Step | Hyp | Ref
| Expression |
1 | | monoord.1 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
2 | | eluzfz2 13273 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
4 | | eleq1 2827 |
. . . . . 6
⊢ (𝑥 = 𝑀 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑀 ∈ (𝑀...𝑁))) |
5 | | fveq2 6783 |
. . . . . . 7
⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) |
6 | 5 | breq2d 5087 |
. . . . . 6
⊢ (𝑥 = 𝑀 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑀))) |
7 | 4, 6 | imbi12d 345 |
. . . . 5
⊢ (𝑥 = 𝑀 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀)))) |
8 | 7 | imbi2d 341 |
. . . 4
⊢ (𝑥 = 𝑀 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀))))) |
9 | | eleq1 2827 |
. . . . . 6
⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑛 ∈ (𝑀...𝑁))) |
10 | | fveq2 6783 |
. . . . . . 7
⊢ (𝑥 = 𝑛 → (𝐹‘𝑥) = (𝐹‘𝑛)) |
11 | 10 | breq2d 5087 |
. . . . . 6
⊢ (𝑥 = 𝑛 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑛))) |
12 | 9, 11 | imbi12d 345 |
. . . . 5
⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛)))) |
13 | 12 | imbi2d 341 |
. . . 4
⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))))) |
14 | | eleq1 2827 |
. . . . . 6
⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝑀...𝑁) ↔ (𝑛 + 1) ∈ (𝑀...𝑁))) |
15 | | fveq2 6783 |
. . . . . . 7
⊢ (𝑥 = (𝑛 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑛 + 1))) |
16 | 15 | breq2d 5087 |
. . . . . 6
⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))) |
17 | 14, 16 | imbi12d 345 |
. . . . 5
⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1))))) |
18 | 17 | imbi2d 341 |
. . . 4
⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))))) |
19 | | eleq1 2827 |
. . . . . 6
⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑁 ∈ (𝑀...𝑁))) |
20 | | fveq2 6783 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (𝐹‘𝑥) = (𝐹‘𝑁)) |
21 | 20 | breq2d 5087 |
. . . . . 6
⊢ (𝑥 = 𝑁 → ((𝐹‘𝑀) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑀) ≤ (𝐹‘𝑁))) |
22 | 19, 21 | imbi12d 345 |
. . . . 5
⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥)) ↔ (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁)))) |
23 | 22 | imbi2d 341 |
. . . 4
⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁))))) |
24 | | fveq2 6783 |
. . . . . . . 8
⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) |
25 | 24 | eleq1d 2824 |
. . . . . . 7
⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑀) ∈ ℝ)) |
26 | | monoord.2 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) |
27 | 26 | ralrimiva 3104 |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ) |
28 | | eluzfz1 13272 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) |
29 | 1, 28 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁)) |
30 | 25, 27, 29 | rspcdva 3563 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ) |
31 | 30 | leidd 11550 |
. . . . 5
⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑀)) |
32 | 31 | a1d 25 |
. . . 4
⊢ (𝜑 → (𝑀 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑀))) |
33 | | peano2fzr 13278 |
. . . . . . . 8
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → 𝑛 ∈ (𝑀...𝑁)) |
34 | 33 | adantl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...𝑁)) |
35 | 34 | expr 457 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑛 ∈ (𝑀...𝑁))) |
36 | 35 | imim1d 82 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → ((𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛)) → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛)))) |
37 | | fveq2 6783 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) |
38 | | fvoveq1 7307 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑛 + 1))) |
39 | 37, 38 | breq12d 5088 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1)))) |
40 | | monoord.3 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
41 | 40 | ralrimiva 3104 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
42 | 41 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...(𝑁 − 1))(𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
43 | | simprl 768 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀)) |
44 | | eluzelz 12601 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → 𝑛 ∈ ℤ) |
45 | 43, 44 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ ℤ) |
46 | | simprr 770 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑛 + 1) ∈ (𝑀...𝑁)) |
47 | | elfzuz3 13262 |
. . . . . . . . . 10
⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) |
48 | 46, 47 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1))) |
49 | | eluzp1m1 12617 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈
(ℤ≥‘(𝑛 + 1))) → (𝑁 − 1) ∈
(ℤ≥‘𝑛)) |
50 | 45, 48, 49 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝑁 − 1) ∈
(ℤ≥‘𝑛)) |
51 | | elfzuzb 13259 |
. . . . . . . 8
⊢ (𝑛 ∈ (𝑀...(𝑁 − 1)) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑛))) |
52 | 43, 50, 51 | sylanbrc 583 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → 𝑛 ∈ (𝑀...(𝑁 − 1))) |
53 | 39, 42, 52 | rspcdva 3563 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))) |
54 | 30 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑀) ∈ ℝ) |
55 | 37 | eleq1d 2824 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑛) ∈ ℝ)) |
56 | 27 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ) |
57 | 55, 56, 34 | rspcdva 3563 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘𝑛) ∈ ℝ) |
58 | | fveq2 6783 |
. . . . . . . . 9
⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1))) |
59 | 58 | eleq1d 2824 |
. . . . . . . 8
⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝑛 + 1)) ∈ ℝ)) |
60 | 59, 56, 46 | rspcdva 3563 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (𝐹‘(𝑛 + 1)) ∈ ℝ) |
61 | | letr 11078 |
. . . . . . 7
⊢ (((𝐹‘𝑀) ∈ ℝ ∧ (𝐹‘𝑛) ∈ ℝ ∧ (𝐹‘(𝑛 + 1)) ∈ ℝ) → (((𝐹‘𝑀) ≤ (𝐹‘𝑛) ∧ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))) |
62 | 54, 57, 60, 61 | syl3anc 1370 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → (((𝐹‘𝑀) ≤ (𝐹‘𝑛) ∧ (𝐹‘𝑛) ≤ (𝐹‘(𝑛 + 1))) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))) |
63 | 53, 62 | mpan2d 691 |
. . . . 5
⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝑛 + 1) ∈ (𝑀...𝑁))) → ((𝐹‘𝑀) ≤ (𝐹‘𝑛) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))) |
64 | 36, 63 | animpimp2impd 843 |
. . . 4
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝜑 → (𝑛 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘(𝑛 + 1)))))) |
65 | 8, 13, 18, 23, 32, 64 | uzind4i 12659 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁)))) |
66 | 1, 65 | mpcom 38 |
. 2
⊢ (𝜑 → (𝑁 ∈ (𝑀...𝑁) → (𝐹‘𝑀) ≤ (𝐹‘𝑁))) |
67 | 3, 66 | mpd 15 |
1
⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁)) |