Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlemfc0 Structured version   Visualization version   GIF version

Theorem ballotlemfc0 31636
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 6667 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝑘))
21breq1d 5073 . . . . . 6 (𝑖 = 𝑘 → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝑘) ≤ 0))
32elrab 3684 . . . . 5 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ↔ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0))
43anbi1i 623 . . . 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 776 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ≤ 0)
6 simprl 767 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → 𝑘 ∈ (1...𝐽))
76adantrr 713 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 𝑘 ∈ (1...𝐽))
8 fzssuz 12938 . . . . . . . . . . . . . 14 (1...𝐽) ⊆ (ℤ‘1)
9 uzssz 12253 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
108, 9sstri 3980 . . . . . . . . . . . . 13 (1...𝐽) ⊆ ℤ
11 zssre 11977 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
1210, 11sstri 3980 . . . . . . . . . . . 12 (1...𝐽) ⊆ ℝ
1312sseli 3967 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
1413ltp1d 11559 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
15 1red 10631 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → 1 ∈ ℝ)
1613, 15readdcld 10659 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℝ)
1713, 16ltnled 10776 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
1814, 17mpbid 233 . . . . . . . . 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 769 . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → 0 < ((𝐹𝐶)‘𝐽))
23 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 𝐽) → 𝑘 = 𝐽)
2423fveq2d 6671 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝑘) = ((𝐹𝐶)‘𝐽))
2524breq2d 5075 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘𝐽)))
26 ballotlemfp1.j . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽 ∈ ℕ)
27 elnnuz 12271 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ‘1))
2826, 27sylib 219 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (ℤ‘1))
29 eluzfz2 12905 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (ℤ‘1) → 𝐽 ∈ (1...𝐽))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ (1...𝐽))
31 eleq1 2905 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
3230, 31syl5ibrcom 248 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 = 𝐽𝑘 ∈ (1...𝐽)))
3332anc2li 556 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = 𝐽 → (𝜑𝑘 ∈ (1...𝐽))))
34 1eluzge0 12281 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (ℤ‘0)
35 fzss1 12936 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ (ℤ‘0) → (1...𝐽) ⊆ (0...𝐽))
3635sseld 3970 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ (ℤ‘0) → (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽)))
3734, 36ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽))
38 0red 10633 . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝐶𝑂)
46 elfzelz 12898 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
4746adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
4839, 40, 41, 42, 43, 45, 47ballotlemfelz 31634 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
4948zred 12076 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
5038, 49ltnled 10776 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽)) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5137, 50sylan2 592 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝐽)) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5233, 51syl6 35 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 = 𝐽 → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0)))
5352imp 407 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5425, 53bitr3d 282 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5522, 54mpbid 233 . . . . . . . . . . . . . 14 ((𝜑𝑘 = 𝐽) → ¬ ((𝐹𝐶)‘𝑘) ≤ 0)
5655ex 413 . . . . . . . . . . . . 13 (𝜑 → (𝑘 = 𝐽 → ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5756con2d 136 . . . . . . . . . . . 12 (𝜑 → (((𝐹𝐶)‘𝑘) ≤ 0 → ¬ 𝑘 = 𝐽))
58 nn1m1nn 11647 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → ∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0)
62 oveq1 7155 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
6362adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
6426nnzd 12075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐽 ∈ ℤ)
65 fzsn 12939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
6664, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐽...𝐽) = {𝐽})
6766adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = {𝐽})
6863, 67eqtr3d 2863 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝐽 = 1) → (1...𝐽) = {𝐽})
6968rexeqdv 3422 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → (∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0))
7061, 69mpbid 233 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0)
71 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑖 = 𝐽 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝐽))
7271breq1d 5073 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝐽 → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝐽) ≤ 0))
7372rexsng 4613 . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝐽) ≤ 0))
7670, 75mpbid 233 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ((𝐹𝐶)‘𝐽) ≤ 0)
7721adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → 0 < ((𝐹𝐶)‘𝐽))
78 0red 10633 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 0 ∈ ℝ)
7939, 40, 41, 42, 43, 44, 64ballotlemfelz 31634 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℤ)
8079zred 12076 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℝ)
8178, 80ltnled 10776 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝐽) ≤ 0))
8281adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝐽) ≤ 0))
8377, 82mpbid 233 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ¬ ((𝐹𝐶)‘𝐽) ≤ 0)
8476, 83pm2.65da 813 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝐽 = 1)
