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Theorem ballotlemfc0 31875
 Description: 𝐹 takes value 0 between negative and positive values. (Contributed by Thierry Arnoux, 24-Nov-2016.)
Hypotheses
Ref Expression
ballotth.m 𝑀 ∈ ℕ
ballotth.n 𝑁 ∈ ℕ
ballotth.o 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotlemfp1.c (𝜑𝐶𝑂)
ballotlemfp1.j (𝜑𝐽 ∈ ℕ)
ballotlemfc0.3 (𝜑 → ∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0)
ballotlemfc0.4 (𝜑 → 0 < ((𝐹𝐶)‘𝐽))
Assertion
Ref Expression
ballotlemfc0 (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 ballotlemfc0
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6646 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝑘))
21breq1d 5041 . . . . . 6 (𝑖 = 𝑘 → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝑘) ≤ 0))
32elrab 3628 . . . . 5 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ↔ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0))
43anbi1i 626 . . . 4 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘) ↔ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘))
5 simprlr 779 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ≤ 0)
6 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → 𝑘 ∈ (1...𝐽))
76adantrr 716 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 𝑘 ∈ (1...𝐽))
8 fzssuz 12946 . . . . . . . . . . . . . 14 (1...𝐽) ⊆ (ℤ‘1)
9 uzssz 12255 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
108, 9sstri 3924 . . . . . . . . . . . . 13 (1...𝐽) ⊆ ℤ
11 zssre 11979 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
1210, 11sstri 3924 . . . . . . . . . . . 12 (1...𝐽) ⊆ ℝ
1312sseli 3911 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
1413ltp1d 11562 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
15 1red 10634 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → 1 ∈ ℝ)
1613, 15readdcld 10662 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℝ)
1713, 16ltnled 10779 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
1814, 17mpbid 235 . . . . . . . . 9 (𝑘 ∈ (1...𝐽) → ¬ (𝑘 + 1) ≤ 𝑘)
197, 18syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ¬ (𝑘 + 1) ≤ 𝑘)
20 simprr 772 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)
21 ballotlemfc0.4 . . . . . . . . . . . . . . . 16 (𝜑 → 0 < ((𝐹𝐶)‘𝐽))
2221adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → 0 < ((𝐹𝐶)‘𝐽))
23 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 𝐽) → 𝑘 = 𝐽)
2423fveq2d 6650 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝑘) = ((𝐹𝐶)‘𝐽))
2524breq2d 5043 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘𝐽)))
26 ballotlemfp1.j . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽 ∈ ℕ)
27 elnnuz 12273 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ‘1))
2826, 27sylib 221 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (ℤ‘1))
29 eluzfz2 12913 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (ℤ‘1) → 𝐽 ∈ (1...𝐽))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ (1...𝐽))
31 eleq1 2877 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
3230, 31syl5ibrcom 250 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 = 𝐽𝑘 ∈ (1...𝐽)))
3332anc2li 559 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = 𝐽 → (𝜑𝑘 ∈ (1...𝐽))))
34 1eluzge0 12283 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (ℤ‘0)
35 fzss1 12944 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ (ℤ‘0) → (1...𝐽) ⊆ (0...𝐽))
3635sseld 3914 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ (ℤ‘0) → (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽)))
3734, 36ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽))
38 0red 10636 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → 0 ∈ ℝ)
39 ballotth.m . . . . . . . . . . . . . . . . . . . . . 22 𝑀 ∈ ℕ
40 ballotth.n . . . . . . . . . . . . . . . . . . . . . 22 𝑁 ∈ ℕ
41 ballotth.o . . . . . . . . . . . . . . . . . . . . . 22 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
42 ballotth.p . . . . . . . . . . . . . . . . . . . . . 22 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
43 ballotth.f . . . . . . . . . . . . . . . . . . . . . 22 𝐹 = (𝑐𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
44 ballotlemfp1.c . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐶𝑂)
4544adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝐶𝑂)
46 elfzelz 12905 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
4746adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
4839, 40, 41, 42, 43, 45, 47ballotlemfelz 31873 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
4948zred 12078 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
5038, 49ltnled 10779 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽)) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5137, 50sylan2 595 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝐽)) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5233, 51syl6 35 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 = 𝐽 → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0)))
5352imp 410 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5425, 53bitr3d 284 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5522, 54mpbid 235 . . . . . . . . . . . . . 14 ((𝜑𝑘 = 𝐽) → ¬ ((𝐹𝐶)‘𝑘) ≤ 0)
5655ex 416 . . . . . . . . . . . . 13 (𝜑 → (𝑘 = 𝐽 → ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5756con2d 136 . . . . . . . . . . . 12 (𝜑 → (((𝐹𝐶)‘𝑘) ≤ 0 → ¬ 𝑘 = 𝐽))
58 nn1m1nn 11649 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ ℕ → (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
5926, 58syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
