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Theorem ballotlemfcc 31158
 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
ballotth.o 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotlemfcc.c (𝜑𝐶𝑂)
ballotlemfcc.j (𝜑𝐽 ∈ ℕ)
ballotlemfcc.3 (𝜑 → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
ballotlemfcc.4 (𝜑 → ((𝐹𝐶)‘𝐽) < 0)
Assertion
Ref Expression
ballotlemfcc (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹𝐶)‘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 ballotlemfcc
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6448 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝑘))
21breq2d 4900 . . . . . 6 (𝑖 = 𝑘 → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝑘)))
32elrab 3572 . . . . 5 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ↔ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)))
43anbi1i 617 . . . 4 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘) ↔ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘))
5 simprl 761 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → 𝑘 ∈ (1...𝐽))
65adantrr 707 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 𝑘 ∈ (1...𝐽))
7 fzssuz 12703 . . . . . . . . . . . . . 14 (1...𝐽) ⊆ (ℤ‘1)
8 uzssz 12016 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
97, 8sstri 3830 . . . . . . . . . . . . 13 (1...𝐽) ⊆ ℤ
10 zssre 11739 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
119, 10sstri 3830 . . . . . . . . . . . 12 (1...𝐽) ⊆ ℝ
1211sseli 3817 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
1312ltp1d 11310 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
14 1red 10379 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → 1 ∈ ℝ)
1512, 14readdcld 10408 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℝ)
1612, 15ltnled 10525 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
1713, 16mpbid 224 . . . . . . . . 9 (𝑘 ∈ (1...𝐽) → ¬ (𝑘 + 1) ≤ 𝑘)
186, 17syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ (𝑘 + 1) ≤ 𝑘)
19 simprr 763 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)
20 ballotlemfcc.4 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝐹𝐶)‘𝐽) < 0)
2120adantr 474 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝐽) < 0)
22 simpr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 𝐽) → 𝑘 = 𝐽)
2322fveq2d 6452 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝑘) = ((𝐹𝐶)‘𝐽))
2423breq1d 4898 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝑘) < 0 ↔ ((𝐹𝐶)‘𝐽) < 0))
25 ballotlemfcc.j . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽 ∈ ℕ)
26 elnnuz 12034 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ‘1))
2725, 26sylib 210 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (ℤ‘1))
28 eluzfz2 12670 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (ℤ‘1) → 𝐽 ∈ (1...𝐽))
2927, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ (1...𝐽))
30 eleq1 2847 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
3129, 30syl5ibrcom 239 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 = 𝐽𝑘 ∈ (1...𝐽)))
3231anc2li 551 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = 𝐽 → (𝜑𝑘 ∈ (1...𝐽))))
33 1eluzge0 12042 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (ℤ‘0)
34 fzss1 12701 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ (ℤ‘0) → (1...𝐽) ⊆ (0...𝐽))
3534sseld 3820 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ (ℤ‘0) → (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽)))
3633, 35ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽))
37 ballotth.m . . . . . . . . . . . . . . . . . . . . . 22 𝑀 ∈ ℕ
38 ballotth.n . . . . . . . . . . . . . . . . . . . . . 22 𝑁 ∈ ℕ
39 ballotth.o . . . . . . . . . . . . . . . . . . . . . 22 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
40 ballotth.p . . . . . . . . . . . . . . . . . . . . . 22 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
41 ballotth.f . . . . . . . . . . . . . . . . . . . . . 22 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
42 ballotlemfcc.c . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐶𝑂)
4342adantr 474 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝐶𝑂)
44 elfzelz 12663 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
4544adantl 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
4637, 38, 39, 40, 41, 43, 45ballotlemfelz 31155 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
4746zred 11838 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
48 0red 10382 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → 0 ∈ ℝ)
4947, 48ltnled 10525 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽)) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5036, 49sylan2 586 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝐽)) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5132, 50syl6 35 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 = 𝐽 → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘))))
5251imp 397 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5324, 52bitr3d 273 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5421, 53mpbid 224 . . . . . . . . . . . . . 14 ((𝜑𝑘 = 𝐽) → ¬ 0 ≤ ((𝐹𝐶)‘𝑘))
5554ex 403 . . . . . . . . . . . . 13 (𝜑 → (𝑘 = 𝐽 → ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5655con2d 132 . . . . . . . . . . . 12 (𝜑 → (0 ≤ ((𝐹𝐶)‘𝑘) → ¬ 𝑘 = 𝐽))
57 nn1m1nn 11400 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ ℕ → (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