85 biortn 933 . . . . . . . . . . . . . . . . . . . . . 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 316 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1)
8887orbi1i 909 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
8986, 88syl6bbr 290 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
9059, 89mpbird 258 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℕ)
91 elnnuz 12271 . . . . . . . . . . . . . . . . . . 19 ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ‘1))
9290, 91sylib 219 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐽 − 1) ∈ (ℤ‘1))
93 elfzp1 12947 . . . . . . . . . . . . . . . . . 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 11643 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ ℂ)
96 1cnd 10625 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℂ)
9795, 96npcand 10990 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
9897oveq2d 7164 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
9998eleq2d 2903 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
10097eqeq2d 2837 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
101100orbi2d 911 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
10294, 99, 1013bitr3d 310 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
103 orcom 866 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1))))
104102, 103syl6bb 288 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
105104biimpd 230 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
106 pm5.6 997 . . . . . . . . . . . . . 14 (((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))) ↔ (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
107105, 106sylibr 235 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
10890nnzd 12075 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℤ)
109 1z 12001 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
110108, 109jctil 520 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
111 elfzelz 12898 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
112111, 109jctir 521 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
113 fzaddel 12931 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
114110, 112, 113syl2an 595 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
115114biimp3a 1462 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
1161153anidm23 1415 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
117 1p1e2 11751 . . . . . . . . . . . . . . . . . . . 20 (1 + 1) = 2
118117a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1 + 1) = 2)
119118, 97oveq12d 7166 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
120119eleq2d 2903 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
121 2eluzge1 12283 . . . . . . . . . . . . . . . . . . 19 2 ∈ (ℤ‘1)
122 fzss1 12936 . . . . . . . . . . . . . . . . . . 19 (2 ∈ (ℤ‘1) → (2...𝐽) ⊆ (1...𝐽))
123121, 122ax-mp 5 . . . . . . . . . . . . . . . . . 18 (2...𝐽) ⊆ (1...𝐽)
124123sseli 3967 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
125120, 124syl6bi 254 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) → (𝑘 + 1) ∈ (1...𝐽)))
126125adantr 481 . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . 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 604 . . . . . . . . . . 11 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) → (𝑘 + 1) ∈ (1...𝐽)))
131130imp 407 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → (𝑘 + 1) ∈ (1...𝐽))
132131adantrr 713 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
133 fveq2 6667 . . . . . . . . . . . . . 14 (𝑖 = (𝑘 + 1) → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘(𝑘 + 1)))
134133breq1d 5073 . . . . . . . . . . . . 13 (𝑖 = (𝑘 + 1) → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
135134elrab 3684 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
136 breq1 5066 . . . . . . . . . . . . 13 (𝑗 = (𝑘 + 1) → (𝑗𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
137136rspccva 3626 . . . . . . . . . . . 12 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}) → (𝑘 + 1) ≤ 𝑘)
138135, 137sylan2br 594 . . . . . . . . . . 11 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0)) → (𝑘 + 1) ≤ 𝑘)
139138expr 457 . . . . . . . . . 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 584 . . . . . . . 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 774 . . . . . . . . . . 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 481 . . . . . . . . . . 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 763 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 𝜑)
146131adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ∈ (1...𝐽))
14735sseld 3970 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → 𝐶𝑂)
150 elfzelz 12898 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (0...𝐽) → (𝑘 + 1) ∈ ℤ)
151150adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → (𝑘 + 1) ∈ ℤ)
15239, 40, 41, 42, 43, 149, 151ballotlemfelz 31634 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℤ)
153152zred 12076 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
154145, 148, 153syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
155 0red 10633 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
156 simplrr 774 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘𝑘) ≤ 0)
1576adantr 481 . . . . . . . . . . . . . . . 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 569 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
16044adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶𝑂)
161 elfznn 12926 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 + 1) ∈ (1...𝐽) → (𝑘 + 1) ∈ ℕ)
162161adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (𝑘 + 1) ∈ ℕ)
16339, 40, 41, 42, 43, 160, 162ballotlemfp1 31635 . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1)))
165164imp 407 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
166159, 165sylan 580 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
167 elfzelz 12898 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
168167zcnd 12077 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
169 1cnd 10625 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
