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Theorem ballotfilemfcc 13233

Description: 𝐹 takes value 0 between positive and negative values. (Contributed by Thierry Arnoux, 2-Apr-2017.)

**Hypotheses**
Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfilem.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfilem.p	⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotlemfcc.c	⊢ (𝜑 → 𝐶 ∈ 𝑂)
ballotlemfcc.j	⊢ (𝜑 → 𝐽 ∈ ℕ)
ballotlemfcc.3	⊢ (𝜑 → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹‘𝐶)‘𝑖))
ballotlemfcc.4	⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) < 0)

**Assertion**
Ref	Expression
ballotfilemfcc	⊢ (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑘) = 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,𝐹 𝐶,𝑖 𝑖,𝐽 𝜑,𝑖,𝑘 𝑘,𝐽 𝐶,𝑘 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑥, 𝑐) 𝐶(𝑥, 𝑐) 𝑃(𝑥, 𝑖, 𝑘, 𝑐) 𝐹(𝑥) 𝐽(𝑥, 𝑐) 𝑀(𝑥) 𝑁(𝑥) 𝑂(𝑥)

Proof of Theorem ballotfilemfcc Dummy variables 𝑗 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5695	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝑘))
2	1	breq2d 4142	. . . . . 6 ⊢ (𝑖 = 𝑘 → (0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
3	2	elrab 2982	. . . . 5 ⊢ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ↔ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
4	3	anbi1i 462	. . . 4 ⊢ ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘) ↔ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘))
5		simprl 535	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) → 𝑘 ∈ (1...𝐽))
6	5	adantrr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → 𝑘 ∈ (1...𝐽))
7		fzssuz 10471	. . . . . . . . . . . . . 14 ⊢ (1...𝐽) ⊆ (ℤ_≥‘1)
8		uzssz 9942	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) ⊆ ℤ
9	7, 8	sstri 3257	. . . . . . . . . . . . 13 ⊢ (1...𝐽) ⊆ ℤ
10		zssre 9651	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℝ
11	9, 10	sstri 3257	. . . . . . . . . . . 12 ⊢ (1...𝐽) ⊆ ℝ
12	11	sseli 3244	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
13	12	ltp1d 9260	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
14		elfzelz 10428	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
15	14	peano2zd 9771	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℤ)
16		zltnle 9690	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
17	14, 15, 16	syl2anc 415	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
18	13, 17	mpbid 147	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝐽) → ¬ (𝑘 + 1) ≤ 𝑘)
19	6, 18	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ¬ (𝑘 + 1) ≤ 𝑘)
20		simprr 537	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)
21		ballotlemfcc.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) < 0)
22	21	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → ((𝐹‘𝐶)‘𝐽) < 0)
23		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → 𝑘 = 𝐽)
24	23	fveq2d 5699	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → ((𝐹‘𝐶)‘𝑘) = ((𝐹‘𝐶)‘𝐽))
25	24	breq1d 4140	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → (((𝐹‘𝐶)‘𝑘) < 0 ↔ ((𝐹‘𝐶)‘𝐽) < 0))
26		ballotlemfcc.j	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐽 ∈ ℕ)
27		elnnuz 9959	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ_≥‘1))
28	26, 27	sylib 122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘1))
29		eluzfz2 10436	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ (ℤ_≥‘1) → 𝐽 ∈ (1...𝐽))
30	28, 29	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ (1...𝐽))
31		eleq1 2301	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
32	30, 31	syl5ibrcom 157	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 = 𝐽 → 𝑘 ∈ (1...𝐽)))
33	32	anc2li 329	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 = 𝐽 → (𝜑 ∧ 𝑘 ∈ (1...𝐽))))
34		1eluzge0 9974	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ (ℤ_≥‘0)
35		fzss1 10469	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ (ℤ_≥‘0) → (1...𝐽) ⊆ (0...𝐽))
36	35	sseld 3247	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ (ℤ_≥‘0) → (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽)))
37	34, 36	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽))
38		ballotth.m	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑀 ∈ ℕ
39		ballotth.n	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑁 ∈ ℕ
40		ballotfilem.o	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
41		ballotfilem.p	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑃 = (𝑥 ∈ (𝒫 𝑂 ∩ Fin) ↦ ((♯‘𝑥) / (♯‘𝑂)))
42		ballotth.f	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
43		ballotlemfcc.c	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐶 ∈ 𝑂)
44	43	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐶 ∈ 𝑂)
45		elfzelz 10428	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
46	45	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
47	38, 39, 40, 41, 42, 44, 46	ballotfilemfelz 13230	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐹‘𝐶)‘𝑘) ∈ ℤ)
48		0zd 9656	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ∈ ℤ)
49		zltnle 9690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹‘𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹‘𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
