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Theorem ballotlemfcc 34474
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 6906 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝑘))
21breq2d 5159 . . . . . 6 (𝑖 = 𝑘 → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝑘)))
32elrab 3694 . . . . 5 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ↔ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)))
43anbi1i 624 . . . 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 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → 𝑘 ∈ (1...𝐽))
65adantrr 717 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 𝑘 ∈ (1...𝐽))
7 fzssuz 13601 . . . . . . . . . . . . . 14 (1...𝐽) ⊆ (ℤ‘1)
8 uzssz 12896 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
97, 8sstri 4004 . . . . . . . . . . . . 13 (1...𝐽) ⊆ ℤ
10 zssre 12617 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
119, 10sstri 4004 . . . . . . . . . . . 12 (1...𝐽) ⊆ ℝ
1211sseli 3990 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
1312ltp1d 12195 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
14 1red 11259 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → 1 ∈ ℝ)
1512, 14readdcld 11287 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℝ)
1612, 15ltnled 11405 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
1713, 16mpbid 232 . . . . . . . . 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 773 . . . . . . . . 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 480 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝐽) < 0)
22 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 𝐽) → 𝑘 = 𝐽)
2322fveq2d 6910 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝑘) = ((𝐹𝐶)‘𝐽))
2423breq1d 5157 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝑘) < 0 ↔ ((𝐹𝐶)‘𝐽) < 0))
25 ballotlemfcc.j . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽 ∈ ℕ)
26 elnnuz 12919 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ‘1))
2725, 26sylib 218 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (ℤ‘1))
28 eluzfz2 13568 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (ℤ‘1) → 𝐽 ∈ (1...𝐽))
2927, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ (1...𝐽))
30 eleq1 2826 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
3129, 30syl5ibrcom 247 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 = 𝐽𝑘 ∈ (1...𝐽)))
3231anc2li 555 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = 𝐽 → (𝜑𝑘 ∈ (1...𝐽))))
33 1eluzge0 12931 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (ℤ‘0)
34 fzss1 13599 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ (ℤ‘0) → (1...𝐽) ⊆ (0...𝐽))
3534sseld 3993 . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝐶𝑂)
44 elfzelz 13560 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
4544adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
4637, 38, 39, 40, 41, 43, 45ballotlemfelz 34471 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
4746zred 12719 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
48 0red 11261 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → 0 ∈ ℝ)
4947, 48ltnled 11405 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽)) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5036, 49sylan2 593 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝐽)) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5132, 50syl6 35 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 = 𝐽 → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘))))
5251imp 406 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝑘) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5324, 52bitr3d 281 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5421, 53mpbid 232 . . . . . . . . . . . . . 14 ((𝜑𝑘 = 𝐽) → ¬ 0 ≤ ((𝐹𝐶)‘𝑘))
5554ex 412 . . . . . . . . . . . . 13 (𝜑 → (𝑘 = 𝐽 → ¬ 0 ≤ ((𝐹𝐶)‘𝑘)))
5655con2d 134 . . . . . . . . . . . 12 (𝜑 → (0 ≤ ((𝐹𝐶)‘𝑘) → ¬ 𝑘 = 𝐽))
57 nn1m1nn 12284 . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
61 oveq1 7437 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
6261adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
6325nnzd 12637 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐽 ∈ ℤ)
64 fzsn 13602 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝐽...𝐽) = {𝐽})
6665adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝐽 = 1) → (𝐽...𝐽) = {𝐽})
6762, 66eqtr3d 2776 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → (1...𝐽) = {𝐽})
6860, 67rexeqtrdv 3326 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖))
69 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑖 = 𝐽 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝐽))
7069breq2d 5159 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝐽 → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7170rexsng 4680 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐽 ∈ ℕ → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7225, 71syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7372adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (∃𝑖 ∈ {𝐽}0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘𝐽)))
7468, 73mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → 0 ≤ ((𝐹𝐶)‘𝐽))
7520adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ((𝐹𝐶)‘𝐽) < 0)
7637, 38, 39, 40, 41, 42, 63ballotlemfelz 34471 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℤ)
7776zred 12719 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℝ)
78 0red 11261 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 0 ∈ ℝ)
7977, 78ltnled 11405 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝐽)))
8079adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (((𝐹𝐶)‘𝐽) < 0 ↔ ¬ 0 ≤ ((𝐹𝐶)‘𝐽)))
8175, 80mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ¬ 0 ≤ ((𝐹𝐶)‘𝐽))
8274, 81pm2.65da 817 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝐽 = 1)
83 biortn 937 . . . . . . . . . . . . . . . . . . . . . 22 𝐽 = 1 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
85 notnotb 315 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1)
8685orbi1i 913 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
8784, 86bitr4di 289 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
8858, 87mpbird 257 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℕ)
89 elnnuz 12919 . . . . . . . . . . . . . . . . . . 19 ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ‘1))
9088, 89sylib 218 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐽 − 1) ∈ (ℤ‘1))
91 elfzp1 13610 . . . . . . . . . . . . . . . . . 18 ((𝐽 − 1) ∈ (ℤ‘1) → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
