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

Theorem ballotlemfcc 30540
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 6189 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝑘))
21breq2d 4663 . . . . . 6 (𝑖 = 𝑘 → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝑘)))
32elrab 3361 . . . . 5 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ↔ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)))
43anbi1i 731 . . . 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 794 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → 𝑘 ∈ (1...𝐽))
65adantrr 753 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 𝑘 ∈ (1...𝐽))
7 fzssuz 12379 . . . . . . . . . . . . . 14 (1...𝐽) ⊆ (ℤ‘1)
8 uzssz 11704 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
97, 8sstri 3610 . . . . . . . . . . . . 13 (1...𝐽) ⊆ ℤ
10 zssre 11381 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
119, 10sstri 3610 . . . . . . . . . . . 12 (1...𝐽) ⊆ ℝ
1211sseli 3597 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
1312ltp1d 10951 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
14 1red 10052 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → 1 ∈ ℝ)
1512, 14readdcld 10066 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℝ)
1612, 15ltnled 10181 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
1713, 16mpbid 222 . . . . . . . . 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 796 . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝐽) < 0)
22 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 𝐽) → 𝑘 = 𝐽)
2322fveq2d 6193 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝑘) = ((𝐹𝐶)‘𝐽))
2423breq1d 4661 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝑘) < 0 ↔ ((𝐹𝐶)‘𝐽) < 0))
25 ballotlemfcc.j . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽 ∈ ℕ)
26 elnnuz 11721 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ‘1))
2725, 26sylib 208 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (ℤ‘1))
28 eluzfz2 12346 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (ℤ‘1) → 𝐽 ∈ (1...𝐽))
2927, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ (1...𝐽))
30 eleq1 2688 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
3129, 30syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 = 𝐽𝑘 ∈ (1...𝐽)))
3231anc2li 580 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = 𝐽 → (𝜑𝑘 ∈ (1...𝐽))))
33 1eluzge0 11729 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (ℤ‘0)
34 fzss1 12377 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ (ℤ‘0) → (1...𝐽) ⊆ (0...𝐽))
3534sseld 3600 . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝐶𝑂)
44 elfzelz 12339 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
4544adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
4637, 38, 39, 40, 41, 43, 45ballotlemfelz 30537 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
4746zred 11479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
48 0red 10038 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → 0 ∈ ℝ)
4947, 48ltnled 10181 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽)) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5036, 49sylan2 491 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝐽)) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5132, 50syl6 35 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 = 𝐽 → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘))))
5251imp 445 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5324, 52bitr3d 270 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5421, 53mpbid 222 . . . . . . . . . . . . . 14 ((𝜑𝑘 = 𝐽) → ¬ 0 ≤ ((𝐹𝐶)‘𝑘))
5554ex 450 . . . . . . . . . . . . 13 (𝜑 → (𝑘 = 𝐽 → ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5655con2d 129 . . . . . . . . . . . 12 (𝜑 → (0 ≤ ((𝐹𝐶)‘𝑘) → ¬ 𝑘 = 𝐽))
57 nn1m1nn 11037 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
61 oveq1 6654 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
6261adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
6325nnzd 11478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐽 ∈ ℤ)
64 fzsn 12380 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐽...𝐽) = {𝐽})
6665adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = {𝐽})
6762, 66eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝐽 = 1) → (1...𝐽) = {𝐽})
6867rexeqdv 3143 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → (∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖) ↔ ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖)))
6960, 68mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖))
70 fveq2 6189 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑖 = 𝐽 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝐽))
7170breq2d 4663 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝐽 → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7271rexsng 4217 . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7569, 74mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → 0 ≤ ((𝐹𝐶)‘𝐽))
7620adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ((𝐹𝐶)‘𝐽) < 0)
7737, 38, 39, 40, 41, 42, 63ballotlemfelz 30537 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℤ)
7877zred 11479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℝ)
79 0red 10038 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 0 ∈ ℝ)
8078, 79ltnled 10181 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝐽)))
8180adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝐽)))
8276, 81mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ¬ 0 ≤ ((𝐹𝐶)‘𝐽))
8375, 82pm2.65da 600 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝐽 = 1)
84 biortn 421 . . . . . . . . . . . . . . . . . . . . . 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 304 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1)
8786orbi1i 542 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
8885, 87syl6bbr 278 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
