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 30528
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 6178 . . . . . . 7 (𝑖 = 𝑘 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝑘))
21breq1d 4654 . . . . . 6 (𝑖 = 𝑘 → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝑘) ≤ 0))
32elrab 3357 . . . . 5 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ↔ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0))
43anbi1i 730 . . . 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 802 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ≤ 0)
6 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → 𝑘 ∈ (1...𝐽))
76adantrr 752 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 𝑘 ∈ (1...𝐽))
8 fzssuz 12367 . . . . . . . . . . . . . 14 (1...𝐽) ⊆ (ℤ‘1)
9 uzssz 11692 . . . . . . . . . . . . . 14 (ℤ‘1) ⊆ ℤ
108, 9sstri 3604 . . . . . . . . . . . . 13 (1...𝐽) ⊆ ℤ
11 zssre 11369 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
1210, 11sstri 3604 . . . . . . . . . . . 12 (1...𝐽) ⊆ ℝ
1312sseli 3591 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℝ)
1413ltp1d 10939 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → 𝑘 < (𝑘 + 1))
15 1red 10040 . . . . . . . . . . . 12 (𝑘 ∈ (1...𝐽) → 1 ∈ ℝ)
1613, 15readdcld 10054 . . . . . . . . . . 11 (𝑘 ∈ (1...𝐽) → (𝑘 + 1) ∈ ℝ)
1713, 16ltnled 10169 . . . . . . . . . 10 (𝑘 ∈ (1...𝐽) → (𝑘 < (𝑘 + 1) ↔ ¬ (𝑘 + 1) ≤ 𝑘))
1814, 17mpbid 222 . . . . . . . . 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 795 . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 = 𝐽) → 𝑘 = 𝐽)
2423fveq2d 6182 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 = 𝐽) → ((𝐹𝐶)‘𝑘) = ((𝐹𝐶)‘𝐽))
2524breq2d 4656 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝑘) ↔ 0 < ((𝐹𝐶)‘𝐽)))
26 ballotlemfp1.j . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽 ∈ ℕ)
27 elnnuz 11709 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 ∈ ℕ ↔ 𝐽 ∈ (ℤ‘1))
2826, 27sylib 208 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐽 ∈ (ℤ‘1))
29 eluzfz2 12334 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (ℤ‘1) → 𝐽 ∈ (1...𝐽))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ (1...𝐽))
31 eleq1 2687 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝐽 → (𝑘 ∈ (1...𝐽) ↔ 𝐽 ∈ (1...𝐽)))
3230, 31syl5ibrcom 237 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 = 𝐽𝑘 ∈ (1...𝐽)))
3332anc2li 579 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = 𝐽 → (𝜑𝑘 ∈ (1...𝐽))))
34 1eluzge0 11717 . . . . . . . . . . . . . . . . . . . 20 1 ∈ (ℤ‘0)
35 fzss1 12365 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ (ℤ‘0) → (1...𝐽) ⊆ (0...𝐽))
3635sseld 3594 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ (ℤ‘0) → (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽)))
3734, 36ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ (0...𝐽))
38 0red 10026 . . . . . . . . . . . . . . . . . . . 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 12327 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
4746adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
4839, 40, 41, 42, 43, 45, 47ballotlemfelz 30526 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
4948zred 11467 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ (0...𝐽)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
5038, 49ltnled 10169 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ (0...𝐽)) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5137, 50sylan2 491 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (1...𝐽)) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5233, 51syl6 35 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 = 𝐽 → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0)))
5352imp 445 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝑘) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5425, 53bitr3d 270 . . . . . . . . . . . . . . 15 ((𝜑𝑘 = 𝐽) → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5522, 54mpbid 222 . . . . . . . . . . . . . 14 ((𝜑𝑘 = 𝐽) → ¬ ((𝐹𝐶)‘𝑘) ≤ 0)
5655ex 450 . . . . . . . . . . . . 13 (𝜑 → (𝑘 = 𝐽 → ¬ ((𝐹𝐶)‘𝑘) ≤ 0))
5756con2d 129 . . . . . . . . . . . 12 (𝜑 → (((𝐹𝐶)‘𝑘) ≤ 0 → ¬ 𝑘 = 𝐽))
58 nn1m1nn 11025 . . . . . . . . . . . . . . . . . . . . 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 6642 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐽 = 1 → (𝐽...𝐽) = (1...𝐽))
6362adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = (1...𝐽))
6426nnzd 11466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐽 ∈ ℤ)
65 fzsn 12368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐽 ∈ ℤ → (𝐽...𝐽) = {𝐽})
6664, 65syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝐽...𝐽) = {𝐽})
6766adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝐽 = 1) → (𝐽...𝐽) = {𝐽})
6863, 67eqtr3d 2656 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝐽 = 1) → (1...𝐽) = {𝐽})
