Step | Hyp | Ref
| Expression |
1 | | ruclem9.7 |
. 2
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
2 | | 2fveq3 6722 |
. . . . . 6
⊢ (𝑘 = 𝑀 → (1st ‘(𝐺‘𝑘)) = (1st ‘(𝐺‘𝑀))) |
3 | 2 | breq2d 5065 |
. . . . 5
⊢ (𝑘 = 𝑀 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ↔ (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑀)))) |
4 | | 2fveq3 6722 |
. . . . . 6
⊢ (𝑘 = 𝑀 → (2nd ‘(𝐺‘𝑘)) = (2nd ‘(𝐺‘𝑀))) |
5 | 4 | breq1d 5063 |
. . . . 5
⊢ (𝑘 = 𝑀 → ((2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀)) ↔ (2nd ‘(𝐺‘𝑀)) ≤ (2nd ‘(𝐺‘𝑀)))) |
6 | 3, 5 | anbi12d 634 |
. . . 4
⊢ (𝑘 = 𝑀 → (((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ∧ (2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀))) ↔ ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑀)) ∧ (2nd ‘(𝐺‘𝑀)) ≤ (2nd ‘(𝐺‘𝑀))))) |
7 | 6 | imbi2d 344 |
. . 3
⊢ (𝑘 = 𝑀 → ((𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ∧ (2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑀)) ∧ (2nd ‘(𝐺‘𝑀)) ≤ (2nd ‘(𝐺‘𝑀)))))) |
8 | | 2fveq3 6722 |
. . . . . 6
⊢ (𝑘 = 𝑛 → (1st ‘(𝐺‘𝑘)) = (1st ‘(𝐺‘𝑛))) |
9 | 8 | breq2d 5065 |
. . . . 5
⊢ (𝑘 = 𝑛 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ↔ (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑛)))) |
10 | | 2fveq3 6722 |
. . . . . 6
⊢ (𝑘 = 𝑛 → (2nd ‘(𝐺‘𝑘)) = (2nd ‘(𝐺‘𝑛))) |
11 | 10 | breq1d 5063 |
. . . . 5
⊢ (𝑘 = 𝑛 → ((2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀)) ↔ (2nd ‘(𝐺‘𝑛)) ≤ (2nd ‘(𝐺‘𝑀)))) |
12 | 9, 11 | anbi12d 634 |
. . . 4
⊢ (𝑘 = 𝑛 → (((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ∧ (2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀))) ↔ ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑛)) ∧ (2nd ‘(𝐺‘𝑛)) ≤ (2nd ‘(𝐺‘𝑀))))) |
13 | 12 | imbi2d 344 |
. . 3
⊢ (𝑘 = 𝑛 → ((𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ∧ (2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑛)) ∧ (2nd ‘(𝐺‘𝑛)) ≤ (2nd ‘(𝐺‘𝑀)))))) |
14 | | 2fveq3 6722 |
. . . . . 6
⊢ (𝑘 = (𝑛 + 1) → (1st ‘(𝐺‘𝑘)) = (1st ‘(𝐺‘(𝑛 + 1)))) |
15 | 14 | breq2d 5065 |
. . . . 5
⊢ (𝑘 = (𝑛 + 1) → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ↔ (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))))) |
16 | | 2fveq3 6722 |
. . . . . 6
⊢ (𝑘 = (𝑛 + 1) → (2nd ‘(𝐺‘𝑘)) = (2nd ‘(𝐺‘(𝑛 + 1)))) |
17 | 16 | breq1d 5063 |
. . . . 5
⊢ (𝑘 = (𝑛 + 1) → ((2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀)) ↔ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺‘𝑀)))) |
18 | 15, 17 | anbi12d 634 |
. . . 4
⊢ (𝑘 = (𝑛 + 1) → (((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ∧ (2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀))) ↔ ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺‘𝑀))))) |
19 | 18 | imbi2d 344 |
. . 3
⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ∧ (2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺‘𝑀)))))) |
20 | | 2fveq3 6722 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (1st ‘(𝐺‘𝑘)) = (1st ‘(𝐺‘𝑁))) |