60 ballotlemfc0.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0)
6160adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → ∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0)
62 oveq1 7143 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
6362adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
6426nnzd 12077 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐽 ∈ ℤ)
65 fzsn 12947 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
6664, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐽...𝐽) = {𝐽})
6766adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = {𝐽})
6863, 67eqtr3d 2835 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝐽 = 1) → (1...𝐽) = {𝐽})
6968rexeqdv 3365 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → (∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0))
7061, 69mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0)
71 fveq2 6646 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑖 = 𝐽 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝐽))
7271breq1d 5041 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝐽 → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝐽) ≤ 0))
7372rexsng 4574 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐽 ∈ ℕ → (∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝐽) ≤ 0))
7426, 73syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝐽) ≤ 0))
7574adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝐽) ≤ 0))
7670, 75mpbid 235 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ((𝐹𝐶)‘𝐽) ≤ 0)
7721adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → 0 < ((𝐹𝐶)‘𝐽))
78 0red 10636 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 0 ∈ ℝ)
7939, 40, 41, 42, 43, 44, 64ballotlemfelz 31873 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℤ)
8079zred 12078 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℝ)
8178, 80ltnled 10779 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝐽) ≤ 0))
8281adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝐽) ≤ 0))
8377, 82mpbid 235 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ¬ ((𝐹𝐶)‘𝐽) ≤ 0)
8476, 83pm2.65da 816 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝐽 = 1)
85 biortn 935 . . . . . . . . . . . . . . . . . . . . . 22 𝐽 = 1 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
8684, 85syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
87 notnotb 318 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1)
8887orbi1i 911 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
8986, 88bitr4di 292 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
9059, 89mpbird 260 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℕ)
91 elnnuz 12273 . . . . . . . . . . . . . . . . . . 19 ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ‘1))
9290, 91sylib 221 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐽 − 1) ∈ (ℤ‘1))
93 elfzp1 12955 . . . . . . . . . . . . . . . . . 18 ((𝐽 − 1) ∈ (ℤ‘1) → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
9492, 93syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1))))
9526nncnd 11644 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ ℂ)
96 1cnd 10628 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℂ)
9795, 96npcand 10993 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
9897oveq2d 7152 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
9998eleq2d 2875 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
10097eqeq2d 2809 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
101100orbi2d 913 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
10294, 99, 1013bitr3d 312 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
103 orcom 867 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1))))
104102, 103syl6bb 290 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
105104biimpd 232 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
106 pm5.6 999 . . . . . . . . . . . . . 14 (((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))) ↔ (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
107105, 106sylibr 237 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
10890nnzd 12077 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℤ)
109 1z 12003 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
110108, 109jctil 523 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
111 elfzelz 12905 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
112111, 109jctir 524 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
113 fzaddel 12939 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
114110, 112, 113syl2an 598 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
115114biimp3a 1466 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
1161153anidm23 1418 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
117 1p1e2 11753 . . . . . . . . . . . . . . . . . . . 20 (1 + 1) = 2
118117a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1 + 1) = 2)
119118, 97oveq12d 7154 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
120119eleq2d 2875 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
121 2eluzge1 12285 . . . . . . . . . . . . . . . . . . 19 2 ∈ (ℤ‘1)
122 fzss1 12944 . . . . . . . . . . . . . . . . . . 19 (2 ∈ (ℤ‘1) → (2...𝐽) ⊆ (1...𝐽))
123121, 122ax-mp 5 . . . . . . . . . . . . . . . . . 18 (2...𝐽) ⊆ (1...𝐽)
124123sseli 3911 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
125120, 124syl6bi 256 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
126125adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
