ballotlemfc0 - Mathbox for Thierry Arnoux

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Theorem ballotlemfc0 33092

Description: 𝐹 takes value 0 between negative and positive values. (Contributed by Thierry Arnoux, 24-Nov-2016.)

**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)))))
ballotlemfp1.c	⊢ (𝜑 → 𝐶 ∈ 𝑂)
ballotlemfp1.j	⊢ (𝜑 → 𝐽 ∈ ℕ)
ballotlemfc0.3	⊢ (𝜑 → ∃𝑖 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑖) ≤ 0)
ballotlemfc0.4	⊢ (𝜑 → 0 < ((𝐹‘𝐶)‘𝐽))

**Assertion**
Ref	Expression
ballotlemfc0	⊢ (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 ballotlemfc0 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6842	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝑘))
2	1	breq1d 5115	. . . . . 6 ⊢ (𝑖 = 𝑘 → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘𝑘) ≤ 0))
3	2	elrab 3645	. . . . 5 ⊢ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ↔ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0))
4	3	anbi1i 624	. . . 4 ⊢ ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘) ↔ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘))
5		simprlr 778	. . . . 5 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) ≤ 0)
6		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) → 𝑘 ∈ (1...𝐽))
7	6	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → 𝑘 ∈ (1...𝐽))
8		fzssuz 13482	. . . . . . . . . . . . . 14 ⊢ (1...𝐽) ⊆ (ℤ_≥‘1)
9		uzssz 12784	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) ⊆ ℤ
10	8, 9	sstri 3953	. . . . . . . . . . . . 13 ⊢ (1...𝐽) ⊆ ℤ
11		zssre 12506	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℝ
12	10, 11	sstri 3953	. . . . . . . . . . . 12 ⊢ (1...𝐽) ⊆ ℝ
13	12	sseli 3940	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
14	13	ltp1d 12085	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
15		1red 11156	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝐽) → 1 ∈ ℝ)
16	13, 15	readdcld 11184	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℝ)
17	13, 16	ltnled 11302	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
18	14, 17	mpbid 231	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝐽) → ¬ (𝑘 + 1) ≤ 𝑘)
19	7, 18	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ¬ (𝑘 + 1) ≤ 𝑘)
20		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)
21		ballotlemfc0.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < ((𝐹‘𝐶)‘𝐽))
22	21	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → 0 < ((𝐹‘𝐶)‘𝐽))
23		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → 𝑘 = 𝐽)
24	23	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → ((𝐹‘𝐶)‘𝑘) = ((𝐹‘𝐶)‘𝐽))
25	24	breq2d 5117	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → (0 < ((𝐹‘𝐶)‘𝑘) ↔ 0 < ((𝐹‘𝐶)‘𝐽)))
26		ballotlemfp1.j	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐽 ∈ ℕ)
27		elnnuz 12807	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ_≥‘1))
28	26, 27	sylib 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘1))
29		eluzfz2 13449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ (ℤ_≥‘1) → 𝐽 ∈ (1...𝐽))
30	28, 29	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ (1...𝐽))
31		eleq1 2825	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
32	30, 31	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑘 = 𝐽 → 𝑘 ∈ (1...𝐽)))
33	32	anc2li 556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 = 𝐽 → (𝜑 ∧ 𝑘 ∈ (1...𝐽))))
34		1eluzge0 12817	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ (ℤ_≥‘0)
35		fzss1 13480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 ∈ (ℤ_≥‘0) → (1...𝐽) ⊆ (0...𝐽))
36	35	sseld 3943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ (ℤ_≥‘0) → (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽)))
37	34, 36	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽))
38		0red 11158	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ∈ ℝ)
39		ballotth.m	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑀 ∈ ℕ
40		ballotth.n	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑁 ∈ ℕ
41		ballotth.o	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
42		ballotth.p	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
43		ballotth.f	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
44		ballotlemfp1.c	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐶 ∈ 𝑂)
45	44	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐶 ∈ 𝑂)
46		elfzelz 13441	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
47	46	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
48	39, 40, 41, 42, 43, 45, 47	ballotlemfelz 33090	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐹‘𝐶)‘𝑘) ∈ ℤ)
