Theorem ballotlemfrcn0 31400

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 11867	. . . . 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 11514	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
5	2, 3, 4	mp2an 688	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℕ
6	5	nnzi 11860	. . . . . 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 31382	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)))
16		elfzelz 12762	. . . . . . . 8 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝑀 + 𝑁)) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
17	15, 16	syl 17	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁))) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
18	17	3adant3 1125	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ ℤ)
19	18, 1	zsubcld 11946	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℤ)
20	2, 3, 8, 9, 10, 11, 12, 13, 14	ballotlemsgt1 31381	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 < ((𝑆‘𝐶)‘𝐽))
21		zltlem1 11889	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ ((𝑆‘𝐶)‘𝐽) ∈ ℤ) → (1 < ((𝑆‘𝐶)‘𝐽) ↔ 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1)))
22	21	biimpa 477	. . . . . 6 ⊢ (((1 ∈ ℤ ∧ ((𝑆‘𝐶)‘𝐽) ∈ ℤ) ∧ 1 < ((𝑆‘𝐶)‘𝐽)) → 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1))
23	1, 18, 20, 22	syl21anc 834	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ≤ (((𝑆‘𝐶)‘𝐽) − 1))
24	18	zred 11941	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ ℝ)
25		1red 10495	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ∈ ℝ)
26	24, 25	resubcld 10922	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℝ)
27		simp1 1129	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐶 ∈ (𝑂 ∖ 𝐸))
28	2, 3, 8, 9, 10, 11, 12, 13	ballotlemiex 31372	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) ∧ ((𝐹‘𝐶)‘(𝐼‘𝐶)) = 0))
29	28	simpld 495	. . . . . . . 8 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)))
30		elfzelz 12762	. . . . . . . 8 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ∈ ℤ)
31	27, 29, 30	3syl 18	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℤ)
32	31	zred 11941	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ∈ ℝ)
33	7	zred 11941	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝑀 + 𝑁) ∈ ℝ)
34		elfzelz 12762	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 𝐽 ∈ ℤ)
35	34	3ad2ant2 1127	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ ℤ)
36		elfzle1 12764	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (1...(𝑀 + 𝑁)) → 1 ≤ 𝐽)
37	36	3ad2ant2 1127	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 1 ≤ 𝐽)
38	35	zred 11941	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ ℝ)
39		simp3 1131	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 < (𝐼‘𝐶))
40	38, 32, 39	ltled 10641	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ≤ (𝐼‘𝐶))
41		elfz4 12755	. . . . . . . . . . 11 ⊢ (((1 ∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ ∧ 𝐽 ∈ ℤ) ∧ (1 ≤ 𝐽 ∧ 𝐽 ≤ (𝐼‘𝐶))) → 𝐽 ∈ (1...(𝐼‘𝐶)))
42	1, 31, 35, 37, 40, 41	syl32anc 1371	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → 𝐽 ∈ (1...(𝐼‘𝐶)))
43	2, 3, 8, 9, 10, 11, 12, 13, 14	ballotlemsel1i 31383	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
44	27, 42, 43	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)))
45		elfzle2 12765	. . . . . . . . 9 ⊢ (((𝑆‘𝐶)‘𝐽) ∈ (1...(𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶))
46	44, 45	syl 17	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶))
47		zlem1lt 11888	. . . . . . . . 9 ⊢ ((((𝑆‘𝐶)‘𝐽) ∈ ℤ ∧ (𝐼‘𝐶) ∈ ℤ) → (((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶)))
48	18, 31, 47	syl2anc 584	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) ≤ (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶)))
49	46, 48	mpbid 233	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))
50	26, 32, 49	ltled 10641	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝐼‘𝐶))
51		elfzle2 12765	. . . . . . 7 ⊢ ((𝐼‘𝐶) ∈ (1...(𝑀 + 𝑁)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
52	27, 29, 51	3syl 18	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (𝐼‘𝐶) ≤ (𝑀 + 𝑁))
53	26, 32, 33, 50, 52	letrd 10650	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁))
54		elfz4 12755	. . . . 5 ⊢ (((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ ∧ (((𝑆‘𝐶)‘𝐽) − 1) ∈ ℤ) ∧ (1 ≤ (((𝑆‘𝐶)‘𝐽) − 1) ∧ (((𝑆‘𝐶)‘𝐽) − 1) ≤ (𝑀 + 𝑁))) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)))
