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Theorem ballotfilemfcc 13154

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
ballotfi.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
ballotfi.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 5672	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝑘))
2	1	breq2d 4123	. . . . . 6 ⊢ (𝑖 = 𝑘 → (0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
3	2	elrab 2975	. . . . 5 ⊢ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ↔ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
4	3	anbi1i 458	. . . 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 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) → 𝑘 ∈ (1...𝐽))
6	5	adantrr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)) → 𝑘 ∈ (1...𝐽))
7		fzssuz 10402	. . . . . . . . . . . . . 14 ⊢ (1...𝐽) ⊆ (ℤ_≥‘1)
8		uzssz 9877	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) ⊆ ℤ
9	7, 8	sstri 3249	. . . . . . . . . . . . 13 ⊢ (1...𝐽) ⊆ ℤ
10		zssre 9586	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℝ
11	9, 10	sstri 3249	. . . . . . . . . . . 12 ⊢ (1...𝐽) ⊆ ℝ
12	11	sseli 3236	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
13	12	ltp1d 9206	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
14		elfzelz 10362	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
15	14	peano2zd 9706	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℤ)
16		zltnle 9625	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ (𝑘 + 1) ∈ ℤ) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
17	14, 15, 16	syl2anc 411	. . . . . . . . . 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 533	. . . . . . . . 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 5676	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → ((𝐹‘𝐶)‘𝑘) = ((𝐹‘𝐶)‘𝐽))
25	24	breq1d 4121	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → (((𝐹‘𝐶)‘𝑘) < 0 ↔ ((𝐹‘𝐶)‘𝐽) < 0))
26		ballotlemfcc.j	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐽 ∈ ℕ)
27		elnnuz 9894	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ_≥‘1))
28	26, 27	sylib 122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘1))
29		eluzfz2 10369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ (ℤ_≥‘1) → 𝐽 ∈ (1...𝐽))
30	28, 29	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ (1...𝐽))
31		eleq1 2297	. . . . . . . . . . . . . . . . . . . 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 9909	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ (ℤ_≥‘0)
35		fzss1 10400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ (ℤ_≥‘0) → (1...𝐽) ⊆ (0...𝐽))
36	35	sseld 3239	. . . . . . . . . . . . . . . . . . . 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		ballotfi.o	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
41		ballotfi.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 10362	. . . . . . . . . . . . . . . . . . . . . 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 13151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐹‘𝐶)‘𝑘) ∈ ℤ)
48		0zd 9591	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ∈ ℤ)
49		zltnle 9625	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐹‘𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹‘𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝑘)))
50	47, 48, 49	syl2anc 411	. . . . . . . . . . . . . . . . . . 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 629	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝐶)‘𝑘) → ¬ 𝑘 = 𝐽))
58		nn1m1nn 9257	. . . . . . . . . . . . . . . . . . . . 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 6059	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
63	62	adantl 277	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
64	26	nnzd 9702	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝐽 ∈ ℤ)
65		fzsn 10403	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2269	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐽 = 1) → (1...𝐽) = {𝐽})
69	61, 68	rexeqtrdv 2752	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹‘𝐶)‘𝑖))
70		fveq2 5672	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝐽))
71	70	breq2d 4123	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝐽 → (0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
72	71	rexsng 3732	. . . . . . . . . . . . . . . . . . . . . . . . . 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 13151	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ)
78		0zd 9591	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ ℤ)
79		zltnle 9625	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐹‘𝐶)‘𝐽) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹‘𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹‘𝐶)‘𝐽)))
80	77, 78, 79	syl2anc 411	. . . . . . . . . . . . . . . . . . . . . . . . 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 667	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 𝐽 = 1)
84		biortn 753	. . . . . . . . . . . . . . . . . . . . . 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 9605	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 1 ∈ ℤ
87		zdceq 9655	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐽 ∈ ℤ ∧ 1 ∈ ℤ) → DECID 𝐽 = 1)
88	64, 86, 87	sylancl 413	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → DECID 𝐽 = 1)
89		notnotbdc 880	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (DECID 𝐽 = 1 → (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1))
90	88, 89	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1))
91	90	orbi1d 799	. . . . . . . . . . . . . . . . . . . . 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 9894	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ_≥‘1))
95	93, 94	sylib 122	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐽 − 1) ∈ (ℤ_≥‘1))
96		elfzp1 10410	. . . . . . . . . . . . . . . . . 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 9253	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ ℂ)
99		1cnd 8292	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℂ)
100	98, 99	npcand 8590	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
101	100	oveq2d 6068	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
102	101	eleq2d 2304	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
103	100	eqeq2d 2246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
104	103	orbi2d 798	. . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . 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 935	. . . . . . . . . . . . . 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 9702	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
112	111, 86	jctil 312	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
113		elfzelz 10362	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
114	113, 86	jctir 313	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
115		fzaddel 10396	. . . . . . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
118	117	3anidm23 1334	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
119		1p1e2 9356	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 + 1) = 2
120	119	a1i 9	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1 + 1) = 2)
121	120, 100	oveq12d 6070	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
122	121	eleq2d 2304	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