50	47, 48, 49	syl2anc 415	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (((𝐹‘𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
51	37, 50	sylan2 286	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐽)) → (((𝐹‘𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
52	33, 51	syl6 33	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 = 𝐽 → (((𝐹‘𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘))))
53	52	imp 124	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → (((𝐹‘𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
54	25, 53	bitr3d 190	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → (((𝐹‘𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
55	22, 54	mpbid 147	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘))
56	55	ex 115	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 = 𝐽 → ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
57	56	con2d 633	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝐶)‘𝑘) → ¬ 𝑘 = 𝐽))
58		nn1m1nn 9322	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ ℕ → (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
59	26, 58	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
60		ballotlemfcc.3	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹‘𝐶)‘𝑖))
61	60	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐽 = 1) → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹‘𝐶)‘𝑖))
62		oveq1 6092	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
63	62	adantl 277	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
64	26	nnzd 9767	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐽 ∈ ℤ)
65		fzsn 10472	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
66	64, 65	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝐽...𝐽) = {𝐽})
67	66	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝐽 = 1) → (𝐽...𝐽) = {𝐽})
68	63, 67	eqtr3d 2273	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐽 = 1) → (1...𝐽) = {𝐽})
69	61, 68	rexeqtrdv 2758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹‘𝐶)‘𝑖))
70		fveq2 5695	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝐽))
71	70	breq2d 4142	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝐽 → (0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
72	71	rexsng 3750	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐽 ∈ ℕ → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
73	26, 72	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
74	73	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
75	69, 74	mpbid 147	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝐽 = 1) → 0 ≤ ((𝐹‘𝐶)‘𝐽))
76	21	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → ((𝐹‘𝐶)‘𝐽) < 0)
77	38, 39, 40, 41, 42, 43, 64	ballotfilemfelz 13230	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ)
78		0zd 9656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ ℤ)
79		zltnle 9690	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝐶)‘𝐽) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹‘𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
80	77, 78, 79	syl2anc 415	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (((𝐹‘𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
81	80	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → (((𝐹‘𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
82	76, 81	mpbid 147	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝐽 = 1) → ¬ 0 ≤ ((𝐹‘𝐶)‘𝐽))
83	75, 82	pm2.65da 671	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 𝐽 = 1)
84		biortn 757	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐽 = 1 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
85	83, 84	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
86		1z 9670	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
87		zdceq 9720	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐽 ∈ ℤ ∧ 1 ∈ ℤ) → DECID 𝐽 = 1)
88	64, 86, 87	sylancl 417	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → DECID 𝐽 = 1)
89		notnotbdc 884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (DECID 𝐽 = 1 → (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1))
90	88, 89	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1))
91	90	orbi1d 803	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
92	85, 91	bitr4d 191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
93	59, 92	mpbird 167	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐽 − 1) ∈ ℕ)
94		elnnuz 9959	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ_≥‘1))
95	93, 94	sylib 122	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐽 − 1) ∈ (ℤ_≥‘1))
96		elfzp1 10479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 − 1) ∈ (ℤ_≥‘1) → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
97	95, 96	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
98	26	nncnd 9318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ ℂ)
99		1cnd 8342	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℂ)
100	98, 99	npcand 8641	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
101	100	oveq2d 6101	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
