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Theorem ruclem9 16235
Description: Lemma for ruc 16240. 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 6898 . . . . . 6 (𝑘 = 𝑀 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑀)))
32breq2d 5157 . . . . 5 (𝑘 = 𝑀 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀))))
4 2fveq3 6898 . . . . . 6 (𝑘 = 𝑀 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑀)))
54breq1d 5155 . . . . 5 (𝑘 = 𝑀 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))
63, 5anbi12d 630 . . . 4 (𝑘 = 𝑀 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))))
76imbi2d 339 . . 3 (𝑘 = 𝑀 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))))
8 2fveq3 6898 . . . . . 6 (𝑘 = 𝑛 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑛)))
98breq2d 5157 . . . . 5 (𝑘 = 𝑛 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛))))
10 2fveq3 6898 . . . . . 6 (𝑘 = 𝑛 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑛)))
1110breq1d 5155 . . . . 5 (𝑘 = 𝑛 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))))
129, 11anbi12d 630 . . . 4 (𝑘 = 𝑛 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀)))))
1312imbi2d 339 . . 3 (𝑘 = 𝑛 → ((𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))))))
14 2fveq3 6898 . . . . . 6 (𝑘 = (𝑛 + 1) → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺‘(𝑛 + 1))))
1514breq2d 5157 . . . . 5 (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
16 2fveq3 6898 . . . . . 6 (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1))))
1716breq1d 5155 . . . . 5 (𝑘 = (𝑛 + 1) → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
1815, 17anbi12d 630 . . . 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 339 . . 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 6898 . . . . . 6 (𝑘 = 𝑁 → (1st ‘(𝐺𝑘)) = (1st ‘(𝐺𝑁)))
2120breq2d 5157 . . . . 5 (𝑘 = 𝑁 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ↔ (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁))))
22 2fveq3 6898 . . . . . 6 (𝑘 = 𝑁 → (2nd ‘(𝐺𝑘)) = (2nd ‘(𝐺𝑁)))
2322breq1d 5155 . . . . 5 (𝑘 = 𝑁 → ((2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀)) ↔ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀))))
2421, 23anbi12d 630 . . . 4 (𝑘 = 𝑁 → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑘)) ∧ (2nd ‘(𝐺𝑘)) ≤ (2nd ‘(𝐺𝑀))) ↔ ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑁)) ∧ (2nd ‘(𝐺𝑁)) ≤ (2nd ‘(𝐺𝑀)))))
2524imbi2d 339 . . 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 16232 . . . . . . 7 (𝜑𝐺:ℕ0⟶(ℝ × ℝ))
31 ruclem9.6 . . . . . . 7 (𝜑𝑀 ∈ ℕ0)
3230, 31ffvelcdmd 7091 . . . . . 6 (𝜑 → (𝐺𝑀) ∈ (ℝ × ℝ))
33 xp1st 8027 . . . . . 6 ((𝐺𝑀) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑀)) ∈ ℝ)
3432, 33syl 17 . . . . 5 (𝜑 → (1st ‘(𝐺𝑀)) ∈ ℝ)
3534leidd 11821 . . . 4 (𝜑 → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)))
36 xp2nd 8028 . . . . . 6 ((𝐺𝑀) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑀)) ∈ ℝ)
3732, 36syl 17 . . . . 5 (𝜑 → (2nd ‘(𝐺𝑀)) ∈ ℝ)
3837leidd 11821 . . . 4 (𝜑 → (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀)))
3935, 38jca 510 . . 3 (𝜑 → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑀)) ∧ (2nd ‘(𝐺𝑀)) ≤ (2nd ‘(𝐺𝑀))))
4026adantr 479 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐹:ℕ⟶ℝ)
4127adantr 479 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦ (((1st𝑥) + (2nd𝑥)) / 2) / 𝑚if(𝑚 < 𝑦, ⟨(1st𝑥), 𝑚⟩, ⟨((𝑚 + (2nd𝑥)) / 2), (2nd𝑥)⟩)))
4230adantr 479 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝐺:ℕ0⟶(ℝ × ℝ))
43 eluznn0 12947 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℕ0)
4431, 43sylan 578 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑀)) → 𝑛 ∈ ℕ0)
4542, 44ffvelcdmd 7091 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) ∈ (ℝ × ℝ))
46 xp1st 8027 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (1st ‘(𝐺𝑛)) ∈ ℝ)
4745, 46syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ∈ ℝ)
48 xp2nd 8028 . . . . . . . . . . 11 ((𝐺𝑛) ∈ (ℝ × ℝ) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
4945, 48syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺𝑛)) ∈ ℝ)
50 nn0p1nn 12557 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
5144, 50syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ ℕ)
5240, 51ffvelcdmd 7091 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