123		2eluzge1 9911	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ (ℤ_≥‘1)
124		fzss1 10400	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ∈ (ℤ_≥‘1) → (2...𝐽) ⊆ (1...𝐽))
125	123, 124	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (2...𝐽) ⊆ (1...𝐽)
126	125	sseli 3236	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . 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 5672	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑘 + 1) → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘(𝑘 + 1)))
136	135	breq2d 4123	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑘 + 1) → (0 ≤ ((𝐹‘𝐶)‘𝑖) ↔ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1))))
137	136	elrab 2975	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1))))
138		breq1 4114	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 ≤ 𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
139	138	rspccva 2922	. . . . . . . . . . . 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 636	. . . . . . . . 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 411	. . . . . . . 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 538	. . . . . . . . . . 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 8277	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
148		simpll 527	. . . . . . . . . . . . . 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 3239	. . . . . . . . . . . . . . 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 10362	. . . . . . . . . . . . . . . . 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 13151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℤ)
156	155	zred 9703	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℝ)
157	148, 151, 156	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℝ)
158		simplrr 538	. . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . 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 10391	. . . . . . . . . . . . . . . . . . . . 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 13152	. . . . . . . . . . . . . . . . . . 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 9704	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
170		1cnd 8292	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
171	169, 170	pncand 8587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
172	171	fveq2d 5676	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐽) → ((𝐹‘𝐶)‘((𝑘 + 1) − 1)) = ((𝐹‘𝐶)‘𝑘))
173	172	oveq1d 6067	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹‘𝐶)‘𝑘) + 1))
174	173	eqeq2d 2246	. . . . . . . . . . . . . . . . 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 9590	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
178		zleltp1 9635	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℤ ∧ ((𝐹‘𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ 0 < (((𝐹‘𝐶)‘𝑘) + 1)))
179	177, 47, 178	sylancr 414	. . . . . . . . . . . . . . . . 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 4115	. . . . . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . . 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 8394	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)))
187	186	adantlrr 483	. . . . . . . . . . 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 1272	. . . . . . . . . 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 671	. . . . . . . . 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 6067	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹‘𝐶)‘𝑘) − 1))
195	194	eqeq2d 2246	. . . . . . . . . . . 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 483	. . . . . . . . 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 421	. . . . . . . 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 4115	. . . . . . . . 9 ⊢ (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1) → (0 ≤ ((𝐹‘𝐶)‘(𝑘 + 1)) ↔ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
201	200	notbid 673	. . . . . . . 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 479	. . . . . . 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 9636	. . . . . . . . 9 ⊢ ((((𝐹‘𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
208	177, 207	mpan2 425	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
209		peano2zm 9617	. . . . . . . . 9 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (((𝐹‘𝐶)‘𝑘) − 1) ∈ ℤ)
210		zltnle 9625	. . . . . . . . 9 ⊢ (((((𝐹‘𝐶)‘𝑘) − 1) ∈ ℤ ∧ 0 ∈ ℤ) → ((((𝐹‘𝐶)‘𝑘) − 1) < 0 ↔ ¬ 0 ≤ (((𝐹‘𝐶)‘𝑘) − 1)))
211	209, 177, 210	sylancl 413	. . . . . . . 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 540	. . . . 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 9703	. . . . . 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 8277	. . . . . 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 8391	. . . . 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 953	. . . 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 3325	. . . . 5 ⊢ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ⊆ (1...𝐽)
222		zssq 9962	. . . . . 6 ⊢ ℤ ⊆ ℚ
223	9, 222	sstri 3249	. . . . 5 ⊢ (1...𝐽) ⊆ ℚ
224	221, 223	sstri 3249	. . . 4 ⊢ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ⊆ ℚ
225	86	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
226	225, 64	fzfigd 10797	. . . . 5 ⊢ (𝜑 → (1...𝐽) ∈ Fin)
227		oveq2 6060	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑞 → (1...𝑖) = (1...𝑞))
228	227	ineq1d 3423	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑞 → ((1...𝑖) ∩ 𝑐) = ((1...𝑞) ∩ 𝑐))
229	228	fveq2d 5676	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑞 → (♯‘((1...𝑖) ∩ 𝑐)) = (♯‘((1...𝑞) ∩ 𝑐)))
230	227	difeq1d 3338	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑞 → ((1...𝑖) ∖ 𝑐) = ((1...𝑞) ∖ 𝑐))
231	230	fveq2d 5676	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑞 → (♯‘((1...𝑖) ∖ 𝑐)) = (♯‘((1...𝑞) ∖ 𝑐)))
232	229, 231	oveq12d 6070	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑞 → ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐))) = ((♯‘((1...𝑞) ∩ 𝑐)) − (♯‘((1...𝑞) ∖ 𝑐))))
233	232	cbvmptv 4208	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))) = (𝑞 ∈ ℤ ↦ ((♯‘((1...𝑞) ∩ 𝑐)) − (♯‘((1...𝑞) ∖ 𝑐))))
234	233	mpteq2i 4199	. . . . . . . . 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 2255	. . . . . . . 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 10362	. . . . . . . . 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 13151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝐽)) → ((𝐹‘𝐶)‘𝑖) ∈ ℤ)
240		zdcle 9656	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ ((𝐹‘𝐶)‘𝑖) ∈ ℤ) → DECID 0 ≤ ((𝐹‘𝐶)‘𝑖))
241	177, 239, 240	sylancr 414	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝐽)) → DECID 0 ≤ ((𝐹‘𝐶)‘𝑖))
242	241	ralrimiva 2617	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ (1...𝐽)DECID 0 ≤ ((𝐹‘𝐶)‘𝑖))
243	226, 242	ssfirab 7199	. . . 4 ⊢ (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ∈ Fin)
244		rabn0r 3537	. . . . 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 11198	. . . 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 1379	. . 3 ⊢ (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)}𝑗 ≤ 𝑘)
248	220, 247	reximddv 2647	. 2 ⊢ (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹‘𝐶)‘𝑖)} ((𝐹‘𝐶)‘𝑘) = 0)
249		elrabi 2972	. . . 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 2640	. 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