6968rexeqdv 3140 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝐽 = 1) → (∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0 ↔ ∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0))
7061, 69mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → ∃𝑖 ∈ {𝐽} ((𝐹𝐶)‘𝑖) ≤ 0)
71 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑖 = 𝐽 → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘𝐽))
7271breq1d 4654 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 = 𝐽 → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘𝐽) ≤ 0))
7372rexsng 4210 . . . . . . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ((𝐹𝐶)‘𝐽) ≤ 0)
7721adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → 0 < ((𝐹𝐶)‘𝐽))
78 0red 10026 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 0 ∈ ℝ)
7939, 40, 41, 42, 43, 44, 64ballotlemfelz 30526 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℤ)
8079zred 11467 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝐹𝐶)‘𝐽) ∈ ℝ)
8178, 80ltnled 10169 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝐽) ≤ 0))
8281adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝐽 = 1) → (0 < ((𝐹𝐶)‘𝐽) ↔ ¬ ((𝐹𝐶)‘𝐽) ≤ 0))
8377, 82mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝐽 = 1) → ¬ ((𝐹𝐶)‘𝐽) ≤ 0)
8476, 83pm2.65da 599 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ¬ 𝐽 = 1)
85 biortn 421 . . . . . . . . . . . . . . . . . . . . . 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 304 . . . . . . . . . . . . . . . . . . . . . 22 (𝐽 = 1 ↔ ¬ ¬ 𝐽 = 1)
8887orbi1i 542 . . . . . . . . . . . . . . . . . . . . 21 ((𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ) ↔ (¬ ¬ 𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ))
8986, 88syl6bbr 278 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐽 − 1) ∈ ℕ ↔ (𝐽 = 1 ∨ (𝐽 − 1) ∈ ℕ)))
9059, 89mpbird 247 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℕ)
91 elnnuz 11709 . . . . . . . . . . . . . . . . . . 19 ((𝐽 − 1) ∈ ℕ ↔ (𝐽 − 1) ∈ (ℤ‘1))
9290, 91sylib 208 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐽 − 1) ∈ (ℤ‘1))
93 elfzp1 12376 . . . . . . . . . . . . . . . . . 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 11021 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽 ∈ ℂ)
96 1cnd 10041 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 1 ∈ ℂ)
9795, 96npcand 10381 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐽 − 1) + 1) = 𝐽)
9897oveq2d 6651 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...((𝐽 − 1) + 1)) = (1...𝐽))
9998eleq2d 2685 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑘 ∈ (1...((𝐽 − 1) + 1)) ↔ 𝑘 ∈ (1...𝐽)))
10097eqeq2d 2630 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑘 = ((𝐽 − 1) + 1) ↔ 𝑘 = 𝐽))
101100orbi2d 737 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = ((𝐽 − 1) + 1)) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
10294, 99, 1013bitr3d 298 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽)))
103 orcom 402 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (1...(𝐽 − 1)) ∨ 𝑘 = 𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1))))
104102, 103syl6bb 276 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (1...𝐽) ↔ (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
105104biimpd 219 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
106 pm5.6 950 . . . . . . . . . . . . . 14 (((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))) ↔ (𝑘 ∈ (1...𝐽) → (𝑘 = 𝐽𝑘 ∈ (1...(𝐽 − 1)))))
107105, 106sylibr 224 . . . . . . . . . . . . 13 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ¬ 𝑘 = 𝐽) → 𝑘 ∈ (1...(𝐽 − 1))))
10890nnzd 11466 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐽 − 1) ∈ ℤ)
109 1z 11392 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
110108, 109jctil 559 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ))
111 elfzelz 12327 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(𝐽 − 1)) → 𝑘 ∈ ℤ)
112111, 109jctir 560 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...(𝐽 − 1)) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ))
113 fzaddel 12360 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ (𝐽 − 1) ∈ ℤ) ∧ (𝑘 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
114110, 112, 113syl2an 494 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 ∈ (1...(𝐽 − 1)) ↔ (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1))))
115114biimp3a 1430 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ (1...(𝐽 − 1)) ∧ 𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
1161153anidm23 1383 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (1...(𝐽 − 1))) → (𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)))
117 1p1e2 11119 . . . . . . . . . . . . . . . . . . . 20 (1 + 1) = 2
118117a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1 + 1) = 2)
119118, 97oveq12d 6653 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1 + 1)...((𝐽 − 1) + 1)) = (2...𝐽))
