Step | Hyp | Ref
| Expression |
1 | | zre 12510 |
. . . . 5
⊢ (𝑗 ∈ ℤ → 𝑗 ∈
ℝ) |
2 | | simplr 768 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑗 ∈ ℝ) |
3 | | fzunt1d.l |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐿 ∈ ℤ) |
4 | 3 | ad2antrr 725 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝐿 ∈ ℤ) |
5 | 4 | zred 12614 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝐿 ∈ ℝ) |
6 | | fzunt1d.n |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | 6 | ad2antrr 725 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑁 ∈ ℤ) |
8 | 7 | zred 12614 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑁 ∈ ℝ) |
9 | | simpr 486 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑗 ≤ 𝐿) |
10 | | fzunt1d.ln |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐿 ≤ 𝑁) |
11 | 10 | ad2antrr 725 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝐿 ≤ 𝑁) |
12 | 2, 5, 8, 9, 11 | letrd 11319 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑗 ≤ 𝑁) |
13 | 12 | ex 414 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (𝑗 ≤ 𝐿 → 𝑗 ≤ 𝑁)) |
14 | 13 | anim2d 613 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
15 | | fzunt1d.k |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ ℤ) |
16 | 15 | ad2antrr 725 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝐾 ∈ ℤ) |
17 | 16 | zred 12614 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝐾 ∈ ℝ) |
18 | | fzunt1d.m |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) |
19 | 18 | ad2antrr 725 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ∈ ℤ) |
20 | 19 | zred 12614 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ∈ ℝ) |
21 | | simplr 768 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℝ) |
22 | | fzunt1d.km |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ≤ 𝑀) |
23 | 22 | ad2antrr 725 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝐾 ≤ 𝑀) |
24 | | simpr 486 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ≤ 𝑗) |
25 | 17, 20, 21, 23, 24 | letrd 11319 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝐾 ≤ 𝑗) |
26 | 25 | ex 414 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (𝑀 ≤ 𝑗 → 𝐾 ≤ 𝑗)) |
27 | 26 | anim1d 612 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → ((𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
28 | 14, 27 | jaod 858 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
29 | | orc 866 |
. . . . . . . . . 10
⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗)) |
30 | | orc 866 |
. . . . . . . . . 10
⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)) |
31 | 29, 30 | jca 513 |
. . . . . . . . 9
⊢ (𝐾 ≤ 𝑗 → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))) |
32 | 31 | ad2antrl 727 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))) |
33 | | simpr 486 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → 𝑗 ∈ ℝ) |
34 | 3 | adantr 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → 𝐿 ∈ ℤ) |
35 | 34 | zred 12614 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → 𝐿 ∈ ℝ) |
36 | 9 | orcd 872 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → (𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗)) |
37 | 18 | ad2antrr 725 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑀 ∈ ℤ) |
38 | 37 | zred 12614 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑀 ∈ ℝ) |
39 | 3 | ad2antrr 725 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝐿 ∈ ℤ) |
40 | 39 | zred 12614 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝐿 ∈ ℝ) |
41 | | simplr 768 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑗 ∈ ℝ) |
42 | | fzunt1d.ml |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ≤ 𝐿) |
43 | 42 | ad2antrr 725 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑀 ≤ 𝐿) |
44 | | simpr 486 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝐿 ≤ 𝑗) |
45 | 38, 40, 41, 43, 44 | letrd 11319 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑀 ≤ 𝑗) |
46 | 45 | olcd 873 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → (𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗)) |
47 | 33, 35, 36, 46 | lecasei 11268 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗)) |
48 | 47 | adantr 482 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗)) |
49 | | simprr 772 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → 𝑗 ≤ 𝑁) |
50 | 49 | olcd 873 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝐿 ∨ 𝑗 ≤ 𝑁)) |
51 | 48, 50 | jca 513 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝐿 ∨ 𝑗 ≤ 𝑁))) |
52 | | orddi 1009 |
. . . . . . . 8
⊢ (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)) ∧ ((𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝐿 ∨ 𝑗 ≤ 𝑁)))) |
53 | 32, 51, 52 | sylanbrc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
54 | 53 | ex 414 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
55 | 28, 54 | impbid 211 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
56 | 1, 55 | sylan2 594 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
57 | 56 | pm5.32da 580 |
. . 3
⊢ (𝜑 → ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
58 | | elfz1 13436 |
. . . . . . 7
⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑗 ∈ (𝐾...𝐿) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿))) |
59 | 15, 3, 58 | syl2anc 585 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝐿) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿))) |
60 | | 3anass 1096 |
. . . . . 6
⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿))) |
61 | 59, 60 | bitrdi 287 |
. . . . 5
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝐿) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿)))) |
62 | | elfz1 13436 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
63 | 18, 6, 62 | syl2anc 585 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
64 | | 3anass 1096 |
. . . . . 6
⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
65 | 63, 64 | bitrdi 287 |
. . . . 5
⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
66 | 61, 65 | orbi12d 918 |
. . . 4
⊢ (𝜑 → ((𝑗 ∈ (𝐾...𝐿) ∨ 𝑗 ∈ (𝑀...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))) |
67 | | elun 4113 |
. . . 4
⊢ (𝑗 ∈ ((𝐾...𝐿) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ (𝐾...𝐿) ∨ 𝑗 ∈ (𝑀...𝑁))) |
68 | | andi 1007 |
. . . 4
⊢ ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
69 | 66, 67, 68 | 3bitr4g 314 |
. . 3
⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝐿) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))) |
70 | | elfz1 13436 |
. . . . 5
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
71 | 15, 6, 70 | syl2anc 585 |
. . . 4
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
72 | | 3anass 1096 |
. . . 4
⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
73 | 71, 72 | bitrdi 287 |
. . 3
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
74 | 57, 69, 73 | 3bitr4d 311 |
. 2
⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝐿) ∪ (𝑀...𝑁)) ↔ 𝑗 ∈ (𝐾...𝑁))) |
75 | 74 | eqrdv 2735 |
1
⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) |