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 = wceq 1398 ∈ wcel 2205 ≠ wne 2414 ∀wral 2522 ∃wrex 2523 {crab 2526 ∖ cdif 3210 ∩ cin 3212 ⊆ wss 3213 ∅c0 3510 𝒫 cpw 3671 {csn 3691 class class class wbr 4111 ↦ cmpt 4173 ‘cfv 5354 (class class class)co 6052 Fincfn 6977 ℝcr 8128 0cc0 8129 1c1 8130 + caddc 8132 < clt 8310 ≤ cle 8311 − cmin 8446 / cdiv 8948 ℕcn 9239 2c2 9290 ℤcz 9579 ℤ_≥cuz 9856 ℚcq 9954 ...cfz 10345 ♯chash 11142

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8220 ax-resscn 8221 ax-1cn 8222 ax-1re 8223 ax-icn 8224 ax-addcl 8225 ax-addrcl 8226 ax-mulcl 8227 ax-mulrcl 8228 ax-addcom 8229 ax-mulcom 8230 ax-addass 8231 ax-mulass 8232 ax-distr 8233 ax-i2m1 8234 ax-0lt1 8235 ax-1rid 8236 ax-0id 8237 ax-rnegex 8238 ax-precex 8239 ax-cnre 8240 ax-pre-ltirr 8241 ax-pre-ltwlin 8242 ax-pre-lttrn 8243 ax-pre-apti 8244 ax-pre-ltadd 8245 ax-pre-mulgt0 8246 ax-pre-mulext 8247

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-irdg 6603 df-frec 6624 df-1o 6649 df-oadd 6653 df-er 6769 df-en 6978 df-dom 6979 df-fin 6980 df-pnf 8312 df-mnf 8313 df-xr 8314 df-ltxr 8315 df-le 8316 df-sub 8448 df-neg 8449 df-reap 8851 df-ap 8858 df-div 8949 df-inn 9240 df-2 9298 df-n0 9499 df-z 9580 df-uz 9857 df-q 9955 df-rp 9990 df-fz 10346 df-ihash 11143

This theorem is referenced by: (None)

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