21 | 20 | breq2d 5065 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ↔ (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑁)))) |
22 | | 2fveq3 6722 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (2nd ‘(𝐺‘𝑘)) = (2nd ‘(𝐺‘𝑁))) |
23 | 22 | breq1d 5063 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀)) ↔ (2nd ‘(𝐺‘𝑁)) ≤ (2nd ‘(𝐺‘𝑀)))) |
24 | 21, 23 | anbi12d 634 |
. . . 4
⊢ (𝑘 = 𝑁 → (((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ∧ (2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀))) ↔ ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑁)) ∧ (2nd ‘(𝐺‘𝑁)) ≤ (2nd ‘(𝐺‘𝑀))))) |
25 | 24 | imbi2d 344 |
. . 3
⊢ (𝑘 = 𝑁 → ((𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑘)) ∧ (2nd ‘(𝐺‘𝑘)) ≤ (2nd ‘(𝐺‘𝑀)))) ↔ (𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑁)) ∧ (2nd ‘(𝐺‘𝑁)) ≤ (2nd ‘(𝐺‘𝑀)))))) |
26 | | ruc.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:ℕ⟶ℝ) |
27 | | ruc.2 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦
⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) |
28 | | ruc.4 |
. . . . . . . 8
⊢ 𝐶 = ({〈0, 〈0,
1〉〉} ∪ 𝐹) |
29 | | ruc.5 |
. . . . . . . 8
⊢ 𝐺 = seq0(𝐷, 𝐶) |
30 | 26, 27, 28, 29 | ruclem6 15796 |
. . . . . . 7
⊢ (𝜑 → 𝐺:ℕ0⟶(ℝ ×
ℝ)) |
31 | | ruclem9.6 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
32 | 30, 31 | ffvelrnd 6905 |
. . . . . 6
⊢ (𝜑 → (𝐺‘𝑀) ∈ (ℝ ×
ℝ)) |
33 | | xp1st 7793 |
. . . . . 6
⊢ ((𝐺‘𝑀) ∈ (ℝ × ℝ) →
(1st ‘(𝐺‘𝑀)) ∈ ℝ) |
34 | 32, 33 | syl 17 |
. . . . 5
⊢ (𝜑 → (1st
‘(𝐺‘𝑀)) ∈
ℝ) |
35 | 34 | leidd 11398 |
. . . 4
⊢ (𝜑 → (1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘𝑀))) |
36 | | xp2nd 7794 |
. . . . . 6
⊢ ((𝐺‘𝑀) ∈ (ℝ × ℝ) →
(2nd ‘(𝐺‘𝑀)) ∈ ℝ) |
37 | 32, 36 | syl 17 |
. . . . 5
⊢ (𝜑 → (2nd
‘(𝐺‘𝑀)) ∈
ℝ) |
38 | 37 | leidd 11398 |
. . . 4
⊢ (𝜑 → (2nd
‘(𝐺‘𝑀)) ≤ (2nd
‘(𝐺‘𝑀))) |
39 | 35, 38 | jca 515 |
. . 3
⊢ (𝜑 → ((1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘𝑀)) ∧ (2nd
‘(𝐺‘𝑀)) ≤ (2nd
‘(𝐺‘𝑀)))) |
40 | 26 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝐹:ℕ⟶ℝ) |
41 | 27 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝐷 = (𝑥 ∈ (ℝ × ℝ), 𝑦 ∈ ℝ ↦
⦋(((1st ‘𝑥) + (2nd ‘𝑥)) / 2) / 𝑚⦌if(𝑚 < 𝑦, 〈(1st ‘𝑥), 𝑚〉, 〈((𝑚 + (2nd ‘𝑥)) / 2), (2nd ‘𝑥)〉))) |
42 | 30 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝐺:ℕ0⟶(ℝ ×
ℝ)) |
43 | | eluznn0 12513 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℕ0
∧ 𝑛 ∈
(ℤ≥‘𝑀)) → 𝑛 ∈ ℕ0) |
44 | 31, 43 | sylan 583 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ ℕ0) |
45 | 42, 44 | ffvelrnd 6905 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑛) ∈ (ℝ ×
ℝ)) |
46 | | xp1st 7793 |
. . . . . . . . . . 11
⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) →
(1st ‘(𝐺‘𝑛)) ∈ ℝ) |
47 | 45, 46 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (1st
‘(𝐺‘𝑛)) ∈
ℝ) |
48 | | xp2nd 7794 |
. . . . . . . . . . 11
⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) →
(2nd ‘(𝐺‘𝑛)) ∈ ℝ) |
49 | 45, 48 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (2nd
‘(𝐺‘𝑛)) ∈
ℝ) |
50 | | nn0p1nn 12129 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ0