53 eqid 2726 . . . . . . . . . 10 (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
54 eqid 2726 . . . . . . . . . 10 (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
5526, 27, 28, 29ruclem8 16234 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
5644, 55syldan 589 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) < (2nd ‘(𝐺𝑛)))
5740, 41, 47, 49, 52, 53, 54, 56ruclem2 16229 . . . . . . . . 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 1139 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ≤ (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
5926, 27, 28, 29ruclem7 16233 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
6044, 59syldan 589 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))))
61 1st2nd2 8034 . . . . . . . . . . . 12 ((𝐺𝑛) ∈ (ℝ × ℝ) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
6245, 61syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺𝑛) = ⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩)
6362oveq1d 7431 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((𝐺𝑛)𝐷(𝐹‘(𝑛 + 1))) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
6460, 63eqtrd 2766 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) = (⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1))))
6564fveq2d 6897 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺‘(𝑛 + 1))) = (1st ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
6658, 65breqtrrd 5173 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1))))
6734adantr 479 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺𝑀)) ∈ ℝ)
68 peano2nn0 12558 . . . . . . . . . . 11 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
6944, 68syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝑛 + 1) ∈ ℕ0)
7042, 69ffvelcdmd 7091 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ))
71 xp1st 8027 . . . . . . . . 9 ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
7270, 71syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
73 letr 11349 . . . . . . . 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 1368 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (1st ‘(𝐺𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
7566, 74mpan2d 692 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) → (1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))))
7664fveq2d 6897 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) = (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))))
7757simp3d 1141 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(⟨(1st ‘(𝐺𝑛)), (2nd ‘(𝐺𝑛))⟩𝐷(𝐹‘(𝑛 + 1)))) ≤ (2nd ‘(𝐺𝑛)))
7876, 77eqbrtrd 5167 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)))
79 xp2nd 8028 . . . . . . . . 9 ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ) → (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
8070, 79syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ)
8137adantr 479 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑀)) → (2nd ‘(𝐺𝑀)) ∈ ℝ)
82 letr 11349 . . . . . . . 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 1368 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
8478, 83mpand 693 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → ((2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀)) → (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀))))
8575, 84anim12d 607 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑀)) → (((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺𝑛)) ∧ (2nd ‘(𝐺𝑛)) ≤ (2nd ‘(𝐺𝑀))) → ((1st ‘(𝐺𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺𝑀)))))
8685expcom 412 . . . 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 12940 . 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 394   = wceq 1534  wcel 2099  csb 3891  cun 3944  ifcif 4523  {csn 4623  cop 4629   class class class wbr 5145   × cxp 5672  wf 6542  cfv 6546  (class class class)co 7416  cmpo 7418  1st c1st 7993  2nd c2nd 7994  cr 11148  0cc0 11149  1c1 11150   + caddc 11152   < clt 11289  cle 11290   / cdiv 11912  cn 12258  2c2 12313  0cn0 12518  cuz 12868  seqcseq 14015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-n0 12519  df-z 12605  df-uz 12869  df-fz 13533  df-seq 14016
This theorem is referenced by:  ruclem10  16236
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