49	48	zred 12607	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐹‘𝐶)‘𝑘) ∈ ℝ)
50	38, 49	ltnled 11302	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0 < ((𝐹‘𝐶)‘𝑘) ↔ ¬ ((𝐹‘𝐶)‘𝑘) ≤ 0))
51	37, 50	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐽)) → (0 < ((𝐹‘𝐶)‘𝑘) ↔ ¬ ((𝐹‘𝐶)‘𝑘) ≤ 0))
52	33, 51	syl6 35	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 = 𝐽 → (0 < ((𝐹‘𝐶)‘𝑘) ↔ ¬ ((𝐹‘𝐶)‘𝑘) ≤ 0)))
53	52	imp 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → (0 < ((𝐹‘𝐶)‘𝑘) ↔ ¬ ((𝐹‘𝐶)‘𝑘) ≤ 0))
54	25, 53	bitr3d 280	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → (0 < ((𝐹‘𝐶)‘𝐽) ↔ ¬ ((𝐹‘𝐶)‘𝑘) ≤ 0))
55	22, 54	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 = 𝐽) → ¬ ((𝐹‘𝐶)‘𝑘) ≤ 0)
56	55	ex 413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 = 𝐽 → ¬ ((𝐹‘𝐶)‘𝑘) ≤ 0))
57	56	con2d 134	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝐹‘𝐶)‘𝑘) ≤ 0 → ¬ 𝑘 = 𝐽))
58		nn1m1nn 12174	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ ℕ → (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
59	26, 58	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
60		ballotlemfc0.3	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ∃𝑖 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑖) ≤ 0)
61	60	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐽 = 1) → ∃𝑖 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑖) ≤ 0)
62		oveq1 7364	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
63	62	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
64	26	nnzd 12526	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝐽 ∈ ℤ)
65		fzsn 13483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
66	64, 65	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝐽...𝐽) = {𝐽})
67	66	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝐽 = 1) → (𝐽...𝐽) = {𝐽})
68	63, 67	eqtr3d 2778	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝐽 = 1) → (1...𝐽) = {𝐽})
69	68	rexeqdv 3314	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝐽 = 1) → (∃𝑖 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ {𝐽} ((𝐹‘𝐶)‘𝑖) ≤ 0))
70	61, 69	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → ∃𝑖 ∈ {𝐽} ((𝐹‘𝐶)‘𝑖) ≤ 0)
71		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘𝐽))
72	71	breq1d 5115	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 = 𝐽 → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘𝐽) ≤ 0))
73	72	rexsng 4635	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐽 ∈ ℕ → (∃𝑖 ∈ {𝐽} ((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘𝐽) ≤ 0))
74	26, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (∃𝑖 ∈ {𝐽} ((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘𝐽) ≤ 0))
75	74	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → (∃𝑖 ∈ {𝐽} ((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘𝐽) ≤ 0))
76	70, 75	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝐽 = 1) → ((𝐹‘𝐶)‘𝐽) ≤ 0)
77	21	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → 0 < ((𝐹‘𝐶)‘𝐽))
78		0red 11158	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ ℝ)
79	39, 40, 41, 42, 43, 44, 64	ballotlemfelz 33090	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℤ)
80	79	zred 12607	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ((𝐹‘𝐶)‘𝐽) ∈ ℝ)
81	78, 80	ltnled 11302	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (0 < ((𝐹‘𝐶)‘𝐽) ↔ ¬ ((𝐹‘𝐶)‘𝐽) ≤ 0))
82	81	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝐽 = 1) → (0 < ((𝐹‘𝐶)‘𝐽) ↔ ¬ ((𝐹‘𝐶)‘𝐽) ≤ 0))
83	77, 82	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝐽 = 1) → ¬ ((𝐹‘𝐶)‘𝐽) ≤ 0)
84	76, 83	pm2.65da 815	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ¬ 𝐽 = 1)
85		biortn 936	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝐽 = 1 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
86	84, 85	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
87		notnotb 314	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1)
88	87	orbi1i 912	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
89	86, 88	bitr4di 288	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
90	59, 89	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐽 − 1) ∈ ℕ)
91		elnnuz 12807	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ_≥‘1))
92	90, 91	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐽 − 1) ∈ (ℤ_≥‘1))
93		elfzp1 13491	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 − 1) ∈ (ℤ_≥‘1) → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
94	92, 93	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
95	26	nncnd 12169	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐽 ∈ ℂ)
