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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlemfcc Structured version   Visualization version   GIF version

Theorem ballotlemfcc 31859
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 6649 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝑘))
21breq2d 5045 . . . . . 6 (𝑖 = 𝑘 → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝑘)))
32elrab 3631 . . . . 5 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ↔ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)))
43anbi1i 626 . . . 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 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → 𝑘 ∈ (1...𝐽))
65adantrr 716 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 𝑘 ∈ (1...𝐽))
7 fzssuz 12947 . . . . . . . . . . . . . 14 (1...𝐽) ⊆ (ℤ‘1)
8 uzssz 12256 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
97, 8sstri 3927 . . . . . . . . . . . . 13 (1...𝐽) ⊆ ℤ
10 zssre 11980 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
119, 10sstri 3927 . . . . . . . . . . . 12 (1...𝐽) ⊆ ℝ
1211sseli 3914 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
1312ltp1d 11563 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
14 1red 10635 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → 1 ∈ ℝ)
1512, 14readdcld 10663 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℝ)
1612, 15ltnled 10780 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
1713, 16mpbid 235 . . . . . . . . 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 772 . . . . . . . . 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 484 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝐽) < 0)
22 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 𝐽) → 𝑘 = 𝐽)
2322fveq2d 6653 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝑘) = ((𝐹𝐶)‘𝐽))
2423breq1d 5043 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝑘) < 0 ↔ ((𝐹𝐶)‘𝐽) < 0))
25 ballotlemfcc.j . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽 ∈ ℕ)
26 elnnuz 12274 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ‘1))
2725, 26sylib 221 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (ℤ‘1))
28 eluzfz2 12914 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (ℤ‘1) → 𝐽 ∈ (1...𝐽))
2927, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ (1...𝐽))
30 eleq1 2880 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
3129, 30syl5ibrcom 250 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 = 𝐽𝑘 ∈ (1...𝐽)))
3231anc2li 559 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = 𝐽 → (𝜑𝑘 ∈ (1...𝐽))))
33 1eluzge0 12284 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (ℤ‘0)
34 fzss1 12945 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ (ℤ‘0) → (1...𝐽) ⊆ (0...𝐽))
3534sseld 3917 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝐶𝑂)
44 elfzelz 12906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
4544adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
4637, 38, 39, 40, 41, 43, 45ballotlemfelz 31856 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
4746zred 12079 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
48 0red 10637 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → 0 ∈ ℝ)
4947, 48ltnled 10780 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽)) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5036, 49sylan2 595 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝐽)) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5132, 50syl6 35 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 = 𝐽 → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘))))
5251imp 410 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5324, 52bitr3d 284 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5421, 53mpbid 235 . . . . . . . . . . . . . 14 ((𝜑𝑘 = 𝐽) → ¬ 0 ≤ ((𝐹𝐶)‘𝑘))
5554ex 416 . . . . . . . . . . . . 13 (𝜑 → (𝑘 = 𝐽 → ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5655con2d 136 . . . . . . . . . . . 12 (𝜑 → (0 ≤ ((𝐹𝐶)‘𝑘) → ¬ 𝑘 = 𝐽))
57 nn1m1nn 11650 . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
61 oveq1 7146 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
6261adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
6325nnzd 12078 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐽 ∈ ℤ)
64 fzsn 12948 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐽...𝐽) = {𝐽})
6665adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = {𝐽})
6762, 66eqtr3d 2838 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝐽 = 1) → (1...𝐽) = {𝐽})
6867rexeqdv 3368 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → (∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖) ↔ ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖)))
6960, 68mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖))
70 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑖 = 𝐽 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝐽))
7170breq2d 5045 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝐽 → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7271rexsng 4577 . . . . . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7569, 74mpbid 235 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → 0 ≤ ((𝐹𝐶)‘𝐽))
7620adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ((𝐹𝐶)‘𝐽) < 0)
7737, 38, 39, 40, 41, 42, 63ballotlemfelz 31856 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℤ)
7877zred 12079 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℝ)
79 0red 10637 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 0 ∈ ℝ)
8078, 79ltnled 10780 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝐽)))
8180adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝐽)))
8276, 81mpbid 235 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ¬ 0 ≤ ((𝐹𝐶)‘𝐽))
8375, 82pm2.65da 816 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝐽 = 1)
84 biortn 935 . . . . . . . . . . . . . . . . . . . . . 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 318 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1)