9290, 91syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
9325nncnd 12279 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ ℂ)
94 1cnd 11253 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℂ)
9593, 94npcand 11621 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
9695oveq2d 7446 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
9796eleq2d 2824 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
9895eqeq2d 2745 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
9998orbi2d 915 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
10092, 97, 993bitr3d 309 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
101 orcom 870 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1))))
102100, 101bitrdi 287 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
103102biimpd 229 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
104 pm5.6 1003 . . . . . . . . . . . . . 14 (((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))) ↔ (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
105103, 104sylibr 234 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
10688nnzd 12637 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℤ)
107 1z 12644 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
108106, 107jctil 519 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
109 elfzelz 13560 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
110109, 107jctir 520 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
111 fzaddel 13594 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
112108, 110, 111syl2an 596 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
113112biimp3a 1468 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
1141133anidm23 1420 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
115 1p1e2 12388 . . . . . . . . . . . . . . . . . . . 20 (1 + 1) = 2
116115a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1 + 1) = 2)
117116, 95oveq12d 7448 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
118117eleq2d 2824 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
119 2eluzge1 12933 . . . . . . . . . . . . . . . . . . 19 2 ∈ (ℤ‘1)
120 fzss1 13599 . . . . . . . . . . . . . . . . . . 19 (2 ∈ (ℤ‘1) → (2...𝐽) ⊆ (1...𝐽))
121119, 120ax-mp 5 . . . . . . . . . . . . . . . . . 18 (2...𝐽) ⊆ (1...𝐽)
122121sseli 3990 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
123118, 122biimtrdi 253 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
124123adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
125114, 124mpd 15 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ (1...𝐽))
126125ex 412 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 + 1) ∈ (1...𝐽)))
127105, 126syld 47 . . . . . . . . . . . 12 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → (𝑘 + 1) ∈ (1...𝐽)))
12856, 127sylan2d 605 . . . . . . . . . . 11 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) → (𝑘 + 1) ∈ (1...𝐽)))
129128imp 406 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → (𝑘 + 1) ∈ (1...𝐽))
130129adantrr 717 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
131 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = (𝑘 + 1) → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘(𝑘 + 1)))
132131breq2d 5159 . . . . . . . . . . . . 13 (𝑖 = (𝑘 + 1) → (0 ≤ ((𝐹𝐶)‘𝑖) ↔ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
133132elrab 3694 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
134 breq1 5150 . . . . . . . . . . . . 13 (𝑗 = (𝑘 + 1) → (𝑗𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
135134rspccva 3620 . . . . . . . . . . . 12 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}) → (𝑘 + 1) ≤ 𝑘)
136133, 135sylan2br 595 . . . . . . . . . . 11 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))) → (𝑘 + 1) ≤ 𝑘)
137136expr 456 . . . . . . . . . 10 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) → (𝑘 + 1) ≤ 𝑘))
138137con3d 152 . . . . . . . . 9 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
13919, 130, 138syl2anc 584 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1))))
14018, 139mpd 15 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))
141 simplrr 778 . . . . . . . . . . 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)
142130adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
143 0red 11261 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
144 simpll 767 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝜑)
145129adantr 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
14634sseld 3993 . . . . . . . . . . . . . . 15 (1 ∈ (ℤ‘0) → ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ (0...𝐽)))
14733, 145, 146mpsyl 68 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (0...𝐽))
14842adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → 𝐶𝑂)
149 elfzelz 13560 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (0...𝐽) → (𝑘 + 1) ∈ ℤ)
150149adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → (𝑘 + 1) ∈ ℤ)
15137, 38, 39, 40, 41, 148, 150ballotlemfelz 34471 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℤ)
152151zred 12719 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
153144, 147, 152syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
154 simplrr 778 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘𝑘))
1555adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
156155, 36syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (0...𝐽))
157128imdistani 568 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
15842adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶𝑂)
159 elfznn 13589 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ ℕ)
160159adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (𝑘 + 1) ∈ ℕ)
16137, 38, 39, 40, 41, 158, 160ballotlemfp1 34472 . . . . . . . . . . . . . . . . . . 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))))
162161simprd 495 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1)))
163162imp 406 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
164157, 163sylan 580 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
165 elfzelz 13560 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
166165zcnd 12720 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
167 1cnd 11253 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
168166, 167pncand 11618 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
169168fveq2d 6910 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → ((𝐹𝐶)‘((𝑘 + 1) − 1)) = ((𝐹𝐶)‘𝑘))