127116, 126mpd 15 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ (1...𝐽))
128127ex 416 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 + 1) ∈ (1...𝐽)))
129107, 128syld 47 . . . . . . . . . . . 12 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → (𝑘 + 1) ∈ (1...𝐽)))
13057, 129sylan2d 607 . . . . . . . . . . 11 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) → (𝑘 + 1) ∈ (1...𝐽)))
131130imp 410 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → (𝑘 + 1) ∈ (1...𝐽))
132131adantrr 716 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
133 fveq2 6646 . . . . . . . . . . . . . 14 (𝑖 = (𝑘 + 1) → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘(𝑘 + 1)))
134133breq1d 5041 . . . . . . . . . . . . 13 (𝑖 = (𝑘 + 1) → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
135134elrab 3628 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
136 breq1 5034 . . . . . . . . . . . . 13 (𝑗 = (𝑘 + 1) → (𝑗𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
137136rspccva 3570 . . . . . . . . . . . 12 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}) → (𝑘 + 1) ≤ 𝑘)
138135, 137sylan2br 597 . . . . . . . . . . 11 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0)) → (𝑘 + 1) ≤ 𝑘)
139138expr 460 . . . . . . . . . 10 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (((𝐹𝐶)‘(𝑘 + 1)) ≤ 0 → (𝑘 + 1) ≤ 𝑘))
140139con3d 155 . . . . . . . . 9 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
14120, 132, 140syl2anc 587 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
14219, 141mpd 15 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ¬ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0)
143 simplrr 777 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)
144132adantr 484 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
145 simpll 766 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝜑)
146131adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
14735sseld 3914 . . . . . . . . . . . . . . 15 (1 ∈ (ℤ‘0) → ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ (0...𝐽)))
14834, 146, 147mpsyl 68 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (0...𝐽))
14944adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → 𝐶𝑂)
150 elfzelz 12905 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (0...𝐽) → (𝑘 + 1) ∈ ℤ)
151150adantl 485 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → (𝑘 + 1) ∈ ℤ)
15239, 40, 41, 42, 43, 149, 151ballotlemfelz 31873 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℤ)
153152zred 12078 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
154145, 148, 153syl2anc 587 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
155 0red 10636 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
156 simplrr 777 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘𝑘) ≤ 0)
1576adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
158157, 37syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (0...𝐽))
159130imdistani 572 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
16044adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶𝑂)
161 elfznn 12934 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ ℕ)
162161adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (𝑘 + 1) ∈ ℕ)
16339, 40, 41, 42, 43, 160, 162ballotlemfp1 31874 . . . . . . . . . . . . . . . . . . 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))))
164163simpld 498 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1)))
165164imp 410 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
166159, 165sylan 583 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
167 elfzelz 12905 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
168167zcnd 12079 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
169 1cnd 10628 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
170168, 169pncand 10990 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
171170fveq2d 6650 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → ((𝐹𝐶)‘((𝑘 + 1) − 1)) = ((𝐹𝐶)‘𝑘))
172171oveq1d 7151 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹𝐶)‘𝑘) − 1))
173172eqeq2d 2809 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)))
174157, 173syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)))
175166, 174mpbid 235 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
176 0z 11983 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
177 zlem1lt 12025 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
17848, 176, 177sylancl 589 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
179178adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
180 breq1 5034 . . . . . . . . . . . . . . . . 17 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1) → (((𝐹𝐶)‘(𝑘 + 1)) < 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
181180adantl 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)) → (((𝐹𝐶)‘(𝑘 + 1)) < 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
182179, 181bitr4d 285 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) < 0))
183145, 158, 175, 182syl21anc 836 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) < 0))
184156, 183mpbid 235 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) < 0)
185154, 155, 184ltled 10780 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0)
186185adantlrr 720 . . . . . . . . . . 11 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0)
187143, 144, 186, 138syl12anc 835 . . . . . . . . . 10 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ≤ 𝑘)
18819adantr 484 . . . . . . . . . 10 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ¬ (𝑘 + 1) ≤ 𝑘)