8958, 88mpbird 247 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℕ)
90 elnnuz 11721 . . . . . . . . . . . . . . . . . . 19 ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ‘1))
9189, 90sylib 208 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐽 − 1) ∈ (ℤ‘1))
92 elfzp1 12388 . . . . . . . . . . . . . . . . . 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 11033 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ ℂ)
95 1cnd 10053 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℂ)
9694, 95npcand 10393 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
9796oveq2d 6663 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
9897eleq2d 2686 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
9996eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
10099orbi2d 738 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
10193, 98, 1003bitr3d 298 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
102 orcom 402 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1))))
103101, 102syl6bb 276 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
104103biimpd 219 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
105 pm5.6 951 . . . . . . . . . . . . . 14 (((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))) ↔ (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
106104, 105sylibr 224 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
10789nnzd 11478 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℤ)
108 1z 11404 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
109107, 108jctil 560 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
110 elfzelz 12339 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
111110, 108jctir 561 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
112 fzaddel 12372 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
113109, 111, 112syl2an 494 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
114113biimp3a 1431 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
1151143anidm23 1384 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
116 1p1e2 11131 . . . . . . . . . . . . . . . . . . . 20 (1 + 1) = 2
117116a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1 + 1) = 2)
118117, 96oveq12d 6665 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
119118eleq2d 2686 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
120 2eluzge1 11731 . . . . . . . . . . . . . . . . . . 19 2 ∈ (ℤ‘1)
121 fzss1 12377 . . . . . . . . . . . . . . . . . . 19 (2 ∈ (ℤ‘1) → (2...𝐽) ⊆ (1...𝐽))
122120, 121ax-mp 5 . . . . . . . . . . . . . . . . . 18 (2...𝐽) ⊆ (1...𝐽)
123122sseli 3597 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
124119, 123syl6bi 243 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
125124adantr 481 . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . 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 499 . . . . . . . . . . 11 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) → (𝑘 + 1) ∈ (1...𝐽)))
130129imp 445 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → (𝑘 + 1) ∈ (1...𝐽))
131130adantrr 753 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
132 fveq2 6189 . . . . . . . . . . . . . 14 (𝑖 = (𝑘 + 1) → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘(𝑘 + 1)))
133132breq2d 4663 . . . . . . . . . . . . 13 (𝑖 = (𝑘 + 1) → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
134133elrab 3361 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
135 breq1 4654 . . . . . . . . . . . . 13 (𝑗 = (𝑘 + 1) → (𝑗𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
136135rspccva 3306 . . . . . . . . . . . 12 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}) → (𝑘 + 1) ≤ 𝑘)
137134, 136sylan2br 493 . . . . . . . . . . 11 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))) → (𝑘 + 1) ≤ 𝑘)
138137expr 643 . . . . . . . . . 10 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) → (𝑘 + 1) ≤ 𝑘))
139138con3d 148 . . . . . . . . 9 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
14019, 131, 139syl2anc 693 . . . . . . . 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 801 . . . . . . . . . . 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 481 . . . . . . . . . . 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 10038 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
145 simpll 790 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝜑)
146130adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
14734sseld 3600 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → 𝐶𝑂)
150 elfzelz 12339 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (0...𝐽) → (𝑘 + 1) ∈ ℤ)
151150adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → (𝑘 + 1) ∈ ℤ)
15237, 38, 39, 40, 41, 149, 151ballotlemfelz 30537 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℤ)
153152zred 11479 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
154145, 148, 153syl2anc 693 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
155 simplrr 801 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘𝑘))
1565adantr 481 . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
15942adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶𝑂)
160 elfznn 12367 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ ℕ)
161160adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (𝑘 + 1) ∈ ℕ)
16237, 38, 39, 40, 41, 159, 161ballotlemfp1 30538 . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1)))
164163imp 445 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
165158, 164sylan 488 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
166 elfzelz 12339 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
167166zcnd 11480 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
168 1cnd 10053 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
169167, 168pncand 10390 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