→ (𝑛 + 1) ∈
ℕ) |
51 | 44, 50 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝑛 + 1) ∈ ℕ) |
52 | 40, 51 | ffvelrnd 6905 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐹‘(𝑛 + 1)) ∈ ℝ) |
53 | | eqid 2737 |
. . . . . . . . . 10
⊢
(1st ‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) = (1st
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) |
54 | | eqid 2737 |
. . . . . . . . . 10
⊢
(2nd ‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) = (2nd
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) |
55 | 26, 27, 28, 29 | ruclem8 15798 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
(1st ‘(𝐺‘𝑛)) < (2nd ‘(𝐺‘𝑛))) |
56 | 44, 55 | syldan 594 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (1st
‘(𝐺‘𝑛)) < (2nd
‘(𝐺‘𝑛))) |
57 | 40, 41, 47, 49, 52, 53, 54, 56 | ruclem2 15793 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((1st
‘(𝐺‘𝑛)) ≤ (1st
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) ∧ (1st
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) < (2nd
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) ∧ (2nd
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) ≤ (2nd ‘(𝐺‘𝑛)))) |
58 | 57 | simp1d 1144 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (1st
‘(𝐺‘𝑛)) ≤ (1st
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1))))) |
59 | 26, 27, 28, 29 | ruclem7 15797 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) |
60 | 44, 59 | syldan 594 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1)))) |
61 | | 1st2nd2 7800 |
. . . . . . . . . . . 12
⊢ ((𝐺‘𝑛) ∈ (ℝ × ℝ) →
(𝐺‘𝑛) = 〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉) |
62 | 45, 61 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑛) = 〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉) |
63 | 62 | oveq1d 7228 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝐺‘𝑛)𝐷(𝐹‘(𝑛 + 1))) = (〈(1st
‘(𝐺‘𝑛)), (2nd
‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) |
64 | 60, 63 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘(𝑛 + 1)) = (〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) |
65 | 64 | fveq2d 6721 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (1st
‘(𝐺‘(𝑛 + 1))) = (1st
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1))))) |
66 | 58, 65 | breqtrrd 5081 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (1st
‘(𝐺‘𝑛)) ≤ (1st
‘(𝐺‘(𝑛 + 1)))) |
67 | 34 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (1st
‘(𝐺‘𝑀)) ∈
ℝ) |
68 | | peano2nn0 12130 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
→ (𝑛 + 1) ∈
ℕ0) |
69 | 44, 68 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝑛 + 1) ∈
ℕ0) |
70 | 42, 69 | ffvelrnd 6905 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (𝐺‘(𝑛 + 1)) ∈ (ℝ ×
ℝ)) |
71 | | xp1st 7793 |
. . . . . . . . 9
⊢ ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ)
→ (1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ) |
72 | 70, 71 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (1st
‘(𝐺‘(𝑛 + 1))) ∈
ℝ) |
