Theorem ballotlemfrcn0 35155

Description: Value of 𝐹 for a reversed counting (𝑅‘𝐶), before the first tie, cannot be zero. (Contributed by Thierry Arnoux, 25-Apr-2017.) (Revised by AV, 6-Oct-2020.)

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))

Assertion

Ref	Expression
ballotlemfrcn0	⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0)

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   𝑖,𝐽

Allowed substitution hints:   𝐶(𝑥, 𝑐)   𝑃(𝑥, 𝑖, 𝑘, 𝑐)   𝑅(𝑥, 𝑘, 𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐹(𝑥)   𝐼(𝑥)   𝐽(𝑥, 𝑐)   𝑀(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlemfrcn0

Dummy variables 𝑣 𝑢 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1zzd 12720	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ∈ ℤ)
2		ballotth.m	. . . . . . . 8 ⊢ 𝑀 ∈ ℕ
3		ballotth.n	. . . . . . . 8 ⊢ 𝑁 ∈ ℕ
4		nnaddcl 12351	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
5	2, 3, 4	mp2an 705	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℕ
6	5	nnzi 12713	. . . . . 6 ⊢ (𝑀 + 𝑁) ∈ ℤ
7	6	a1i 11	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝑀 + 𝑁) ∈ ℤ)
8		ballotth.o	. . . . . . . . 9 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
9		ballotth.p	. . . . . . . . 9 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
10		ballotth.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
11		ballotth.e	. . . . . . . . 9 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
12		ballotth.mgtn	. . . . . . . . 9 ⊢ 𝑁 < 𝑀
13		ballotth.i	. . . . . . . . 9 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
14		ballotth.s	. . . . . . . . 9 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
15	2, 3, 8, 9, 10, 11, 12, 13, 14	ballotlemsdom 35137	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
16	15	elfzelzd 13650	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
17	16	3adant3 1150	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
18	17, 1	zsubcld 12801	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℤ)
19	2, 3, 8, 9, 10, 11, 12, 13, 14	ballotlemsgt1 35136	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 < ((𝑆‘𝐶)‘𝐽))
20		zltlem1 12742	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ ((𝑆‘𝐶)‘𝐽) ∈ ℤ) → (1 < ((𝑆‘𝐶)‘𝐽) ↔ 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1)))
21	20	biimpa 482	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ ((𝑆‘𝐶)‘𝐽) ∈ ℤ) ∧ 1 < ((𝑆‘𝐶)‘𝐽)) → 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1))
22	1, 17, 19, 21	syl21anc 851	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1))
23	17	zred 12796	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ ℝ)
24		1red 11302	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ∈ ℝ)
25	23, 24	resubcld 11737	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℝ)
26		simp1 1154	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐶 ∈ (𝑂 ∖ 𝐸))
27	2, 3, 8, 9, 10, 11, 12, 13	ballotlemiex 35127	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
28	27	simpld 500	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
29		elfzelz 13649	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ)
30	26, 28, 29	3syl 19	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℤ)
31	30	zred 12796	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℝ)
32	7	zred 12796	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝑀 + 𝑁) ∈ ℝ)
33		elfzelz 13649	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 𝐽 ∈ ℤ)
34	33	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ ℤ)
35		elfzle1 13653	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 1 ≤ 𝐽)
36	35	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ≤ 𝐽)
37	34	zred 12796	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ ℝ)
38		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 < (𝐼‘𝐶))
39	37, 31, 38	ltled 11451	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ≤ (𝐼‘𝐶))
40	1, 30, 34, 36, 39	elfzd 13640	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ (1...(𝐼‘𝐶)))
41	2, 3, 8, 9, 10, 11, 12, 13, 14	ballotlemsel1i 35138	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
42	26, 40, 41	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
43		elfzle2 13654	. . . . . . . . 9 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶))
44	42, 43	syl 18	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶))
45		zlem1lt 12741	. . . . . . . . 9 ⊢ ((((𝑆‘𝐶)‘𝐽) ∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ) → (((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶)))
46	17, 30, 45	syl2anc 596	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶)))
47	44, 46	mpbid 235	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))
48	25, 31, 47	ltled 11451	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝐼‘𝐶))