170169fveq2d 6193 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → ((𝐹𝐶)‘((𝑘 + 1) − 1)) = ((𝐹𝐶)‘𝑘))
171170oveq1d 6662 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹𝐶)‘𝑘) + 1))
172171eqeq2d 2631 . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
175 0z 11385 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
176 zleltp1 11425 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ ((𝐹𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
177175, 46, 176sylancr 695 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
178177adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
179 breq2 4655 . . . . . . . . . . . . . . . . 17 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1) → (0 < ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
180179adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 < ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
181178, 180bitr4d 271 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘(𝑘 + 1))))
182145, 157, 174, 181syl21anc 1324 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘(𝑘 + 1))))
183155, 182mpbid 222 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 < ((𝐹𝐶)‘(𝑘 + 1)))
184144, 154, 183ltled 10182 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))
185184adantlrr 757 . . . . . . . . . . 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 1323 . . . . . . . . . 10 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ≤ 𝑘)
18718, 186mtand 691 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ (𝑘 + 1) ∈ 𝐶)
188162simpld 475 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1)))
189188imp 445 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
190158, 189sylan 488 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
1915adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
192170oveq1d 6662 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹𝐶)‘𝑘) − 1))
193192eqeq2d 2631 . . . . . . . . . . . 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 222 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
196195adantlrr 757 . . . . . . . . 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 702 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
198 breq2 4655 . . . . . . . . 9 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1) → (0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
199198notbid 308 . . . . . . . 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 222 . . . . . 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 487 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
204203adantrr 753 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
205 zlem1lt 11426 . . . . . . . . 9 ((((𝐹𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
206175, 205mpan2 707 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
207 zre 11378 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((𝐹𝐶)‘𝑘) ∈ ℝ)
208 1red 10052 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → 1 ∈ ℝ)
209207, 208resubcld 10455 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) − 1) ∈ ℝ)
210 0red 10038 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → 0 ∈ ℝ)
211209, 210ltnled 10181 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((((𝐹𝐶)‘𝑘) − 1) < 0 ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
212206, 211bitrd 268 . . . . . . 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 247 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ≤ 0)
215 simprlr 803 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 0 ≤ ((𝐹𝐶)‘𝑘))
216204zred 11479 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
217 0red 10038 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 0 ∈ ℝ)
218216, 217letri3d 10176 . . . . 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 957 . . . 4 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
2204, 219sylan2b 492 . . 3 ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
221 ssrab2 3685 . . . . . 6 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ (1...𝐽)
222221, 11sstri 3610 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ
223222a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ)
224 fzfi 12766 . . . . . 6 (1...𝐽) ∈ Fin
225 ssfi 8177 . . . . . 6 (((1...𝐽) ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ (1...𝐽)) → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin)
226224, 221, 225mp2an 708 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin
227226a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin)
228 rabn0 3956 . . . . 5 ({𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅ ↔ ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
22959, 228sylibr 224 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅)
230 fimaxre 10965 . . . 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 1325 . . 3 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)
232220, 231reximddv 3017 . 2 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ((𝐹𝐶)‘𝑘) = 0)
233 elrabi 3357 . . . 4 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} → 𝑘 ∈ (1...𝐽))
234233anim1i 592 . . 3 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ((𝐹𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) = 0))
235234reximi2 3009 . 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 196  wo 383  wa 384   = wceq 1482  wcel 1989  wne 2793  wral 2911  wrex 2912  {crab 2915  cdif 3569  cin 3571  wss 3572  c0 3913  𝒫 cpw 4156  {csn 4175   class class class wbr 4651  cmpt 4727  cfv 5886  (class class class)co 6647  Fincfn 7952  cr 9932  0cc0 9933  1c1 9934   + caddc 9936   < clt 10071  cle 10072  cmin 10263   / cdiv 10681  cn 11017  2c2 11067  cz 11374  cuz 11684  ...cfz 12323  #chash 13112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-card 8762  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-n0 11290  df-z 11375  df-uz 11685  df-fz 12324  df-hash 13113
This theorem is referenced by:  ballotlem1c  30554
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