102	101	eleq2d 2308	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
103	100	eqeq2d 2250	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
104	103	orbi2d 802	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
105	97, 102, 104	3bitr3d 218	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
106		orcom 740	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽 ∨ 𝑘 ∈ (1...(𝐽 − 1))))
107	105, 106	bitrdi 196	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽 ∨ 𝑘 ∈ (1...(𝐽 − 1)))))
108	107	biimpd 144	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽 ∨ 𝑘 ∈ (1...(𝐽 − 1)))))
109		pm5.6r 939	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽 ∨ 𝑘 ∈ (1...(𝐽 − 1)))) → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
110	108, 109	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
111	93	nnzd 9767	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
112	111, 86	jctil 312	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
113		elfzelz 10428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
114	113, 86	jctir 313	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
115		fzaddel 10465	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
116	112, 114, 115	syl2an 289	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
117	116	biimp3a 1386	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
118	117	3anidm23 1338	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
119		1p1e2 9421	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 + 1) = 2
120	119	a1i 9	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1 + 1) = 2)
121	120, 100	oveq12d 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
122	121	eleq2d 2308	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
123		2eluzge1 9976	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ (ℤ_≥‘1)
124		fzss1 10469	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ∈ (ℤ_≥‘1) → (2...𝐽) ⊆ (1...𝐽))
125	123, 124	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (2...𝐽) ⊆ (1...𝐽)
126	125	sseli 3244	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
127	122, 126	biimtrdi 163	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
128	127	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
129	118, 128	mpd 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ (1...𝐽))
130	129	ex 115	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 + 1) ∈ (1...𝐽)))
131	110, 130	syld 45	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → (𝑘 + 1) ∈ (1...𝐽)))
132	57, 131	sylan2d 294	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) → (𝑘 + 1) ∈ (1...𝐽)))
133	132	imp 124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) → (𝑘 + 1) ∈ (1...𝐽))
134	133	adantrr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
135		fveq2 5695	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑘 + 1) → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘(𝑘 + 1)))
136	135	breq2d 4142	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑘 + 1) → (0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1))))
137	136	elrab 2982	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1))))
138		breq1 4133	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 ≤ 𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
139	138	rspccva 2928	. . . . . . . . . . . 12 ⊢ ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}) → (𝑘 + 1) ≤ 𝑘)
140	137, 139	sylan2br 288	. . . . . . . . . . 11 ⊢ ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)))) → (𝑘 + 1) ≤ 𝑘)
141	140	expr 375	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)) → (𝑘 + 1) ≤ 𝑘))
142	141	con3d 640	. . . . . . . . 9 ⊢ ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1))))
143	20, 134, 142	syl2anc 415	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1))))
144	19, 143	mpd 13	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ¬ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)))
145		simplrr 542	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)
146	134	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
147		0red 8327	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
148		simpll 531	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝜑)
149	133	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
150	35	sseld 3247	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ (0...𝐽)))
151	34, 149, 150	mpsyl 65	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (0...𝐽))
152	43	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → 𝐶 ∈ 𝑂)
153		elfzelz 10428	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ (0...𝐽) → (𝑘 + 1) ∈ ℤ)
154	153	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → (𝑘 + 1) ∈ ℤ)
155	38, 39, 40, 41, 42, 152, 154	ballotfilemfelz 13230	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℤ)
156	155	zred 9768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℝ)
157	148, 151, 156	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℝ)
158		simplrr 542	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹‘𝐶)‘𝑘))
159	5	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
160	159, 37	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (0...𝐽))