5825, 57syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
59 ballotlemfcc.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
6059adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
61 oveq1 6931 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
6261adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
6325nnzd 11837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐽 ∈ ℤ)
64 fzsn 12704 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐽...𝐽) = {𝐽})
6665adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = {𝐽})
6762, 66eqtr3d 2816 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝐽 = 1) → (1...𝐽) = {𝐽})
6867rexeqdv 3341 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → (∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖) ↔ ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖)))
6960, 68mpbid 224 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖))
70 fveq2 6448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑖 = 𝐽 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝐽))
7170breq2d 4900 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝐽 → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7271rexsng 4445 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐽 ∈ ℕ → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7325, 72syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7473adantr 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7569, 74mpbid 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → 0 ≤ ((𝐹𝐶)‘𝐽))
7620adantr 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ((𝐹𝐶)‘𝐽) < 0)
7737, 38, 39, 40, 41, 42, 63ballotlemfelz 31155 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℤ)
7877zred 11838 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℝ)
79 0red 10382 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 0 ∈ ℝ)
8078, 79ltnled 10525 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝐽)))
8180adantr 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝐽)))
8276, 81mpbid 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ¬ 0 ≤ ((𝐹𝐶)‘𝐽))
8375, 82pm2.65da 807 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝐽 = 1)
84 biortn 924 . . . . . . . . . . . . . . . . . . . . . 22 𝐽 = 1 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
8583, 84syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
86 notnotb 307 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1)
8786orbi1i 900 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
8885, 87syl6bbr 281 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
8958, 88mpbird 249 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℕ)
90 elnnuz 12034 . . . . . . . . . . . . . . . . . . 19 ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ‘1))
9189, 90sylib 210 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐽 − 1) ∈ (ℤ‘1))
92 elfzp1 12712 . . . . . . . . . . . . . . . . . 18 ((𝐽 − 1) ∈ (ℤ‘1) → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
9391, 92syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
9425nncnd 11396 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ ℂ)
95 1cnd 10373 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℂ)
9694, 95npcand 10740 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
9796oveq2d 6940 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
9897eleq2d 2845 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
9996eqeq2d 2788 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
10099orbi2d 902 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
10193, 98, 1003bitr3d 301 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
102 orcom 859 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1))))
103101, 102syl6bb 279 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
104103biimpd 221 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
105 pm5.6 987 . . . . . . . . . . . . . 14 (((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))) ↔ (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
106104, 105sylibr 226 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
10789nnzd 11837 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℤ)
108 1z 11763 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
109107, 108jctil 515 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
110 elfzelz 12663 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
111110, 108jctir 516 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
112 fzaddel 12696 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
113109, 111, 112syl2an 589 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
114113biimp3a 1542 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
1151143anidm23 1493 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
116 1p1e2 11511 . . . . . . . . . . . . . . . . . . . 20 (1 + 1) = 2
117116a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1 + 1) = 2)
118117, 96oveq12d 6942 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
119118eleq2d 2845 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
120 2eluzge1 12044 . . . . . . . . . . . . . . . . . . 19 2 ∈ (ℤ‘1)
121 fzss1 12701 . . . . . . . . . . . . . . . . . . 19 (2 ∈ (ℤ‘1) → (2...𝐽) ⊆ (1...𝐽))
122120, 121ax-mp 5 . . . . . . . . . . . . . . . . . 18 (2...𝐽) ⊆ (1...𝐽)
123122sseli 3817 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
124119, 123syl6bi 245 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