49		elfzle2 13654	. . . . . . 7 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
50	26, 28, 49	3syl 19	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
51	25, 31, 32, 48, 50	letrd 11460	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁))
52	1, 7, 18, 22, 51	elfzd 13640	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)))
53		biid 264	. . . . . . . . 9 ⊢ ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))
54	47, 53	sylibr 237	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))
55	2, 3, 8, 9, 10, 11, 12, 13	ballotlemi 35126	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
56	55	breq2d 5115	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )))
57	56	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )))
58	54, 57	mpbid 235	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
59		ltso 11383	. . . . . . . . . 10 ⊢ < Or ℝ
60	59	a1i 11	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → < Or ℝ)
61	2, 3, 8, 9, 10, 11, 12, 13	ballotlemsup 35130	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ∃𝑧 ∈ ℝ (∀𝑤 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ¬ 𝑤 < 𝑧 ∧ ∀𝑤 ∈ ℝ (𝑧 < 𝑤 → ∃𝑦 ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}𝑦 < 𝑤)))
62	60, 61	inflb 9475	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((((𝑆‘𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} → ¬ (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )))
63	62	con2d 135	. . . . . . 7 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ) → ¬ (((𝑆‘𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}))
64	26, 58, 63	sylc 66	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ¬ (((𝑆‘𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0})
65		fveqeq2 6892	. . . . . . 7 ⊢ (𝑘 = (((𝑆‘𝐶)‘𝐽) − 1) → (((𝐹‘𝐶)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
66	65	elrab 3645	. . . . . 6 ⊢ ((((𝑆‘𝐶)‘𝐽) − 1) ∈ {𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0} ↔ ((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
67	64, 66	sylnib 331	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ¬ ((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
68		imnan 405	. . . . 5 ⊢ (((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) → ¬ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0) ↔ ¬ ((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
69	67, 68	sylibr 237	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)) → ¬ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
70	52, 69	mpd 16	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ¬ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)
71	70	neqned 2963	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ≠ 0)
72		ballotth.r	. . . . . . . . . 10 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
73	2, 3, 8, 9, 10, 11, 12, 13, 14, 72	ballotlemro 35148	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
74	73	adantr 486	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
75		elfzelz 13649	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
76	75	adantl 487	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
77	2, 3, 8, 9, 10, 74, 76	ballotlemfelz 35116	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℤ)
78	77	zcnd 12797	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ)
79	78	negeq0d 11654	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) = 0 ↔ -((𝐹‘(𝑅‘𝐶))‘𝐽) = 0))
80		eqid 2761	. . . . . . 7 ⊢ (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
81	2, 3, 8, 9, 10, 11, 12, 13, 14, 72, 80	ballotlemfrceq 35154	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽))
82	81	eqeq1d 2763	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0 ↔ -((𝐹‘(𝑅‘𝐶))‘𝐽) = 0))
83	79, 82	bitr4d 285	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
84	83	necon3bid 3000	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ≠ 0))
85	26, 40, 84	syl2anc 596	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ≠ 0))
86	71, 85	mpbird 260	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413   ∖ cdif 3896   ∩ cin 3898  ifcif 4482  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186   Or wor 5558   “ cima 5654  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  Fincfn 8966  infcinf 9426  ℝcr 11192  0cc0 11193  1c1 11194   + caddc 11196   < clt 11336   ≤ cle 11337   − cmin 11534  -cneg 11535   / cdiv 11966  ℕcn 12328  ℤcz 12686  ...cfz 13632  ♯chash 14467

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  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 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-n0 12600  df-z 12687  df-uz 12959  df-rp 13114  df-fz 13633  df-hash 14468

This theorem is used by:  ballotlemirc  35157