Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ruclem9 Structured version   Visualization version   GIF version

Theorem ruclem9 14887
 Description: Lemma for ruc 14892. The first components of the 𝐺 sequence are increasing, and the second components are decreasing. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1 (𝜑𝐹:ℕ⟶ℝ)
ruc.2 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
ruc.4 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
ruc.5 𝐺 = seq0(𝐷, 𝐶)
ruclem9.6 (𝜑𝑀 ∈ ℕ0)
ruclem9.7 (𝜑𝑁 ∈ (ℤ𝑀))
Assertion
Ref Expression
ruclem9 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑚,𝐺,𝑥,𝑦   𝑚,𝑀,𝑥,𝑦   𝑚,𝑁,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐶(𝑥,𝑦,𝑚)   𝐷(𝑥,𝑦,𝑚)

Proof of Theorem ruclem9
Dummy variables 𝑛 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ruclem9.7 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
2 fveq2 6150 . . . . . . 7 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
32fveq2d 6154 . . . . . 6 (𝑘 = 𝑀 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑀)))
43breq2d 4630 . . . . 5 (𝑘 = 𝑀 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀))))
52fveq2d 6154 . . . . . 6 (𝑘 = 𝑀 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑀)))
65breq1d 4628 . . . . 5 (𝑘 = 𝑀 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))
74, 6anbi12d 746 . . . 4 (𝑘 = 𝑀 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))))
87imbi2d 330 . . 3 (𝑘 = 𝑀 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))))
9 fveq2 6150 . . . . . . 7 (𝑘 = 𝑛 → (𝐺𝑘) = (𝐺𝑛))
109fveq2d 6154 . . . . . 6 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
1110breq2d 4630 . . . . 5 (𝑘 = 𝑛 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛))))
129fveq2d 6154 . . . . . 6 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
1312breq1d 4628 . . . . 5 (𝑘 = 𝑛 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))))
1411, 13anbi12d 746 . . . 4 (𝑘 = 𝑛 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀)))))
1514imbi2d 330 . . 3 (𝑘 = 𝑛 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))))))
16 fveq2 6150 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
1716fveq2d 6154 . . . . . 6 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
1817breq2d 4630 . . . . 5 (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
1916fveq2d 6154 . . . . . 6 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
2019breq1d 4628 . . . . 5 (𝑘 = (𝑛 + 1) → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
2118, 20anbi12d 746 . . . 4 (𝑘 = (𝑛 + 1) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀)))))
2221imbi2d 330 . . 3 (𝑘 = (𝑛 + 1) → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))))
23 fveq2 6150 . . . . . . 7 (𝑘 = 𝑁 → (𝐺𝑘) = (𝐺𝑁))
2423fveq2d 6154 . . . . . 6 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
2524breq2d 4630 . . . . 5 (𝑘 = 𝑁 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁))))
2623fveq2d 6154 . . . . . 6 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
2726breq1d 4628 . . . . 5 (𝑘 = 𝑁 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))
2825, 27anbi12d 746 . . . 4 (𝑘 = 𝑁 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀)))))
2928imbi2d 330 . . 3 (𝑘 = 𝑁 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))))
30 ruc.1 . . . . . . . . 9 (𝜑𝐹:ℕ⟶ℝ)
31 ruc.2 . . . . . . . . 9 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
32 ruc.4 . . . . . . . . 9 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
33 ruc.5 . . . . . . . . 9 𝐺 = seq0(𝐷, 𝐶)
3430, 31, 32, 33ruclem6 14884 . . . . . . . 8 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
35 ruclem9.6 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3634, 35ffvelrnd 6317 . . . . . . 7 (𝜑 → (𝐺𝑀) ∈ (ℝ × ℝ))
37 xp1st 7146 . . . . . . 7 ((𝐺𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑀)) ∈ ℝ)
3836, 37syl 17 . . . . . 6 (𝜑 → (1st ‘(𝐺𝑀)) ∈ ℝ)
3938leidd 10539 . . . . 5 (𝜑 → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)))
40 xp2nd 7147 . . . . . . 7 ((𝐺𝑀) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑀)) ∈ ℝ)
4136, 40syl 17 . . . . . 6 (𝜑 → (2nd ‘(𝐺𝑀)) ∈ ℝ)
4241leidd 10539 . . . . 5 (𝜑 → (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))
4339, 42jca 554 . . . 4 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))
4443a1i 11 . . 3 (𝑀 ∈ ℤ → (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))))
4530adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐹:ℕ⟶ℝ)
4631adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
4734adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐺:ℕ0⟶(ℝ × ℝ))
48 eluznn0 11701 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℕ0)