125124adantr 474 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
126115, 125mpd 15 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ (1...𝐽))
127126ex 403 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 + 1) ∈ (1...𝐽)))
128106, 127syld 47 . . . . . . . . . . . 12 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → (𝑘 + 1) ∈ (1...𝐽)))
12956, 128sylan2d 598 . . . . . . . . . . 11 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) → (𝑘 + 1) ∈ (1...𝐽)))
130129imp 397 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → (𝑘 + 1) ∈ (1...𝐽))
131130adantrr 707 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
132 fveq2 6448 . . . . . . . . . . . . . 14 (𝑖 = (𝑘 + 1) → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘(𝑘 + 1)))
133132breq2d 4900 . . . . . . . . . . . . 13 (𝑖 = (𝑘 + 1) → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
134133elrab 3572 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
135 breq1 4891 . . . . . . . . . . . . 13 (𝑗 = (𝑘 + 1) → (𝑗𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
136135rspccva 3510 . . . . . . . . . . . 12 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}) → (𝑘 + 1) ≤ 𝑘)
137134, 136sylan2br 588 . . . . . . . . . . 11 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))) → (𝑘 + 1) ≤ 𝑘)
138137expr 450 . . . . . . . . . 10 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) → (𝑘 + 1) ≤ 𝑘))
139138con3d 150 . . . . . . . . 9 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
14019, 131, 139syl2anc 579 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
14118, 140mpd 15 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))
142 simplrr 768 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)
143131adantr 474 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
144 0red 10382 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
145 simpll 757 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝜑)
146130adantr 474 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
14734sseld 3820 . . . . . . . . . . . . . . 15 (1 ∈ (ℤ‘0) → ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ (0...𝐽)))
14833, 146, 147mpsyl 68 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (0...𝐽))
14942adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → 𝐶𝑂)
150 elfzelz 12663 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (0...𝐽) → (𝑘 + 1) ∈ ℤ)
151150adantl 475 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → (𝑘 + 1) ∈ ℤ)
15237, 38, 39, 40, 41, 149, 151ballotlemfelz 31155 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℤ)
153152zred 11838 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
154145, 148, 153syl2anc 579 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
155 simplrr 768 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘𝑘))
1565adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
157156, 36syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (0...𝐽))
158129imdistani 564 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
15942adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶𝑂)
160 elfznn 12691 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ ℕ)
161160adantl 475 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (𝑘 + 1) ∈ ℕ)
16237, 38, 39, 40, 41, 159, 161ballotlemfp1 31156 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1)) ∧ ((𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))))
163162simprd 491 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1)))
164163imp 397 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
165158, 164sylan 575 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
166 elfzelz 12663 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
167166zcnd 11839 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
168 1cnd 10373 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
169167, 168pncand 10737 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
170169fveq2d 6452 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → ((𝐹𝐶)‘((𝑘 + 1) − 1)) = ((𝐹𝐶)‘𝑘))
171170oveq1d 6939 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹𝐶)‘𝑘) + 1))
172171eqeq2d 2788 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)))
173156, 172syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)))
174165, 173mpbid 224 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
175 0z 11743 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
176 zleltp1 11784 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ ((𝐹𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
177175, 46, 176sylancr 581 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
178177adantr 474 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
179 breq2 4892 . . . . . . . . . . . . . . . . 17 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1) → (0 < ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
180179adantl 475 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 < ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
181178, 180bitr4d 274 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘(𝑘 + 1))))
182145, 157, 174, 181syl21anc 828 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘(𝑘 + 1))))
183155, 182mpbid 224 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 < ((𝐹𝐶)‘(𝑘 + 1)))
184144, 154, 183ltled 10526 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))
185184adantlrr 711 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))
