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Theorem ruclem9 15011
Description: Lemma for ruc 15016. 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 6229 . . . . . . 7 (𝑘 = 𝑀 → (𝐺𝑘) = (𝐺𝑀))
32fveq2d 6233 . . . . . 6 (𝑘 = 𝑀 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑀)))
43breq2d 4697 . . . . 5 (𝑘 = 𝑀 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀))))
52fveq2d 6233 . . . . . 6 (𝑘 = 𝑀 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑀)))
65breq1d 4695 . . . . 5 (𝑘 = 𝑀 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))
74, 6anbi12d 747 . . . 4 (𝑘 = 𝑀 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))))
87imbi2d 329 . . 3 (𝑘 = 𝑀 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))))
9 fveq2 6229 . . . . . . 7 (𝑘 = 𝑛 → (𝐺𝑘) = (𝐺𝑛))
109fveq2d 6233 . . . . . 6 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
1110breq2d 4697 . . . . 5 (𝑘 = 𝑛 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛))))
129fveq2d 6233 . . . . . 6 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
1312breq1d 4695 . . . . 5 (𝑘 = 𝑛 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))))
1411, 13anbi12d 747 . . . 4 (𝑘 = 𝑛 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀)))))
1514imbi2d 329 . . 3 (𝑘 = 𝑛 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))))))
16 fveq2 6229 . . . . . . 7 (𝑘 = (𝑛 + 1) → (𝐺𝑘) = (𝐺‘(𝑛 + 1)))
1716fveq2d 6233 . . . . . 6 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
1817breq2d 4697 . . . . 5 (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
1916fveq2d 6233 . . . . . 6 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
2019breq1d 4695 . . . . 5 (𝑘 = (𝑛 + 1) → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
2118, 20anbi12d 747 . . . 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 329 . . 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 6229 . . . . . . 7 (𝑘 = 𝑁 → (𝐺𝑘) = (𝐺𝑁))
2423fveq2d 6233 . . . . . 6 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
2524breq2d 4697 . . . . 5 (𝑘 = 𝑁 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁))))
2623fveq2d 6233 . . . . . 6 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
2726breq1d 4695 . . . . 5 (𝑘 = 𝑁 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))
2825, 27anbi12d 747 . . . 4 (𝑘 = 𝑁 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀)))))
2928imbi2d 329 . . 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 15008 . . . . . . . 8 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
35 ruclem9.6 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3634, 35ffvelrnd 6400 . . . . . . 7 (𝜑 → (𝐺𝑀) ∈ (ℝ × ℝ))
37 xp1st 7242 . . . . . . 7 ((𝐺𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑀)) ∈ ℝ)
3836, 37syl 17 . . . . . 6 (𝜑 → (1st ‘(𝐺𝑀)) ∈ ℝ)
3938leidd 10632 . . . . 5 (𝜑 → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)))
40 xp2nd 7243 . . . . . . 7 ((𝐺𝑀) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑀)) ∈ ℝ)
4136, 40syl 17 . . . . . 6 (𝜑 → (2nd ‘(𝐺𝑀)) ∈ ℝ)
4241leidd 10632 . . . . 5 (𝜑 → (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))
4339, 42jca 553 . . . 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 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐹:ℕ⟶ℝ)
4631adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
4734adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐺:ℕ0⟶(ℝ × ℝ))
48 eluznn0 11795 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℕ0)
4935, 48sylan 487 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℕ0)
5047, 49ffvelrnd 6400 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) ∈ (ℝ × ℝ))
51 xp1st 7242 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
5250, 51syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ∈ ℝ)
53 xp2nd 7243 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
5450, 53syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
55 nn0p1nn 11370 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
5649, 55syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ ℕ)
5745, 56ffvelrnd 6400 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
58 eqid 2651 . . . . . . . . . 10 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
59 eqid 2651 . . . . . . . . . 10 (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
6030, 31, 32, 33ruclem8 15010 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
6149, 60syldan 486 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
6245, 46, 52, 54, 57, 58, 59, 61ruclem2 15005 . . . . . . . . 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 1093 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ≤ (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
6430, 31, 32, 33ruclem7 15009 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
6549, 64syldan 486 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
66 1st2nd2 7249 . . . . . . . . . . . 12 ((𝐺𝑛) ∈ (ℝ × ℝ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
6750, 66syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
6867oveq1d 6705 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
6965, 68eqtrd 2685 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
7069fveq2d 6233 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
7163, 70breqtrrd 4713 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1))))
7238adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑀)) ∈ ℝ)
73 peano2nn0 11371 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
7449, 73syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ ℕ0)
7547, 74ffvelrnd 6400 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
76 xp1st 7242 . . . . . . . . 9 ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
7775, 76syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
78 letr 10169 . . . . . . . 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 1366 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
8071, 79mpan2d 710 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
8169fveq2d 6233 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
8262simp3d 1095 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2nd ‘(𝐺𝑛)))
8381, 82eqbrtrd 4707 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)))
84 xp2nd 7243 . . . . . . . . 9 ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
8575, 84syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
8641adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺𝑀)) ∈ ℝ)
87 letr 10169 . . . . . . . 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 1366 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
8983, 88mpand 711 . . . . . 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 450 . . . 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 11784 . 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 383   = wceq 1523  wcel 2030  csb 3566  cun 3605  ifcif 4119  {csn 4210  cop 4216   class class class wbr 4685   × cxp 5141  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  cle 10113   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  seqcseq 12841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842
This theorem is referenced by:  ruclem10  15012
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