161	132	imdistani 449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
162	43	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶 ∈ 𝑂)
163		elfznn 10460	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ ℕ)
164	163	adantl 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (𝑘 + 1) ∈ ℕ)
165	38, 39, 40, 41, 42, 162, 164	ballotfilemfp1 13231	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1)) ∧ ((𝑘 + 1) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1))))
166	165	simprd 114	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1)))
167	166	imp 124	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1))
168	161, 167	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1))
169	14	zcnd 9769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
170		1cnd 8342	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
171	169, 170	pncand 8638	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
172	171	fveq2d 5699	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐽) → ((𝐹‘𝐶)‘((𝑘 + 1) − 1)) = ((𝐹‘𝐶)‘𝑘))
173	172	oveq1d 6100	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹‘𝐶)‘𝑘) + 1))
174	173	eqeq2d 2250	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1) ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1)))
175	159, 174	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1) ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1)))
176	168, 175	mpbid 147	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1))
177		0z 9655	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
178		zleltp1 9700	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℤ ∧ ((𝐹‘𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ 0 < (((𝐹‘𝐶)‘𝑘) + 1)))
179	177, 47, 178	sylancr 418	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ 0 < (((𝐹‘𝐶)‘𝑘) + 1)))
180	179	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ 0 < (((𝐹‘𝐶)‘𝑘) + 1)))
181		breq2 4134	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1) → (0 < ((𝐹‘𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹‘𝐶)‘𝑘) + 1)))
182	181	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1)) → (0 < ((𝐹‘𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹‘𝐶)‘𝑘) + 1)))
183	180, 182	bitr4d 191	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ 0 < ((𝐹‘𝐶)‘(𝑘 + 1))))
184	148, 160, 176, 183	syl21anc 1277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ 0 < ((𝐹‘𝐶)‘(𝑘 + 1))))
185	158, 184	mpbid 147	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 < ((𝐹‘𝐶)‘(𝑘 + 1)))
186	147, 157, 185	ltled 8445	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)))
187	186	adantlrr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)))
188	145, 146, 187, 140	syl12anc 1276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ≤ 𝑘)
189	19, 188	mtand 675	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ¬ (𝑘 + 1) ∈ 𝐶)
190	165	simpld 112	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1)))
191	190	imp 124	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1))
192	161, 191	sylan 283	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1))
193	5	adantr 276	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
194	172	oveq1d 6100	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹‘𝐶)‘𝑘) − 1))
195	194	eqeq2d 2250	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1)))
196	193, 195	syl 14	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1)))
197	192, 196	mpbid 147	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1))
198	197	adantlrr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1))
199	189, 198	mpdan 425	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1))
200		breq2 4134	. . . . . . . . 9 ⊢ (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1) → (0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)) ↔ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
201	200	notbid 677	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1) → (¬ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)) ↔ ¬ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
202	199, 201	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → (¬ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)) ↔ ¬ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
203	144, 202	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ¬ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1))
204	5, 37	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) → 𝑘 ∈ (0...𝐽))
205	204, 47	syldan 282	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) → ((𝐹‘𝐶)‘𝑘) ∈ ℤ)
206	205	adantrr 483	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) ∈ ℤ)
207		zlem1lt 9701	. . . . . . . . 9 ⊢ ((((𝐹‘𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
208	177, 207	mpan2 429	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
209		peano2zm 9682	. . . . . . . . 9 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (((𝐹‘𝐶)‘𝑘) − 1) ∈ ℤ)
210		zltnle 9690	. . . . . . . . 9 ⊢ (((((𝐹‘𝐶)‘𝑘) − 1) ∈ ℤ ∧ 0 ∈ ℤ) → ((((𝐹‘𝐶)‘𝑘) − 1) < 0 ↔ ¬ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
211	209, 177, 210	sylancl 417	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → ((((𝐹‘𝐶)‘𝑘) − 1) < 0 ↔ ¬ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