186142, 143, 185, 137syl12anc 827 . . . . . . . . . 10 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ≤ 𝑘)
18718, 186mtand 806 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ (𝑘 + 1) ∈ 𝐶)
188162simpld 490 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1)))
189188imp 397 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
190158, 189sylan 575 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
1915adantr 474 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
192170oveq1d 6939 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹𝐶)‘𝑘) − 1))
193192eqeq2d 2788 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)))
194191, 193syl 17 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)))
195190, 194mpbid 224 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
196195adantlrr 711 . . . . . . . . 9 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
197187, 196mpdan 677 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
198 breq2 4892 . . . . . . . . 9 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1) → (0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
199198notbid 310 . . . . . . . 8 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1) → (¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
200197, 199syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
201141, 200mpbid 224 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1))
2025, 36syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → 𝑘 ∈ (0...𝐽))
203202, 46syldan 585 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
204203adantrr 707 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
205 zlem1lt 11785 . . . . . . . . 9 ((((𝐹𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
206175, 205mpan2 681 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
207 zre 11736 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((𝐹𝐶)‘𝑘) ∈ ℝ)
208 1red 10379 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → 1 ∈ ℝ)
209207, 208resubcld 10805 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) − 1) ∈ ℝ)
210 0red 10382 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → 0 ∈ ℝ)
211209, 210ltnled 10525 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((((𝐹𝐶)‘𝑘) − 1) < 0 ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
212206, 211bitrd 271 . . . . . . 7 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
213204, 212syl 17 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
214201, 213mpbird 249 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ≤ 0)
215 simprlr 770 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 0 ≤ ((𝐹𝐶)‘𝑘))
216204zred 11838 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
217 0red 10382 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 0 ∈ ℝ)
218216, 217letri3d 10520 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (((𝐹𝐶)‘𝑘) = 0 ↔ (((𝐹𝐶)‘𝑘) ≤ 0 ∧ 0 ≤ ((𝐹𝐶)‘𝑘))))
219214, 215, 218mpbir2and 703 . . . 4 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
2204, 219sylan2b 587 . . 3 ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
221 ssrab2 3908 . . . . . 6 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ (1...𝐽)
222221, 11sstri 3830 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ
223222a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ)
224 fzfi 13094 . . . . . 6 (1...𝐽) ∈ Fin
225 ssfi 8470 . . . . . 6 (((1...𝐽) ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ (1...𝐽)) → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin)
226224, 221, 225mp2an 682 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin
227226a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin)
228 rabn0 4188 . . . . 5 ({𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅ ↔ ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
22959, 228sylibr 226 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅)
230 fimaxre 11324 . . . 4 (({𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ ∧ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅) → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)
231223, 227, 229, 230syl3anc 1439 . . 3 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)
232220, 231reximddv 3199 . 2 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ((𝐹𝐶)‘𝑘) = 0)
233 elrabi 3567 . . . 4 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} → 𝑘 ∈ (1...𝐽))
234233anim1i 608 . . 3 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ((𝐹𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) = 0))
235234reximi2 3191 . 2 (∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ((𝐹𝐶)‘𝑘) = 0 → ∃𝑘 ∈ (1...𝐽)((𝐹𝐶)‘𝑘) = 0)
236232, 235syl 17 1 (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹𝐶)‘𝑘) = 0)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∃wrex 3091  {crab 3094   ∖ cdif 3789   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  𝒫 cpw 4379  {csn 4398   class class class wbr 4888   ↦ cmpt 4967  ‘cfv 6137  (class class class)co 6924  Fincfn 8243  ℝcr 10273  0cc0 10274  1c1 10275   + caddc 10277   < clt 10413   ≤ cle 10414   − cmin 10608   / cdiv 11034  ℕcn 11378  2c2 11434  ℤcz 11732  ℤ≥cuz 11996  ...cfz 12647  ♯chash 13439 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-2 11442  df-n0 11647  df-z 11733  df-uz 11997  df-fz 12648  df-hash 13440 This theorem is referenced by:  ballotlem1c  31172
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