73 | | letr 10926 |
. . . . . . . 8
⊢
(((1st ‘(𝐺‘𝑀)) ∈ ℝ ∧ (1st
‘(𝐺‘𝑛)) ∈ ℝ ∧
(1st ‘(𝐺‘(𝑛 + 1))) ∈ ℝ) →
(((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑛)) ∧ (1st ‘(𝐺‘𝑛)) ≤ (1st ‘(𝐺‘(𝑛 + 1)))) → (1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))))) |
74 | 67, 47, 72, 73 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (((1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘𝑛)) ∧ (1st
‘(𝐺‘𝑛)) ≤ (1st
‘(𝐺‘(𝑛 + 1)))) → (1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘(𝑛 + 1))))) |
75 | 66, 74 | mpan2d 694 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘𝑛)) → (1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘(𝑛 + 1))))) |
76 | 64 | fveq2d 6721 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (2nd
‘(𝐺‘(𝑛 + 1))) = (2nd
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1))))) |
77 | 57 | simp3d 1146 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (2nd
‘(〈(1st ‘(𝐺‘𝑛)), (2nd ‘(𝐺‘𝑛))〉𝐷(𝐹‘(𝑛 + 1)))) ≤ (2nd ‘(𝐺‘𝑛))) |
78 | 76, 77 | eqbrtrd 5075 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (2nd
‘(𝐺‘(𝑛 + 1))) ≤ (2nd
‘(𝐺‘𝑛))) |
79 | | xp2nd 7794 |
. . . . . . . . 9
⊢ ((𝐺‘(𝑛 + 1)) ∈ (ℝ × ℝ)
→ (2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ) |
80 | 70, 79 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (2nd
‘(𝐺‘(𝑛 + 1))) ∈
ℝ) |
81 | 37 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (2nd
‘(𝐺‘𝑀)) ∈
ℝ) |
82 | | letr 10926 |
. . . . . . . 8
⊢
(((2nd ‘(𝐺‘(𝑛 + 1))) ∈ ℝ ∧ (2nd
‘(𝐺‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐺‘𝑀)) ∈ ℝ) → (((2nd
‘(𝐺‘(𝑛 + 1))) ≤ (2nd
‘(𝐺‘𝑛)) ∧ (2nd
‘(𝐺‘𝑛)) ≤ (2nd
‘(𝐺‘𝑀))) → (2nd
‘(𝐺‘(𝑛 + 1))) ≤ (2nd
‘(𝐺‘𝑀)))) |
83 | 80, 49, 81, 82 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (((2nd
‘(𝐺‘(𝑛 + 1))) ≤ (2nd
‘(𝐺‘𝑛)) ∧ (2nd
‘(𝐺‘𝑛)) ≤ (2nd
‘(𝐺‘𝑀))) → (2nd
‘(𝐺‘(𝑛 + 1))) ≤ (2nd
‘(𝐺‘𝑀)))) |
84 | 78, 83 | mpand 695 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((2nd
‘(𝐺‘𝑛)) ≤ (2nd
‘(𝐺‘𝑀)) → (2nd
‘(𝐺‘(𝑛 + 1))) ≤ (2nd
‘(𝐺‘𝑀)))) |
85 | 75, 84 | anim12d 612 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → (((1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘𝑛)) ∧ (2nd
‘(𝐺‘𝑛)) ≤ (2nd
‘(𝐺‘𝑀))) → ((1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘(𝑛 + 1))) ∧ (2nd
‘(𝐺‘(𝑛 + 1))) ≤ (2nd
‘(𝐺‘𝑀))))) |
86 | 85 | expcom 417 |
. . . 4
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝜑 → (((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑛)) ∧ (2nd ‘(𝐺‘𝑛)) ≤ (2nd ‘(𝐺‘𝑀))) → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺‘𝑀)))))) |
87 | 86 | a2d 29 |
. . 3
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑛)) ∧ (2nd ‘(𝐺‘𝑛)) ≤ (2nd ‘(𝐺‘𝑀)))) → (𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘(𝑛 + 1))) ∧ (2nd ‘(𝐺‘(𝑛 + 1))) ≤ (2nd ‘(𝐺‘𝑀)))))) |
88 | 7, 13, 19, 25, 39, 87 | uzind4i 12506 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → ((1st ‘(𝐺‘𝑀)) ≤ (1st ‘(𝐺‘𝑁)) ∧ (2nd ‘(𝐺‘𝑁)) ≤ (2nd ‘(𝐺‘𝑀))))) |
89 | 1, 88 | mpcom 38 |
1
⊢ (𝜑 → ((1st
‘(𝐺‘𝑀)) ≤ (1st
‘(𝐺‘𝑁)) ∧ (2nd
‘(𝐺‘𝑁)) ≤ (2nd
‘(𝐺‘𝑀)))) |