Theorem ballotlemfrcn0 34514

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 12506	. . . . 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 12151	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
5	2, 3, 4	mp2an 692	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℕ
6	5	nnzi 12499	. . . . . 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 34496	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
16	15	elfzelzd 13428	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
17	16	3adant3 1132	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
18	17, 1	zsubcld 12585	. . . . 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 34495	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 < ((𝑆‘𝐶)‘𝐽))
20		zltlem1 12528	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ ((𝑆‘𝐶)‘𝐽) ∈ ℤ) → (1 < ((𝑆‘𝐶)‘𝐽) ↔ 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1)))
21	20	biimpa 476	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ ((𝑆‘𝐶)‘𝐽) ∈ ℤ) ∧ 1 < ((𝑆‘𝐶)‘𝐽)) → 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1))
22	1, 17, 19, 21	syl21anc 837	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1))
23	17	zred 12580	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ ℝ)
24		1red 11116	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ∈ ℝ)
25	23, 24	resubcld 11548	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℝ)
26		simp1 1136	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐶 ∈ (𝑂 ∖ 𝐸))
27	2, 3, 8, 9, 10, 11, 12, 13	ballotlemiex 34486	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
28	27	simpld 494	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
29		elfzelz 13427	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ)
30	26, 28, 29	3syl 18	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℤ)
31	30	zred 12580	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℝ)
32	7	zred 12580	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝑀 + 𝑁) ∈ ℝ)
33		elfzelz 13427	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 𝐽 ∈ ℤ)
34	33	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ ℤ)
35		elfzle1 13430	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 1 ≤ 𝐽)
36	35	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ≤ 𝐽)
37	34	zred 12580	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ ℝ)
38		simp3 1138	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 < (𝐼‘𝐶))
39	37, 31, 38	ltled 11264	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ≤ (𝐼‘𝐶))
40	1, 30, 34, 36, 39	elfzd 13418	. . . . . . . . . 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 34497	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
42	26, 40, 41	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
43		elfzle2 13431	. . . . . . . . 9 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶))
44	42, 43	syl 17	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶))
45		zlem1lt 12527	. . . . . . . . 9 ⊢ ((((𝑆‘𝐶)‘𝐽) ∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ) → (((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶)))
46	17, 30, 45	syl2anc 584	. . . . . . . 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 232	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))
48	25, 31, 47	ltled 11264	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝐼‘𝐶))
49		elfzle2 13431	. . . . . . 7 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
50	26, 28, 49	3syl 18	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
51	25, 31, 32, 48, 50	letrd 11273	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁))
52	1, 7, 18, 22, 51	elfzd 13418	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)))
53		biid 261	. . . . . . . . 9 ⊢ ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))
54	47, 53	sylibr 234	. . . . . . . 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 34485	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
56	55	breq2d 5104	. . . . . . . . 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 1133	. . . . . . . 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 232	. . . . . . 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 11196	. . . . . . . . . 10 ⊢ < Or ℝ
60	59	a1i 11	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → < Or ℝ)
61	2, 3, 8, 9, 10, 11, 12, 13	ballotlemsup 34489	. . . . . . . . 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 9380	. . . . . . . 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 134	. . . . . . 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 65	. . . . . 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 6831	. . . . . . 7 ⊢ (𝑘 = (((𝑆‘𝐶)‘𝐽) − 1) → (((𝐹‘𝐶)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
66	65	elrab 3648	. . . . . 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 328	. . . . 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 399	. . . . 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 234	. . . 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 15	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ¬ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)
71	70	neqned 2932	. 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 34507	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
74	73	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
75		elfzelz 13427	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
76	75	adantl 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
77	2, 3, 8, 9, 10, 74, 76	ballotlemfelz 34475	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℤ)
78	77	zcnd 12581	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ)
79	78	negeq0d 11467	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) = 0 ↔ -((𝐹‘(𝑅‘𝐶))‘𝐽) = 0))
80		eqid 2729	. . . . . . 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 34513	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽))
82	81	eqeq1d 2731	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0 ↔ -((𝐹‘(𝑅‘𝐶))‘𝐽) = 0))
83	79, 82	bitr4d 282	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
84	83	necon3bid 2969	. . 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 584	. 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 257	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3394   ∖ cdif 3900   ∩ cin 3902  ifcif 4476  𝒫 cpw 4551   class class class wbr 5092   ↦ cmpt 5173   Or wor 5526   “ cima 5622  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  Fincfn 8872  infcinf 9331  ℝcr 11008  0cc0 11009  1c1 11010   + caddc 11012   < clt 11149   ≤ cle 11150   − cmin 11347  -cneg 11348   / cdiv 11777  ℕcn 12128  ℤcz 12471  ...cfz 13410  ♯chash 14237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-fz 13411  df-hash 14238

This theorem is referenced by:  ballotlemirc  34516