8786orbi1i 911 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
8885, 87syl6bbr 292 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
8958, 88mpbird 260 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℕ)
90 elnnuz 12274 . . . . . . . . . . . . . . . . . . 19 ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ‘1))
9189, 90sylib 221 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐽 − 1) ∈ (ℤ‘1))
92 elfzp1 12956 . . . . . . . . . . . . . . . . . 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 11645 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ ℂ)
95 1cnd 10629 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℂ)
9694, 95npcand 10994 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
9796oveq2d 7155 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
9897eleq2d 2878 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
9996eqeq2d 2812 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
10099orbi2d 913 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
10193, 98, 1003bitr3d 312 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
102 orcom 867 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1))))
103101, 102syl6bb 290 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
104103biimpd 232 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
105 pm5.6 999 . . . . . . . . . . . . . 14 (((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))) ↔ (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
106104, 105sylibr 237 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
10789nnzd 12078 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℤ)
108 1z 12004 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
109107, 108jctil 523 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
110 elfzelz 12906 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
111110, 108jctir 524 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
112 fzaddel 12940 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
113109, 111, 112syl2an 598 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
114113biimp3a 1466 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
1151143anidm23 1418 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
116 1p1e2 11754 . . . . . . . . . . . . . . . . . . . 20 (1 + 1) = 2
117116a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1 + 1) = 2)
118117, 96oveq12d 7157 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
119118eleq2d 2878 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
120 2eluzge1 12286 . . . . . . . . . . . . . . . . . . 19 2 ∈ (ℤ‘1)
121 fzss1 12945 . . . . . . . . . . . . . . . . . . 19 (2 ∈ (ℤ‘1) → (2...𝐽) ⊆ (1...𝐽))
122120, 121ax-mp 5 . . . . . . . . . . . . . . . . . 18 (2...𝐽) ⊆ (1...𝐽)
123122sseli 3914 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
124119, 123syl6bi 256 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
125124adantr 484 . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . 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 607 . . . . . . . . . . 11 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) → (𝑘 + 1) ∈ (1...𝐽)))
130129imp 410 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → (𝑘 + 1) ∈ (1...𝐽))
131130adantrr 716 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
132 fveq2 6649 . . . . . . . . . . . . . 14 (𝑖 = (𝑘 + 1) → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘(𝑘 + 1)))
133132breq2d 5045 . . . . . . . . . . . . 13 (𝑖 = (𝑘 + 1) → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
134133elrab 3631 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
135 breq1 5036 . . . . . . . . . . . . 13 (𝑗 = (𝑘 + 1) → (𝑗𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
136135rspccva 3573 . . . . . . . . . . . 12 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}) → (𝑘 + 1) ≤ 𝑘)
137134, 136sylan2br 597 . . . . . . . . . . 11 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))) → (𝑘 + 1) ≤ 𝑘)
138137expr 460 . . . . . . . . . 10 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) → (𝑘 + 1) ≤ 𝑘))
139138con3d 155 . . . . . . . . 9 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
14019, 131, 139syl2anc 587 . . . . . . . 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 777 . . . . . . . . . . 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 484 . . . . . . . . . . 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 10637 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
145 simpll 766 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝜑)
146130adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
14734sseld 3917 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → 𝐶𝑂)
150 elfzelz 12906 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (0...𝐽) → (𝑘 + 1) ∈ ℤ)
151150adantl 485 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → (𝑘 + 1) ∈ ℤ)
15237, 38, 39, 40, 41, 149, 151ballotlemfelz 31856 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℤ)
153152zred 12079 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
154145, 148, 153syl2anc 587 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
155 simplrr 777 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘𝑘))
1565adantr 484 . . . . . . . . . . . . . . . 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 572 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
15942adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶𝑂)
160 elfznn 12935 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ ℕ)
161160adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (𝑘 + 1) ∈ ℕ)
16237, 38, 39, 40, 41, 159, 161ballotlemfp1 31857 . . . . . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1)))
164163imp 410 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
165158, 164sylan 583 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
166 elfzelz 12906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
167166zcnd 12080 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