189187, 188condan 817 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (𝑘 + 1) ∈ 𝐶)
190163simprd 499 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1)))
191190imp 410 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
192159, 191sylan 583 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
1936adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
194171oveq1d 7151 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹𝐶)‘𝑘) + 1))
195194eqeq2d 2809 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)))
196193, 195syl 17 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) ↔ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1)))
197192, 196mpbid 235 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
198197adantlrr 720 . . . . . . . . 9 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
199189, 198mpdan 686 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
200 breq1 5034 . . . . . . . . 9 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1) → (((𝐹𝐶)‘(𝑘 + 1)) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
201200notbid 321 . . . . . . . 8 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1) → (¬ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0 ↔ ¬ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
202199, 201syl 17 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (¬ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0 ↔ ¬ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
203142, 202mpbid 235 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ¬ (((𝐹𝐶)‘𝑘) + 1) ≤ 0)
2046, 37syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → 𝑘 ∈ (0...𝐽))
205204, 48syldan 594 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
206205adantrr 716 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
207 zleltp1 12024 . . . . . . . . 9 ((0 ∈ ℤ ∧ ((𝐹𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
208176, 207mpan 689 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
209 0red 10636 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → 0 ∈ ℝ)
210 zre 11976 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((𝐹𝐶)‘𝑘) ∈ ℝ)
211 1red 10634 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → 1 ∈ ℝ)
212210, 211readdcld 10662 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) + 1) ∈ ℝ)
213209, 212ltnled 10779 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (0 < (((𝐹𝐶)‘𝑘) + 1) ↔ ¬ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
214208, 213bitrd 282 . . . . . . 7 (((𝐹𝐶)‘𝑘) ∈ ℤ → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ ¬ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
215206, 214syl 17 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ ¬ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
216203, 215mpbird 260 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 0 ≤ ((𝐹𝐶)‘𝑘))
217206zred 12078 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
218 0red 10636 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 0 ∈ ℝ)
219217, 218letri3d 10774 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (((𝐹𝐶)‘𝑘) = 0 ↔ (((𝐹𝐶)‘𝑘) ≤ 0 ∧ 0 ≤ ((𝐹𝐶)‘𝑘))))
2205, 216, 219mpbir2and 712 . . . 4 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
2214, 220sylan2b 596 . . 3 ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
222 ssrab2 4007 . . . . . 6 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ (1...𝐽)
223222, 12sstri 3924 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ ℝ
224223a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ ℝ)
225 fzfi 13338 . . . . . 6 (1...𝐽) ∈ Fin
226 ssfi 8725 . . . . . 6 (((1...𝐽) ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ (1...𝐽)) → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin)
227225, 222, 226mp2an 691 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin
228227a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin)
229 rabn0 4293 . . . . 5 ({𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ≠ ∅ ↔ ∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0)
23060, 229sylibr 237 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ≠ ∅)
231 fimaxre 11576 . . . 4 (({𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ ℝ ∧ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ≠ ∅) → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)
232224, 228, 230, 231syl3anc 1368 . . 3 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)
233221, 232reximddv 3234 . 2 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ((𝐹𝐶)‘𝑘) = 0)
234 elrabi 3623 . . . 4 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} → 𝑘 ∈ (1...𝐽))
235234anim1i 617 . . 3 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∧ ((𝐹𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) = 0))
236235reximi2 3207 . 2 (∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ((𝐹𝐶)‘𝑘) = 0 → ∃𝑘 ∈ (1...𝐽)((𝐹𝐶)‘𝑘) = 0)
237233, 236syl 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 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {csn 4525   class class class wbr 5031   ↦ cmpt 5111  ‘cfv 6325  (class class class)co 7136  Fincfn 8495  ℝcr 10528  0cc0 10529  1c1 10530   + caddc 10532   < clt 10667   ≤ cle 10668   − cmin 10862   / cdiv 11289  ℕcn 11628  2c2 11683  ℤcz 11972  ℤ≥cuz 12234  ...cfz 12888  ♯chash 13689 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-dju 9317  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-hash 13690 This theorem is referenced by:  ballotlem5  31882  ballotlemic  31889
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