96		1cnd 11150	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℂ)
97	95, 96	npcand 11516	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
98	97	oveq2d 7373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
99	98	eleq2d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
100	97	eqeq2d 2747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
101	100	orbi2d 914	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
102	94, 99, 101	3bitr3d 308	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
103		orcom 868	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽 ∨ 𝑘 ∈ (1...(𝐽 − 1))))
104	102, 103	bitrdi 286	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽 ∨ 𝑘 ∈ (1...(𝐽 − 1)))))
105	104	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽 ∨ 𝑘 ∈ (1...(𝐽 − 1)))))
106		pm5.6 1000	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))) ↔ (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽 ∨ 𝑘 ∈ (1...(𝐽 − 1)))))
107	105, 106	sylibr 233	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
108	90	nnzd 12526	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐽 − 1) ∈ ℤ)
109		1z 12533	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℤ
110	108, 109	jctil 520	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
111		elfzelz 13441	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
112	111, 109	jctir 521	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
113		fzaddel 13475	. . . . . . . . . . . . . . . . . 18 ⊢ (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
114	110, 112, 113	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
115	114	biimp3a 1469	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
116	115	3anidm23 1421	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
117		1p1e2 12278	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 + 1) = 2
118	117	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1 + 1) = 2)
119	118, 97	oveq12d 7375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
120	119	eleq2d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
121		2eluzge1 12819	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ (ℤ_≥‘1)
122		fzss1 13480	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ∈ (ℤ_≥‘1) → (2...𝐽) ⊆ (1...𝐽))
123	121, 122	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (2...𝐽) ⊆ (1...𝐽)
124	123	sseli 3940	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
125	120, 124	syl6bi 252	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
126	125	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
127	116, 126	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ (1...𝐽))
128	127	ex 413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 + 1) ∈ (1...𝐽)))
129	107, 128	syld 47	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → (𝑘 + 1) ∈ (1...𝐽)))
130	57, 129	sylan2d 605	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) → (𝑘 + 1) ∈ (1...𝐽)))
131	130	imp 407	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) → (𝑘 + 1) ∈ (1...𝐽))
132	131	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
133		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑘 + 1) → ((𝐹‘𝐶)‘𝑖) = ((𝐹‘𝐶)‘(𝑘 + 1)))
134	133	breq1d 5115	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑘 + 1) → (((𝐹‘𝐶)‘𝑖) ≤ 0 ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0))
135	134	elrab 3645	. . . . . . . . . . . 12 ⊢ ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0))
136		breq1 5108	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 ≤ 𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
137	136	rspccva 3580	. . . . . . . . . . . 12 ⊢ ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}) → (𝑘 + 1) ≤ 𝑘)
138	135, 137	sylan2br 595	. . . . . . . . . . 11 ⊢ ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0)) → (𝑘 + 1) ≤ 𝑘)
139	138	expr 457	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0 → (𝑘 + 1) ≤ 𝑘))
140	139	con3d 152	. . . . . . . . 9 ⊢ ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0))
141	20, 132, 140	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0))
142	19, 141	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ¬ ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0)
143		simplrr 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)
144	132	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
145		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝜑)
146	131	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
147	35	sseld 3943	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (ℤ_≥‘0) → ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ (0...𝐽)))