168 1cnd 10629 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
169167, 168pncand 10991 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
170169fveq2d 6653 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → ((𝐹𝐶)‘((𝑘 + 1) − 1)) = ((𝐹𝐶)‘𝑘))
171170oveq1d 7154 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹𝐶)‘𝑘) + 1))
172171eqeq2d 2812 . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
175 0z 11984 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
176 zleltp1 12025 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ ((𝐹𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
177175, 46, 176sylancr 590 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
178177adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
179 breq2 5037 . . . . . . . . . . . . . . . . 17 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1) → (0 < ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
180179adantl 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 < ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
181178, 180bitr4d 285 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘(𝑘 + 1))))
182145, 157, 174, 181syl21anc 836 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘(𝑘 + 1))))
183155, 182mpbid 235 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 < ((𝐹𝐶)‘(𝑘 + 1)))
184144, 154, 183ltled 10781 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))
185184adantlrr 720 . . . . . . . . . . 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 835 . . . . . . . . . 10 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ≤ 𝑘)
18718, 186mtand 815 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ (𝑘 + 1) ∈ 𝐶)
188162simpld 498 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1)))
189188imp 410 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
190158, 189sylan 583 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
1915adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
192170oveq1d 7154 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹𝐶)‘𝑘) − 1))
193192eqeq2d 2812 . . . . . . . . . . . 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 235 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
196195adantlrr 720 . . . . . . . . 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 686 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
198 breq2 5037 . . . . . . . . 9 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1) → (0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
199198notbid 321 . . . . . . . 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 235 . . . . . 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 594 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
204203adantrr 716 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
205 zlem1lt 12026 . . . . . . . . 9 ((((𝐹𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
206175, 205mpan2 690 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
207 zre 11977 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((𝐹𝐶)‘𝑘) ∈ ℝ)
208 1red 10635 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → 1 ∈ ℝ)
209207, 208resubcld 11061 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) − 1) ∈ ℝ)
210 0red 10637 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → 0 ∈ ℝ)
211209, 210ltnled 10780 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((((𝐹𝐶)‘𝑘) − 1) < 0 ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
212206, 211bitrd 282 . . . . . . 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 260 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ≤ 0)
215 simprlr 779 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 0 ≤ ((𝐹𝐶)‘𝑘))
216204zred 12079 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
217 0red 10637 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 0 ∈ ℝ)
218216, 217letri3d 10775 . . . . 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 712 . . . 4 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
2204, 219sylan2b 596 . . 3 ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
221 ssrab2 4010 . . . . . 6 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ (1...𝐽)
222221, 11sstri 3927 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ
223222a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ)
224 fzfi 13339 . . . . . 6 (1...𝐽) ∈ Fin
225 ssfi 8726 . . . . . 6 (((1...𝐽) ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ (1...𝐽)) → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin)
226224, 221, 225mp2an 691 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin
227226a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin)
228 rabn0 4296 . . . . 5 ({𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅ ↔ ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
22959, 228sylibr 237 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅)
230 fimaxre 11577 . . . 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 1368 . . 3 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)
232220, 231reximddv 3237 . 2 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ((𝐹𝐶)‘𝑘) = 0)
233 elrabi 3626 . . . 4 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} → 𝑘 ∈ (1...𝐽))
234233anim1i 617 . . 3 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ((𝐹𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) = 0))
235234reximi2 3210 . 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 209  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {crab 3113  cdif 3881  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500  {csn 4528   class class class wbr 5033  cmpt 5113  cfv 6328  (class class class)co 7139  Fincfn 8496  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   < clt 10668  cle 10669  cmin 10863   / cdiv 11290  cn 11629  2c2 11684  cz 11973  cuz 12235  ...cfz 12889  chash 13690
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-hash 13691
This theorem is referenced by:  ballotlem1c  31873
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