120119eleq2d 2685 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑘 + 1) ∈ ((1 + 1)...((𝐽 − 1) + 1)) ↔ (𝑘 + 1) ∈ (2...𝐽)))
121 2eluzge1 11719 . . . . . . . . . . . . . . . . . . 19 2 ∈ (ℤ‘1)
122 fzss1 12365 . . . . . . . . . . . . . . . . . . 19 (2 ∈ (ℤ‘1) → (2...𝐽) ⊆ (1...𝐽))
123121, 122ax-mp 5 . . . . . . . . . . . . . . . . . 18 (2...𝐽) ⊆ (1...𝐽)
124123sseli 3591 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (2...𝐽) → (𝑘 + 1) ∈ (1...𝐽))
125120, 124syl6bi 243 . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . 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 499 . . . . . . . . . . 11 (𝜑 → ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) → (𝑘 + 1) ∈ (1...𝐽)))
131130imp 445 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → (𝑘 + 1) ∈ (1...𝐽))
132131adantrr 752 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (𝑘 + 1) ∈ (1...𝐽))
133 fveq2 6178 . . . . . . . . . . . . . 14 (𝑖 = (𝑘 + 1) → ((𝐹𝐶)‘𝑖) = ((𝐹𝐶)‘(𝑘 + 1)))
134133breq1d 4654 . . . . . . . . . . . . 13 (𝑖 = (𝑘 + 1) → (((𝐹𝐶)‘𝑖) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
135134elrab 3357 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ↔ ((𝑘 + 1) ∈ (1...𝐽) ∧ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
136 breq1 4647 . . . . . . . . . . . . 13 (𝑗 = (𝑘 + 1) → (𝑗𝑘 ↔ (𝑘 + 1) ≤ 𝑘))
137136rspccva 3303 . . . . . . . . . . . 12 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ (𝑘 + 1) ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}) → (𝑘 + 1) ≤ 𝑘)
138135, 137sylan2br 493 . . . . . . . . . . 11 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ ((𝑘 + 1) ∈ (1...𝐽) ∧ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0)) → (𝑘 + 1) ≤ 𝑘)
139138expr 642 . . . . . . . . . 10 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (((𝐹𝐶)‘(𝑘 + 1)) ≤ 0 → (𝑘 + 1) ≤ 𝑘))
140139con3d 148 . . . . . . . . 9 ((∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ≤ 𝑘 → ¬ ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0))
14120, 132, 140syl2anc 692 . . . . . . . 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 800 . . . . . . . . . . 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 789 . . . . . . . . . . . . . 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 3594 . . . . . . . . . . . . . . 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 12327 . . . . . . . . . . . . . . . . 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 30526 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℤ)
153152zred 11467 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘 + 1) ∈ (0...𝐽)) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
154145, 148, 153syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ∈ ℝ)
155 0red 10026 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → 0 ∈ ℝ)
156 simplrr 800 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → (𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)))
16044adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → 𝐶𝑂)
161 elfznn 12355 . . . . . . . . . . . . . . . . . . . . 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 30527 . . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → (¬ (𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1)))
165164imp 445 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
166159, 165sylan 488 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1))
167 elfzelz 12327 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℤ)
168167zcnd 11468 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 𝑘 ∈ ℂ)
169 1cnd 10041 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...𝐽) → 1 ∈ ℂ)
170168, 169pncand 10378 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (1...𝐽) → ((𝑘 + 1) − 1) = 𝑘)
171170fveq2d 6182 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...𝐽) → ((𝐹𝐶)‘((𝑘 + 1) − 1)) = ((𝐹𝐶)‘𝑘))
172171oveq1d 6650 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) − 1) = (((𝐹𝐶)‘𝑘) − 1))
173172eqeq2d 2630 . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1))
176 0z 11373 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
177 zlem1lt 11414 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝐶)‘𝑘) ∈ ℤ ∧ 0 ∈ ℤ) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) − 1) < 0))
17848, 176, 177sylancl 693 . . . . . . . . . . . . . . . . 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 4647 . . . . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝐽)) ∧ ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) − 1)) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) < 0))
183145, 158, 175, 182syl21anc 1323 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → (((𝐹𝐶)‘𝑘) ≤ 0 ↔ ((𝐹𝐶)‘(𝑘 + 1)) < 0))
184156, 183mpbid 222 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) < 0)
185154, 155, 184ltled 10170 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ ¬ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) ≤ 0)