148	34, 146, 147	mpsyl 68	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (0...𝐽))
149	44	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → 𝐶 ∈ 𝑂)
150		elfzelz 13441	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ (0...𝐽) → (𝑘 + 1) ∈ ℤ)
151	150	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → (𝑘 + 1) ∈ ℤ)
152	39, 40, 41, 42, 43, 149, 151	ballotlemfelz 33090	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℤ)
153	152	zred 12607	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℝ)
154	145, 148, 153	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) ∈ ℝ)
155		0red 11158	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
156		simplrr 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘𝑘) ≤ 0)
157	6	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
158	157, 37	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (0...𝐽))
159	130	imdistani 569	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
160	44	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶 ∈ 𝑂)
161		elfznn 13470	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ ℕ)
162	161	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (𝑘 + 1) ∈ ℕ)
163	39, 40, 41, 42, 43, 160, 162	ballotlemfp1 33091	. . . . . . . . . . . . . . . . . . 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))))
164	163	simpld 495	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1)))
165	164	imp 407	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1))
166	159, 165	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1))
167		elfzelz 13441	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
168	167	zcnd 12608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
169		1cnd 11150	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
170	168, 169	pncand 11513	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
171	170	fveq2d 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐽) → ((𝐹‘𝐶)‘((𝑘 + 1) − 1)) = ((𝐹‘𝐶)‘𝑘))
172	171	oveq1d 7372	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹‘𝐶)‘𝑘) − 1))
173	172	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1)))
174	157, 173	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1)))
175	166, 174	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1))
176		0z 12510	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
177		zlem1lt 12555	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹‘𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
178	48, 176, 177	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
179	178	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1)) → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
180		breq1 5108	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1) → (((𝐹‘𝐶)‘(𝑘 + 1)) < 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
181	180	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1)) → (((𝐹‘𝐶)‘(𝑘 + 1)) < 0 ↔ (((𝐹‘𝐶)‘𝑘) − 1) < 0))
182	179, 181	bitr4d 281	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) − 1)) → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) < 0))
183	145, 158, 175, 182	syl21anc 836	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (((𝐹‘𝐶)‘𝑘) ≤ 0 ↔ ((𝐹‘𝐶)‘(𝑘 + 1)) < 0))
184	156, 183	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) < 0)
185	154, 155, 184	ltled 11303	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0)
186	185	adantlrr 719	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0)
187	143, 144, 186, 138	syl12anc 835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ≤ 𝑘)
188	19	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ¬ (𝑘 + 1) ≤ 𝑘)
189	187, 188	condan 816	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → (𝑘 + 1) ∈ 𝐶)
190	163	simprd 496	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1)))
191	190	imp 407	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1))
192	159, 191	sylan 580	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1))
193	6	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
194	171	oveq1d 7372	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝐽) → (((𝐹‘𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹‘𝐶)‘𝑘) + 1))
195	194	eqeq2d 2747	. . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ (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 231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1))
198	197	adantlrr 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1))
199	189, 198	mpdan 685	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1))
200		breq1 5108	. . . . . . . . 9 ⊢ (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1) → (((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0 ↔ (((𝐹‘𝐶)‘𝑘) + 1) ≤ 0))