212	208, 211	bitrd 188	. . . . . . 7 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ ¬ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
213	206, 212	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ ¬ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
214	203, 213	mpbird 167	. . . . 5 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) ≤ 0)
215		simprlr 544	. . . . 5 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → 0 ≤ ((𝐹‘𝐶)‘𝑘))
216	206	zred 9768	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) ∈ ℝ)
217		0red 8327	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → 0 ∈ ℝ)
218	216, 217	letri3d 8441	. . . . 5 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → (((𝐹‘𝐶)‘𝑘) = 0 ↔ (((𝐹‘𝐶)‘𝑘) ≤ 0 ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))))
219	214, 215, 218	mpbir2and 957	. . . 4 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) = 0)
220	4, 219	sylan2b 287	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) = 0)
221		ssrab2 3333	. . . . 5 ⊢ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ⊆ (1...𝐽)
222		zssq 10027	. . . . . 6 ⊢ ℤ ⊆ ℚ
223	9, 222	sstri 3257	. . . . 5 ⊢ (1...𝐽) ⊆ ℚ
224	221, 223	sstri 3257	. . . 4 ⊢ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ⊆ ℚ
225	86	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
226	225, 64	fzfigd 10868	. . . . 5 ⊢ (𝜑 → (1...𝐽) ∈ Fin)
227		oveq2 6093	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑞 → (1...𝑖) = (1...𝑞))
228	227	ineq1d 3431	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑞 → ((1...𝑖) ∩ 𝑐) = ((1...𝑞) ∩ 𝑐))
229	228	fveq2d 5699	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑞 → (♯‘((1...𝑖) ∩ 𝑐)) = (♯‘((1...𝑞) ∩ 𝑐)))
230	227	difeq1d 3346	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑞 → ((1...𝑖) ∖ 𝑐) = ((1...𝑞) ∖ 𝑐))
231	230	fveq2d 5699	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑞 → (♯‘((1...𝑖) ∖ 𝑐)) = (♯‘((1...𝑞) ∖ 𝑐)))
232	229, 231	oveq12d 6103	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑞 → ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))) = ((♯‘((1...𝑞) ∩ 𝑐)) − (♯‘((1...𝑞) ∖ 𝑐))))
233	232	cbvmptv 4227	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))) = (𝑞 ∈ ℤ ↦ ((♯‘((1...𝑞) ∩ 𝑐)) − (♯‘((1...𝑞) ∖ 𝑐))))
234	233	mpteq2i 4218	. . . . . . . . 9 ⊢ (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))))) = (𝑐 ∈ 𝑂 ↦ (𝑞 ∈ ℤ ↦ ((♯‘((1...𝑞) ∩ 𝑐)) − (♯‘((1...𝑞) ∖ 𝑐)))))
235	42, 234	eqtri 2259	. . . . . . . 8 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑞 ∈ ℤ ↦ ((♯‘((1...𝑞) ∩ 𝑐)) − (♯‘((1...𝑞) ∖ 𝑐)))))
236	43	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝐽)) → 𝐶 ∈ 𝑂)
237		elfzelz 10428	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝐽) → 𝑖 ∈ ℤ)
238	237	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝐽)) → 𝑖 ∈ ℤ)
239	38, 39, 40, 41, 235, 236, 238	ballotfilemfelz 13230	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝐽)) → ((𝐹‘𝐶)‘𝑖) ∈ ℤ)
240		zdcle 9721	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ ((𝐹‘𝐶)‘𝑖) ∈ ℤ) → DECID 0 ≤ ((𝐹‘𝐶)‘𝑖))
241	177, 239, 240	sylancr 418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝐽)) → DECID 0 ≤ ((𝐹‘𝐶)‘𝑖))
242	241	ralrimiva 2623	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ (1...𝐽)DECID 0 ≤ ((𝐹‘𝐶)‘𝑖))
243	226, 242	ssfirab 7244	. . . 4 ⊢ (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ∈ Fin)
244		rabn0r 3548	. . . . 5 ⊢ (∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹‘𝐶)‘𝑖) → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ≠ ∅)
245	60, 244	syl 14	. . . 4 ⊢ (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ≠ ∅)
246		fimaxq 11270	. . . 4 ⊢ (({𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ⊆ ℚ ∧ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ≠ ∅) → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)
247	224, 243, 245, 246	mp3an2i 1383	. . 3 ⊢ (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)
248	220, 247	reximddv 2653	. 2 ⊢ (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ((𝐹‘𝐶)‘𝑘) = 0)
249		elrabi 2979	. . . 4 ⊢ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} → 𝑘 ∈ (1...𝐽))
250	249	anim1i 340	. . 3 ⊢ ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ∧ ((𝐹‘𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) = 0))
251	250	reximi2 2646	. 2 ⊢ (∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ((𝐹‘𝐶)‘𝑘) = 0 → ∃𝑘 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑘) = 0)
252	248, 251	syl 14	1 ⊢ (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑘) = 0)

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∀wral 2528 ∃wrex 2529 {crab 2532 ∖ cdif 3217 ∩ cin 3219 ⊆ wss 3220 ∅c0 3520 𝒫 cpw 3688 {csn 3709 class class class wbr 4130 ↦ cmpt 4192 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 ℝcr 8178 0cc0 8179 1c1 8180 + caddc 8182 < clt 8360 ≤ cle 8361 − cmin 8497 / cdiv 9002 ℕcn 9304 2c2 9355 ℤcz 9644 ℤ_≥cuz 9921 ℚcq 10019 ...cfz 10411 ♯chash 11214

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-fz 10412 df-ihash 11215

This theorem is used by: ballotfilem1c 13251

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