186185adantlrr 756 . . . . . . . . . . 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 1322 . . . . . . . . . 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 834 . . . . . . . . 9 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → (𝑘 + 1) ∈ 𝐶)
190163simprd 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) → ((𝑘 + 1) ∈ 𝐶 → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1)))
191190imp 445 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑘 + 1) ∈ (1...𝐽)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1))
192159, 191sylan 488 . . . . . . . . . . 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 6650 . . . . . . . . . . . . 13 (𝑘 ∈ (1...𝐽) → (((𝐹𝐶)‘((𝑘 + 1) − 1)) + 1) = (((𝐹𝐶)‘𝑘) + 1))
195194eqeq2d 2630 . . . . . . . . . . . 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 222 . . . . . . . . . 10 (((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) ∧ (𝑘 + 1) ∈ 𝐶) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
198197adantlrr 756 . . . . . . . . 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 701 . . . . . . . 8 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1))
200 breq1 4647 . . . . . . . . 9 (((𝐹𝐶)‘(𝑘 + 1)) = (((𝐹𝐶)‘𝑘) + 1) → (((𝐹𝐶)‘(𝑘 + 1)) ≤ 0 ↔ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
201200notbid 308 . . . . . . . 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 222 . . . . . 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 487 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
206205adantrr 752 . . . . . . 7 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℤ)
207 zleltp1 11413 . . . . . . . . 9 ((0 ∈ ℤ ∧ ((𝐹𝐶)‘𝑘) ∈ ℤ) → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
208176, 207mpan 705 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (0 ≤ ((𝐹𝐶)‘𝑘) ↔ 0 < (((𝐹𝐶)‘𝑘) + 1)))
209 0red 10026 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → 0 ∈ ℝ)
210 zre 11366 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → ((𝐹𝐶)‘𝑘) ∈ ℝ)
211 1red 10040 . . . . . . . . . 10 (((𝐹𝐶)‘𝑘) ∈ ℤ → 1 ∈ ℝ)
212210, 211readdcld 10054 . . . . . . . . 9 (((𝐹𝐶)‘𝑘) ∈ ℤ → (((𝐹𝐶)‘𝑘) + 1) ∈ ℝ)
213209, 212ltnled 10169 . . . . . . . 8 (((𝐹𝐶)‘𝑘) ∈ ℤ → (0 < (((𝐹𝐶)‘𝑘) + 1) ↔ ¬ (((𝐹𝐶)‘𝑘) + 1) ≤ 0))
214208, 213bitrd 268 . . . . . . 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 247 . . . . 5 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 0 ≤ ((𝐹𝐶)‘𝑘))
217206zred 11467 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) ∈ ℝ)
218 0red 10026 . . . . . 6 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → 0 ∈ ℝ)
219217, 218letri3d 10164 . . . . 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 956 . . . 4 ((𝜑 ∧ ((𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) ≤ 0) ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
2214, 220sylan2b 492 . . 3 ((𝜑 ∧ (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∧ ∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)) → ((𝐹𝐶)‘𝑘) = 0)
222 ssrab2 3679 . . . . . 6 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ (1...𝐽)
223222, 12sstri 3604 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ ℝ
224223a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ ℝ)
225 fzfi 12754 . . . . . 6 (1...𝐽) ∈ Fin
226 ssfi 8165 . . . . . 6 (((1...𝐽) ∈ Fin ∧ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ⊆ (1...𝐽)) → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin)
227225, 222, 226mp2an 707 . . . . 5 {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin
228227a1i 11 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∈ Fin)
229 rabn0 3949 . . . . 5 ({𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ≠ ∅ ↔ ∃𝑖 ∈ (1...𝐽)((𝐹𝐶)‘𝑖) ≤ 0)
23060, 229sylibr 224 . . . 4 (𝜑 → {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ≠ ∅)
231 fimaxre 10953 . . . 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 1324 . . 3 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}∀𝑗 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0}𝑗𝑘)
233221, 232reximddv 3015 . 2 (𝜑 → ∃𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ((𝐹𝐶)‘𝑘) = 0)
234 elrabi 3353 . . . 4 (𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} → 𝑘 ∈ (1...𝐽))
235234anim1i 591 . . 3 ((𝑘 ∈ {𝑖 ∈ (1...𝐽) ∣ ((𝐹𝐶)‘𝑖) ≤ 0} ∧ ((𝐹𝐶)‘𝑘) = 0) → (𝑘 ∈ (1...𝐽) ∧ ((𝐹𝐶)‘𝑘) = 0))
236235reximi2 3007 . 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 196  wo 383  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  {crab 2913  cdif 3564  cin 3566  wss 3567  c0 3907  𝒫 cpw 4149  {csn 4168   class class class wbr 4644  cmpt 4720  cfv 5876  (class class class)co 6635  Fincfn 7940  cr 9920  0cc0 9921  1c1 9922   + caddc 9924   < clt 10059  cle 10060  cmin 10251   / cdiv 10669  cn 11005  2c2 11055  cz 11362  cuz 11672  ...cfz 12311  #chash 13100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-hash 13101
This theorem is referenced by:  ballotlem5  30535  ballotlemic  30542
  Copyright terms: Public domain W3C validator