170169oveq1d 7445 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹𝐶)‘𝑘) + 1))
171170eqeq2d 2745 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)))
172155, 171syl 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)))
173164, 172mpbid 232 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
174 0z 12621 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
175 zleltp1 12665 . . . . . . . . . . . . . . . . . 18 ((0 ∈ ℤ ∧ ((𝐹𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
176174, 46, 175sylancr 587 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
177176adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
178 breq2 5151 . . . . . . . . . . . . . . . . 17 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1) → (0 < ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
179178adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 < ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
180177, 179bitr4d 282 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘(𝑘 + 1))))
181144, 156, 173, 180syl21anc 838 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘(𝑘 + 1))))
182154, 181mpbid 232 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 < ((𝐹𝐶)‘(𝑘 + 1)))
183143, 153, 182ltled 11406 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))
184183adantlrr 721 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)))
185141, 142, 184, 136syl12anc 837 . . . . . . . . . 10 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ≤ 𝑘)
18618, 185mtand 816 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ (𝑘 + 1) ∈ 𝐶)
187161simpld 494 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1)))
188187imp 406 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
189157, 188sylan 580 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
1905adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
191169oveq1d 7445 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹𝐶)‘𝑘) − 1))
192191eqeq2d 2745 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)))
193190, 192syl 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)))
194189, 193mpbid 232 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
195194adantlrr 721 . . . . . . . . 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))
196186, 195mpdan 687 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
197 breq2 5151 . . . . . . . . 9 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1) → (0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) ↔ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
198197notbid 318 . . . . . . . 8 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1) → (¬ 0 ≤ ((𝐹𝐶)‘(𝑘 + 1)) ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
199196, 198syl 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)))
200140, 199mpbid 232 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1))
2015, 36syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → 𝑘 ∈ (0...𝐽))
202201, 46syldan 591 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘))) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
203202adantrr 717 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
204 zlem1lt 12666 . . . . . . . . 9 ((((𝐹𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
205174, 204mpan2 691 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
206 zre 12614 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((𝐹𝐶)‘𝑘) ∈ ℝ)
207 1red 11259 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → 1 ∈ ℝ)
208206, 207resubcld 11688 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) − 1) ∈ ℝ)
209 0red 11261 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → 0 ∈ ℝ)
210208, 209ltnled 11405 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((((𝐹𝐶)‘𝑘) − 1) < 0 ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
211205, 210bitrd 279 . . . . . . 7 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
212203, 211syl 17 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ¬ 0 ≤ (((𝐹𝐶)‘𝑘) − 1)))
213200, 212mpbird 257 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ≤ 0)
214 simprlr 780 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 0 ≤ ((𝐹𝐶)‘𝑘))
215203zred 12719 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
216 0red 11261 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → 0 ∈ ℝ)
217215, 216letri3d 11400 . . . . 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))))
218213, 214, 217mpbir2and 713 . . . 4 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ 0 ≤ ((𝐹𝐶)‘𝑘)) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
2194, 218sylan2b 594 . . 3 ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
220 ssrab2 4089 . . . . . 6 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ (1...𝐽)
221220, 11sstri 4004 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ
222221a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ ℝ)
223 fzfi 14009 . . . . . 6 (1...𝐽) ∈ Fin
224 ssfi 9211 . . . . . 6 (((1...𝐽) ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ⊆ (1...𝐽)) → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin)
225223, 220, 224mp2an 692 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin
226225a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∈ Fin)
227 rabn0 4394 . . . . 5 ({𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅ ↔ ∃𝑖 ∈ (1...𝐽)0 ≤ ((𝐹𝐶)‘𝑖))
22859, 227sylibr 234 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ≠ ∅)
229 fimaxre 12209 . . . 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)
230222, 226, 228, 229syl3anc 1370 . . 3 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)}𝑗𝑘)
231219, 230reximddv 3168 . 2 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ((𝐹𝐶)‘𝑘) = 0)
232 elrabi 3689 . . . 4 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} → 𝑘 ∈ (1...𝐽))
233232anim1i 615 . . 3 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ∧ ((𝐹𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) = 0))
234233reximi2 3076 . 2 (∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ 0 ≤ ((𝐹𝐶)‘𝑖)} ((𝐹𝐶)‘𝑘) = 0 → ∃𝑘 ∈ (1...𝐽)((𝐹𝐶)‘𝑘) = 0)
235231, 234syl 17 1 (𝜑 → ∃𝑘 ∈ (1...𝐽)((𝐹𝐶)‘𝑘) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  {crab 3432  cdif 3959  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  cmpt 5230  cfv 6562  (class class class)co 7430  Fincfn 8983  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  cn 12263  2c2 12318  cz 12610  cuz 12875  ...cfz 13543  chash 14365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366
This theorem is referenced by:  ballotlem1c  34488
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