170168, 169pncand 10987 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
171170fveq2d 6671 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → ((𝐹𝐶)‘((𝑘 + 1) − 1)) = ((𝐹𝐶)‘𝑘))
172171oveq1d 7163 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹𝐶)‘𝑘) − 1))
173172eqeq2d 2837 . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
176 0z 11981 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
177 zlem1lt 12023 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
17848, 176, 177sylancl 586 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (0...𝐽)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
179178adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
180 breq1 5066 . . . . . . . . . . . . . . . . 17 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1) → (((𝐹𝐶)‘(𝑘 + 1)) < 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
181180adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)) → (((𝐹𝐶)‘(𝑘 + 1)) < 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
182179, 181bitr4d 283 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) < 0))
183145, 158, 175, 182syl21anc 835 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) < 0))
184156, 183mpbid 233 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) < 0)
185154, 155, 184ltled 10777 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0)
186185adantlrr 717 . . . . . . . . . . 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 834 . . . . . . . . . 10 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (𝑘 + 1) ≤ 𝑘)
18819adantr 481 . . . . . . . . . 10 (((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ¬ (𝑘 + 1) ≤ 𝑘)
189187, 188condan 814 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (𝑘 + 1) ∈ 𝐶)
190163simprd 496 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1)))
191190imp 407 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
192159, 191sylan 580 . . . . . . . . . . 11 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
1936adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → 𝑘 ∈ (1...𝐽))
194171oveq1d 7163 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹𝐶)‘𝑘) + 1))
195194eqeq2d 2837 . . . . . . . . . . . 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 233 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
198197adantlrr 717 . . . . . . . . 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 683 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
200 breq1 5066 . . . . . . . . 9 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1) → (((𝐹𝐶)‘(𝑘 + 1)) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
201200notbid 319 . . . . . . . 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 233 . . . . . 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 591 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
206205adantrr 713 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
207 zleltp1 12022 . . . . . . . . 9 ((0 ∈ ℤ ∧ ((𝐹𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
208176, 207mpan 686 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
209 0red 10633 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → 0 ∈ ℝ)
210 zre 11974 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((𝐹𝐶)‘𝑘) ∈ ℝ)
211 1red 10631 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → 1 ∈ ℝ)
212210, 211readdcld 10659 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) + 1) ∈ ℝ)
213209, 212ltnled 10776 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (0 < (((𝐹𝐶)‘𝑘) + 1) ↔ ¬ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
214208, 213bitrd 280 . . . . . . 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 258 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 0 ≤ ((𝐹𝐶)‘𝑘))
217206zred 12076 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
218 0red 10633 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 0 ∈ ℝ)
219217, 218letri3d 10771 . . . . 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 709 . . . 4 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
2214, 220sylan2b 593 . . 3 ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
222 ssrab2 4060 . . . . . 6 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ (1...𝐽)
223222, 12sstri 3980 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ ℝ
224223a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ ℝ)
225 fzfi 13330 . . . . . 6 (1...𝐽) ∈ Fin
226 ssfi 8727 . . . . . 6 (((1...𝐽) ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ (1...𝐽)) → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin)
227225, 222, 226mp2an 688 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin
228227a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin)
229 rabn0 4343 . . . . 5 ({𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ≠ ∅ ↔ ∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0)
23060, 229sylibr 235 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ≠ ∅)
231 fimaxre 11573 . . . 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 1365 . . 3 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)
233221, 232reximddv 3280 . 2 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ((𝐹𝐶)‘𝑘) = 0)
234 elrabi 3679 . . . 4 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} → 𝑘 ∈ (1...𝐽))
235234anim1i 614 . . 3 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∧ ((𝐹𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) = 0))
236235reximi2 3249 . 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 207  wa 396  wo 843   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  {crab 3147  cdif 3937  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564   class class class wbr 5063  cmpt 5143  cfv 6352  (class class class)co 7148  Fincfn 8498  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664  cle 10665  cmin 10859   / cdiv 11286  cn 11627  2c2 11681  cz 11970  cuz 12232  ...cfz 12882  chash 13680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-hash 13681
This theorem is referenced by:  ballotlem5  31643  ballotlemic  31650
  Copyright terms: Public domain W3C validator