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Theorem ruclem9 16038
Description: Lemma for ruc 16043. 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 2fveq3 6824 . . . . . 6 (𝑘 = 𝑀 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑀)))
32breq2d 5101 . . . . 5 (𝑘 = 𝑀 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀))))
4 2fveq3 6824 . . . . . 6 (𝑘 = 𝑀 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑀)))
54breq1d 5099 . . . . 5 (𝑘 = 𝑀 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))
63, 5anbi12d 631 . . . 4 (𝑘 = 𝑀 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))))
76imbi2d 340 . . 3 (𝑘 = 𝑀 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))))
8 2fveq3 6824 . . . . . 6 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
98breq2d 5101 . . . . 5 (𝑘 = 𝑛 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛))))
10 2fveq3 6824 . . . . . 6 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
1110breq1d 5099 . . . . 5 (𝑘 = 𝑛 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))))
129, 11anbi12d 631 . . . 4 (𝑘 = 𝑛 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀)))))
1312imbi2d 340 . . 3 (𝑘 = 𝑛 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))))))
14 2fveq3 6824 . . . . . 6 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
1514breq2d 5101 . . . . 5 (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
16 2fveq3 6824 . . . . . 6 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
1716breq1d 5099 . . . . 5 (𝑘 = (𝑛 + 1) → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
1815, 17anbi12d 631 . . . 4 (𝑘 = (𝑛 + 1) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀)))))
1918imbi2d 340 . . 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))))))
20 2fveq3 6824 . . . . . 6 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
2120breq2d 5101 . . . . 5 (𝑘 = 𝑁 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁))))
22 2fveq3 6824 . . . . . 6 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
2322breq1d 5099 . . . . 5 (𝑘 = 𝑁 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))
2421, 23anbi12d 631 . . . 4 (𝑘 = 𝑁 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀)))))
2524imbi2d 340 . . 3 (𝑘 = 𝑁 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))))
26 ruc.1 . . . . . . . 8 (𝜑𝐹:ℕ⟶ℝ)
27 ruc.2 . . . . . . . 8 (𝜑𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
28 ruc.4 . . . . . . . 8 𝐶 = ({⟨0, ⟨0, 1⟩⟩} ∪ 𝐹)
29 ruc.5 . . . . . . . 8 𝐺 = seq0(𝐷, 𝐶)
3026, 27, 28, 29ruclem6 16035 . . . . . . 7 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
31 ruclem9.6 . . . . . . 7 (𝜑𝑀 ∈ ℕ0)
3230, 31ffvelcdmd 7012 . . . . . 6 (𝜑 → (𝐺𝑀) ∈ (ℝ × ℝ))
33 xp1st 7923 . . . . . 6 ((𝐺𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑀)) ∈ ℝ)
3432, 33syl 17 . . . . 5 (𝜑 → (1st ‘(𝐺𝑀)) ∈ ℝ)
3534leidd 11634 . . . 4 (𝜑 → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)))
36 xp2nd 7924 . . . . . 6 ((𝐺𝑀) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑀)) ∈ ℝ)
3732, 36syl 17 . . . . 5 (𝜑 → (2nd ‘(𝐺𝑀)) ∈ ℝ)
3837leidd 11634 . . . 4 (𝜑 → (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))
3935, 38jca 512 . . 3 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))
4026adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐹:ℕ⟶ℝ)
4127adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
4230adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐺:ℕ0⟶(ℝ × ℝ))
43 eluznn0 12750 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℕ0)
4431, 43sylan 580 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℕ0)
4542, 44ffvelcdmd 7012 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) ∈ (ℝ × ℝ))
46 xp1st 7923 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
4745, 46syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ∈ ℝ)
48 xp2nd 7924 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4945, 48syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
50 nn0p1nn 12365 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
5144, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ ℕ)
5240, 51ffvelcdmd 7012 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
53 eqid 2736 . . . . . . . . . 10 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
54 eqid 2736 . . . . . . . . . 10 (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5526, 27, 28, 29ruclem8 16037 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
5644, 55syldan 591 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
5740, 41, 47, 49, 52, 53, 54, 56ruclem2 16032 . . . . . . . . 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))))
5857simp1d 1141 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ≤ (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5926, 27, 28, 29ruclem7 16036 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
6044, 59syldan 591 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
61 1st2nd2 7930 . . . . . . . . . . . 12 ((𝐺𝑛) ∈ (ℝ × ℝ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
6245, 61syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
6362oveq1d 7344 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
6460, 63eqtrd 2776 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
6564fveq2d 6823 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
6658, 65breqtrrd 5117 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1))))
6734adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑀)) ∈ ℝ)
68 peano2nn0 12366 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
6944, 68syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ ℕ0)
7042, 69ffvelcdmd 7012 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
71 xp1st 7923 . . . . . . . . 9 ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
7270, 71syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
73 letr 11162 . . . . . . . 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)))))
7467, 47, 72, 73syl3anc 1370 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
7566, 74mpan2d 691 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
7664fveq2d 6823 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
7757simp3d 1143 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2nd ‘(𝐺𝑛)))
7876, 77eqbrtrd 5111 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)))
79 xp2nd 7924 . . . . . . . . 9 ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
8070, 79syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
8137adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺𝑀)) ∈ ℝ)
82 letr 11162 . . . . . . . 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))))
8380, 49, 81, 82syl3anc 1370 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
8478, 83mpand 692 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
8575, 84anim12d 609 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀)))))
8685expcom 414 . . . 4 (𝑛 ∈ (ℤ𝑀) → (𝜑 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))))
8786a2d 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))))))
887, 13, 19, 25, 39, 87uzind4i 12743 . 2 (𝑁 ∈ (ℤ𝑀) → (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀)))))
891, 88mpcom 38 1 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  csb 3842  cun 3895  ifcif 4472  {csn 4572  cop 4578   class class class wbr 5089   × cxp 5612  wf 6469  cfv 6473  (class class class)co 7329  cmpo 7331  1st c1st 7889  2nd c2nd 7890  cr 10963  0cc0 10964  1c1 10965   + caddc 10967   < clt 11102  cle 11103   / cdiv 11725  cn 12066  2c2 12121  0cn0 12326  cuz 12675  seqcseq 13814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-er 8561  df-en 8797  df-dom 8798  df-sdom 8799  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-div 11726  df-nn 12067  df-2 12129  df-n0 12327  df-z 12413  df-uz 12676  df-fz 13333  df-seq 13815
This theorem is referenced by:  ruclem10  16039
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