Proof of Theorem fzind
Step | Hyp | Ref
| Expression |
1 | | breq1 3990 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑀 → (𝑥 ≤ 𝑁 ↔ 𝑀 ≤ 𝑁)) |
2 | 1 | anbi2d 461 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑀 → ((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))) |
3 | | fzind.1 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑀 → (𝜑 ↔ 𝜓)) |
4 | 2, 3 | imbi12d 233 |
. . . . . . . . 9
⊢ (𝑥 = 𝑀 → (((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) → 𝜑) ↔ ((𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓))) |
5 | | breq1 3990 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑥 ≤ 𝑁 ↔ 𝑦 ≤ 𝑁)) |
6 | 5 | anbi2d 461 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁))) |
7 | | fzind.2 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
8 | 6, 7 | imbi12d 233 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) → 𝜑) ↔ ((𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁) → 𝜒))) |
9 | | breq1 3990 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑦 + 1) → (𝑥 ≤ 𝑁 ↔ (𝑦 + 1) ≤ 𝑁)) |
10 | 9 | anbi2d 461 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑦 + 1) → ((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁))) |
11 | | fzind.3 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜃)) |
12 | 10, 11 | imbi12d 233 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 + 1) → (((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) → 𝜑) ↔ ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜃))) |
13 | | breq1 3990 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐾 → (𝑥 ≤ 𝑁 ↔ 𝐾 ≤ 𝑁)) |
14 | 13 | anbi2d 461 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐾 → ((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝐾 ≤ 𝑁))) |
15 | | fzind.4 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐾 → (𝜑 ↔ 𝜏)) |
16 | 14, 15 | imbi12d 233 |
. . . . . . . . 9
⊢ (𝑥 = 𝐾 → (((𝑁 ∈ ℤ ∧ 𝑥 ≤ 𝑁) → 𝜑) ↔ ((𝑁 ∈ ℤ ∧ 𝐾 ≤ 𝑁) → 𝜏))) |
17 | | fzind.5 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓) |
18 | 17 | 3expib 1201 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → ((𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → 𝜓)) |
19 | | zre 9205 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℤ → 𝑦 ∈
ℝ) |
20 | | zre 9205 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
ℝ) |
21 | | p1le 8754 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑦 + 1) ≤ 𝑁) → 𝑦 ≤ 𝑁) |
22 | 21 | 3expia 1200 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑁 → 𝑦 ≤ 𝑁)) |
23 | 19, 20, 22 | syl2an 287 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑦 + 1) ≤ 𝑁 → 𝑦 ≤ 𝑁)) |
24 | 23 | imdistanda 446 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℤ → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → (𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁))) |
25 | 24 | imim1d 75 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℤ → (((𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁) → 𝜒) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜒))) |
26 | 25 | 3ad2ant2 1014 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (((𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁) → 𝜒) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜒))) |
27 | | zltp1le 9255 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑦 < 𝑁 ↔ (𝑦 + 1) ≤ 𝑁)) |
28 | 27 | adantlr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑁 ∈ ℤ) → (𝑦 < 𝑁 ↔ (𝑦 + 1) ≤ 𝑁)) |
29 | 28 | expcom 115 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℤ → ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (𝑦 < 𝑁 ↔ (𝑦 + 1) ≤ 𝑁))) |
30 | 29 | pm5.32d 447 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℤ → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑦 < 𝑁) ↔ ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁))) |
31 | 30 | adantl 275 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑦 < 𝑁) ↔ ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁))) |
32 | | fzind.6 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁)) → (𝜒 → 𝜃)) |
33 | 32 | expcom 115 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦 ∧ 𝑦 < 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝜒 → 𝜃))) |
34 | 33 | 3expa 1198 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑦 < 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝜒 → 𝜃))) |
35 | 34 | com12 30 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ 𝑦 < 𝑁) → (𝜒 → 𝜃))) |
36 | 31, 35 | sylbird 169 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁) → (𝜒 → 𝜃))) |
37 | 36 | ex 114 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁) → (𝜒 → 𝜃)))) |
38 | 37 | com23 78 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈ ℤ → (((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) ∧ (𝑦 + 1) ≤ 𝑁) → (𝑁 ∈ ℤ → (𝜒 → 𝜃)))) |
39 | 38 | expd 256 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ℤ → ((𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → ((𝑦 + 1) ≤ 𝑁 → (𝑁 ∈ ℤ → (𝜒 → 𝜃))))) |
40 | 39 | 3impib 1196 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → ((𝑦 + 1) ≤ 𝑁 → (𝑁 ∈ ℤ → (𝜒 → 𝜃)))) |
41 | 40 | com23 78 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (𝑁 ∈ ℤ → ((𝑦 + 1) ≤ 𝑁 → (𝜒 → 𝜃)))) |
42 | 41 | impd 252 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → (𝜒 → 𝜃))) |
43 | 42 | a2d 26 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜒) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜃))) |
44 | 26, 43 | syld 45 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑀 ≤ 𝑦) → (((𝑁 ∈ ℤ ∧ 𝑦 ≤ 𝑁) → 𝜒) → ((𝑁 ∈ ℤ ∧ (𝑦 + 1) ≤ 𝑁) → 𝜃))) |
45 | 4, 8, 12, 16, 18, 44 | uzind 9312 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → ((𝑁 ∈ ℤ ∧ 𝐾 ≤ 𝑁) → 𝜏)) |
46 | 45 | expcomd 1434 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → (𝐾 ≤ 𝑁 → (𝑁 ∈ ℤ → 𝜏))) |
47 | 46 | 3expb 1199 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾)) → (𝐾 ≤ 𝑁 → (𝑁 ∈ ℤ → 𝜏))) |
48 | 47 | expcom 115 |
. . . . 5
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → (𝑀 ∈ ℤ → (𝐾 ≤ 𝑁 → (𝑁 ∈ ℤ → 𝜏)))) |
49 | 48 | com23 78 |
. . . 4
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾) → (𝐾 ≤ 𝑁 → (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → 𝜏)))) |
50 | 49 | 3impia 1195 |
. . 3
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → 𝜏))) |
51 | 50 | impd 252 |
. 2
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝜏)) |
52 | 51 | impcom 124 |
1
⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → 𝜏) |