “ cima 5679 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 Fincfn 8943 infcinf 9440 ℂcc 11112 ℝcr 11113 0cc0 11114 1c1 11115 + caddc 11117 < clt 11253 ≤ cle 11254 − cmin 11449 -cneg 11450 / cdiv 11876 ℕcn 12217 ℤcz 12563 ℤ_≥cuz 12827 ...cfz 13489 ♯chash 14295

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-fz 13490 df-hash 14296

This theorem is referenced by: ballotlemfrcn0 33827

W3C validator