4935, 48sylan 488 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℕ0)
5047, 49ffvelrnd 6317 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) ∈ (ℝ × ℝ))
51 xp1st 7146 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
5250, 51syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ∈ ℝ)
53 xp2nd 7147 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
5450, 53syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
55 nn0p1nn 11277 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
5649, 55syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ ℕ)
5745, 56ffvelrnd 6317 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
58 eqid 2626 . . . . . . . . . 10 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
59 eqid 2626 . . . . . . . . . 10 (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
6030, 31, 32, 33ruclem8 14886 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
6149, 60syldan 487 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
6245, 46, 52, 54, 57, 58, 59, 61ruclem2 14881 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((1st ‘(𝐺𝑛)) ≤ (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ∧ (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) < (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ∧ (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2nd ‘(𝐺𝑛))))
6362simp1d 1071 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ≤ (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
6430, 31, 32, 33ruclem7 14885 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
6549, 64syldan 487 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
66 1st2nd2 7153 . . . . . . . . . . . 12 ((𝐺𝑛) ∈ (ℝ × ℝ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
6750, 66syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
6867oveq1d 6620 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
6965, 68eqtrd 2660 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
7069fveq2d 6154 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
7163, 70breqtrrd 4646 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1))))
7238adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑀)) ∈ ℝ)
73 peano2nn0 11278 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
7449, 73syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ ℕ0)
7547, 74ffvelrnd 6317 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
76 xp1st 7146 . . . . . . . . 9 ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
7775, 76syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
78 letr 10076 . . . . . . . 8 (((1st ‘(𝐺𝑀)) ∈ ℝ ∧ (1st ‘(𝐺𝑛)) ∈ ℝ ∧ (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
7972, 52, 77, 78syl3anc 1323 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
8071, 79mpan2d 709 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
8169fveq2d 6154 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
8262simp3d 1073 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2nd ‘(𝐺𝑛)))
8381, 82eqbrtrd 4640 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)))
84 xp2nd 7147 . . . . . . . . 9 ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
8575, 84syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
8641adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺𝑀)) ∈ ℝ)
87 letr 10076 . . . . . . . 8 (((2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ ∧ (2nd ‘(𝐺𝑛)) ∈ ℝ ∧ (2nd ‘(𝐺𝑀)) ∈ ℝ) → (((2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
8885, 54, 86, 87syl3anc 1323 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
8983, 88mpand 710 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
9080, 89anim12d 585 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀)))))
9190expcom 451 . . . 4 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))))
9291a2d 29 . . 3 (𝑛 ∈ (ℤ𝑀) → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀)))) → (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))))
938, 15, 22, 29, 44, 92uzind4 11690 . 2 (𝑁 ∈ (ℤ𝑀) → (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀)))))
941, 93mpcom 38 1 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992  ⦋csb 3519   ∪ cun 3558  ifcif 4063  {csn 4153  ⟨cop 4159   class class class wbr 4618   × cxp 5077  ⟶wf 5846  ‘cfv 5850  (class class class)co 6605   ↦ cmpt2 6607  1st c1st 7114  2nd c2nd 7115  ℝcr 9880  0cc0 9881  1c1 9882   + caddc 9884   < clt 10019   ≤ cle 10020   / cdiv 10629  ℕcn 10965  2c2 11015  ℕ0cn0 11237  ℤcz 11322  ℤ≥cuz 11631  seqcseq 12738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-seq 12739 This theorem is referenced by:  ruclem10  14888
 Copyright terms: Public domain W3C validator