201	200	notbid 317	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘(𝑘 + 1)) = (((𝐹‘𝐶)‘𝑘) + 1) → (¬ ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0 ↔ ¬ (((𝐹‘𝐶)‘𝑘) + 1) ≤ 0))
202	199, 201	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → (¬ ((𝐹‘𝐶)‘(𝑘 + 1)) ≤ 0 ↔ ¬ (((𝐹‘𝐶)‘𝑘) + 1) ≤ 0))
203	142, 202	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ¬ (((𝐹‘𝐶)‘𝑘) + 1) ≤ 0)
204	6, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) → 𝑘 ∈ (0...𝐽))
205	204, 48	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0)) → ((𝐹‘𝐶)‘𝑘) ∈ ℤ)
206	205	adantrr 715	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) ∈ ℤ)
207		zleltp1 12554	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ ((𝐹‘𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ 0 < (((𝐹‘𝐶)‘𝑘) + 1)))
208	176, 207	mpan 688	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ 0 < (((𝐹‘𝐶)‘𝑘) + 1)))
209		0red 11158	. . . . . . . . 9 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → 0 ∈ ℝ)
210		zre 12503	. . . . . . . . . 10 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → ((𝐹‘𝐶)‘𝑘) ∈ ℝ)
211		1red 11156	. . . . . . . . . 10 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → 1 ∈ ℝ)
212	210, 211	readdcld 11184	. . . . . . . . 9 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (((𝐹‘𝐶)‘𝑘) + 1) ∈ ℝ)
213	209, 212	ltnled 11302	. . . . . . . 8 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (0 < (((𝐹‘𝐶)‘𝑘) + 1) ↔ ¬ (((𝐹‘𝐶)‘𝑘) + 1) ≤ 0))
214	208, 213	bitrd 278	. . . . . . 7 ⊢ (((𝐹‘𝐶)‘𝑘) ∈ ℤ → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ ¬ (((𝐹‘𝐶)‘𝑘) + 1) ≤ 0))
215	206, 214	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → (0 ≤ ((𝐹‘𝐶)‘𝑘) ↔ ¬ (((𝐹‘𝐶)‘𝑘) + 1) ≤ 0))
216	203, 215	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → 0 ≤ ((𝐹‘𝐶)‘𝑘))
217	206	zred 12607	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) ∈ ℝ)
218		0red 11158	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → 0 ∈ ℝ)
219	217, 218	letri3d 11297	. . . . 5 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → (((𝐹‘𝐶)‘𝑘) = 0 ↔ (((𝐹‘𝐶)‘𝑘) ≤ 0 ∧ 0 ≤ ((𝐹‘𝐶)‘𝑘))))
220	5, 216, 219	mpbir2and 711	. . . 4 ⊢ ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) = 0)
221	4, 220	sylan2b 594	. . 3 ⊢ ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)) → ((𝐹‘𝐶)‘𝑘) = 0)
222		ssrab2 4037	. . . . . 6 ⊢ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ⊆ (1...𝐽)
223	222, 12	sstri 3953	. . . . 5 ⊢ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ⊆ ℝ
224	223	a1i 11	. . . 4 ⊢ (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ⊆ ℝ)
225		fzfi 13877	. . . . . 6 ⊢ (1...𝐽) ∈ Fin
226		ssfi 9117	. . . . . 6 ⊢ (((1...𝐽) ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ⊆ (1...𝐽)) → {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ∈ Fin)
227	225, 222, 226	mp2an 690	. . . . 5 ⊢ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ∈ Fin
228	227	a1i 11	. . . 4 ⊢ (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ∈ Fin)
229		rabn0 4345	. . . . 5 ⊢ ({𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ≠ ∅ ↔ ∃𝑖 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑖) ≤ 0)
230	60, 229	sylibr 233	. . . 4 ⊢ (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ≠ ∅)
231		fimaxre 12099	. . . 4 ⊢ (({𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ⊆ ℝ ∧ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ≠ ∅) → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)
232	224, 228, 230, 231	syl3anc 1371	. . 3 ⊢ (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0}𝑗 ≤ 𝑘)
233	221, 232	reximddv 3168	. 2 ⊢ (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ((𝐹‘𝐶)‘𝑘) = 0)
234		elrabi 3639	. . . 4 ⊢ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} → 𝑘 ∈ (1...𝐽))
235	234	anim1i 615	. . 3 ⊢ ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ∧ ((𝐹‘𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹‘𝐶)‘𝑘) = 0))
236	235	reximi2 3082	. 2 ⊢ (∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹‘𝐶)‘𝑖) ≤ 0} ((𝐹‘𝐶)‘𝑘) = 0 → ∃𝑘 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑘) = 0)
237	233, 236	syl 17	1 ⊢ (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹‘𝐶)‘𝑘) = 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 ∖ cdif 3907 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℤcz 12499 ℤ_≥cuz 12763 ...cfz 13424 ♯chash 14230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-hash 14231

This theorem is referenced by: ballotlem5 33099 ballotlemic 33106

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