55	1, 7, 19, 23, 53, 54	syl32anc 1371	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) ∈ (1...(𝑀 + 𝑁)))
56		biid 262	. . . . . . . . 9 ⊢ ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))
57	49, 56	sylibr 235	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → (((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶))
58	2, 3, 8, 9, 10, 11, 12, 13	ballotlemi 31371	. . . . . . . . . 10 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝐼‘𝐶) = inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < ))
59	58	breq2d 4980	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → ((((𝑆‘𝐶)‘𝐽) − 1) < (𝐼‘𝐶) ↔ (((𝑆‘𝐶)‘𝐽) − 1) < inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝐶)‘𝑘) = 0}, ℝ, < )))
60	59	3ad2ant1 1126	. . . . . . . 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, < )))
61	57, 60	mpbid 233	. . . . . . 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, < ))
62		ltso 10574	. . . . . . . . . 10 ⊢ < Or ℝ
63	62	a1i 11	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → < Or ℝ)
64	2, 3, 8, 9, 10, 11, 12, 13	ballotlemsup 31375	. . . . . . . . 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)))
65	63, 64	inflb 8806	. . . . . . . 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, < )))
66	65	con2d 136	. . . . . . 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}))
67	27, 61, 66	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})
68		fveqeq2 6554	. . . . . . 7 ⊢ (𝑘 = (((𝑆‘𝐶)‘𝐽) − 1) → (((𝐹‘𝐶)‘𝑘) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
69	68	elrab 3621	. . . . . 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))
70	67, 69	sylnib 329	. . . . 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))
71		imnan 400	. . . . 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))
72	70, 71	sylibr 235	. . . 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))
73	55, 72	mpd 15	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ¬ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0)
74	73	neqned 2993	. 2 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ≠ 0)
75		ballotth.r	. . . . . . . . . 10 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
76	2, 3, 8, 9, 10, 11, 12, 13, 14, 75	ballotlemro 31393	. . . . . . . . 9 ⊢ (𝐶 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝐶) ∈ 𝑂)
77	76	adantr 481	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (𝑅‘𝐶) ∈ 𝑂)
78		elfzelz 12762	. . . . . . . . 9 ⊢ (𝐽 ∈ (1...(𝐼‘𝐶)) → 𝐽 ∈ ℤ)
79	78	adantl 482	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → 𝐽 ∈ ℤ)
80	2, 3, 8, 9, 10, 77, 79	ballotlemfelz 31361	. . . . . . 7 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℤ)
81	80	zcnd 11942	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ∈ ℂ)
82	81	negeq0d 10843	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) = 0 ↔ -((𝐹‘(𝑅‘𝐶))‘𝐽) = 0))
83		eqid 2797	. . . . . . 7 ⊢ (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢)))) = (𝑢 ∈ Fin, 𝑣 ∈ Fin ↦ ((♯‘(𝑣 ∩ 𝑢)) − (♯‘(𝑣 ∖ 𝑢))))
84	2, 3, 8, 9, 10, 11, 12, 13, 14, 75, 83	ballotlemfrceq 31399	. . . . . 6 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = -((𝐹‘(𝑅‘𝐶))‘𝐽))
85	84	eqeq1d 2799	. . . . 5 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0 ↔ -((𝐹‘(𝑅‘𝐶))‘𝐽) = 0))
86	82, 85	bitr4d 283	. . . 4 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) = 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) = 0))
87	86	necon3bid 3030	. . 3 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝐼‘𝐶))) → (((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0 ↔ ((𝐹‘𝐶)‘(((𝑆‘𝐶)‘𝐽) − 1)) ≠ 0))
88	27, 42, 87	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))
89	74, 88	mpbird 258	1 ⊢ ((𝐶 ∈ (𝑂 ∖ 𝐸) ∧ 𝐽 ∈ (1...(𝑀 + 𝑁)) ∧ 𝐽 < (𝐼‘𝐶)) → ((𝐹‘(𝑅‘𝐶))‘𝐽) ≠ 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∀wral 3107  {crab 3111   ∖ cdif 3862   ∩ cin 3864  ifcif 4387  𝒫 cpw 4459   class class class wbr 4968   ↦ cmpt 5047   Or wor 5368   “ cima 5453  ‘cfv 6232  (class class class)co 7023   ∈ cmpo 7025  Fincfn 8364  infcinf 8758  ℝcr 10389  0cc0 10390  1c1 10391   + caddc 10393   < clt 10528   ≤ cle 10529   − cmin 10723  -cneg 10724   / cdiv 11151  ℕcn 11492  ℤcz 11835  ...cfz 12746  ♯chash 13544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-inf 8760  df-dju 9183  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-n0 11752  df-z 11836  df-uz 12098  df-rp 12244  df-fz 12747  df-hash 13545

This theorem is referenced by:  ballotlemirc  31402