| Step | Hyp | Ref
| Expression |
| 1 | | zre 12617 |
. . . . 5
⊢ (𝑗 ∈ ℤ → 𝑗 ∈
ℝ) |
| 2 | | simplr 769 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑗 ∈ ℝ) |
| 3 | | fzunt1d.l |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐿 ∈ ℤ) |
| 4 | 3 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝐿 ∈ ℤ) |
| 5 | 4 | zred 12722 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝐿 ∈ ℝ) |
| 6 | | fzunt1d.n |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 7 | 6 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑁 ∈ ℤ) |
| 8 | 7 | zred 12722 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑁 ∈ ℝ) |
| 9 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑗 ≤ 𝐿) |
| 10 | | fzunt1d.ln |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐿 ≤ 𝑁) |
| 11 | 10 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝐿 ≤ 𝑁) |
| 12 | 2, 5, 8, 9, 11 | letrd 11418 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → 𝑗 ≤ 𝑁) |
| 13 | 12 | ex 412 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (𝑗 ≤ 𝐿 → 𝑗 ≤ 𝑁)) |
| 14 | 13 | anim2d 612 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 15 | | fzunt1d.k |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ ℤ) |
| 16 | 15 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝐾 ∈ ℤ) |
| 17 | 16 | zred 12722 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝐾 ∈ ℝ) |
| 18 | | fzunt1d.m |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 19 | 18 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ∈ ℤ) |
| 20 | 19 | zred 12722 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ∈ ℝ) |
| 21 | | simplr 769 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝑗 ∈ ℝ) |
| 22 | | fzunt1d.km |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ≤ 𝑀) |
| 23 | 22 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝐾 ≤ 𝑀) |
| 24 | | simpr 484 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝑀 ≤ 𝑗) |
| 25 | 17, 20, 21, 23, 24 | letrd 11418 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑀 ≤ 𝑗) → 𝐾 ≤ 𝑗) |
| 26 | 25 | ex 412 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (𝑀 ≤ 𝑗 → 𝐾 ≤ 𝑗)) |
| 27 | 26 | anim1d 611 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → ((𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 28 | 14, 27 | jaod 860 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 29 | | orc 868 |
. . . . . . . . . 10
⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗)) |
| 30 | | orc 868 |
. . . . . . . . . 10
⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)) |
| 31 | 29, 30 | jca 511 |
. . . . . . . . 9
⊢ (𝐾 ≤ 𝑗 → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))) |
| 32 | 31 | ad2antrl 728 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))) |
| 33 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → 𝑗 ∈ ℝ) |
| 34 | 3 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → 𝐿 ∈ ℤ) |
| 35 | 34 | zred 12722 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → 𝐿 ∈ ℝ) |
| 36 | 9 | orcd 874 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝑗 ≤ 𝐿) → (𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗)) |
| 37 | 18 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑀 ∈ ℤ) |
| 38 | 37 | zred 12722 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑀 ∈ ℝ) |
| 39 | 3 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝐿 ∈ ℤ) |
| 40 | 39 | zred 12722 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝐿 ∈ ℝ) |
| 41 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑗 ∈ ℝ) |
| 42 | | fzunt1d.ml |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ≤ 𝐿) |
| 43 | 42 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑀 ≤ 𝐿) |
| 44 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝐿 ≤ 𝑗) |
| 45 | 38, 40, 41, 43, 44 | letrd 11418 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → 𝑀 ≤ 𝑗) |
| 46 | 45 | olcd 875 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ 𝐿 ≤ 𝑗) → (𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗)) |
| 47 | 33, 35, 36, 46 | lecasei 11367 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗)) |
| 48 | 47 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗)) |
| 49 | | simprr 773 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → 𝑗 ≤ 𝑁) |
| 50 | 49 | olcd 875 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝐿 ∨ 𝑗 ≤ 𝑁)) |
| 51 | 48, 50 | jca 511 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝑗 ≤ 𝐿 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝐿 ∨ 𝑗 ≤ 𝑁))) |
| 52 | | orddi 1012 |
. . . . . . . 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 583 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℝ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 54 | 53 | ex 412 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
| 55 | 28, 54 | impbid 212 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℝ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 56 | 1, 55 | sylan2 593 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 57 | 56 | pm5.32da 579 |
. . 3
⊢ (𝜑 → ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
| 58 | | elfz1 13552 |
. . . . . . 7
⊢ ((𝐾 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑗 ∈ (𝐾...𝐿) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿))) |
| 59 | 15, 3, 58 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝐿) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿))) |
| 60 | | 3anass 1095 |
. . . . . 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 13552 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 63 | 18, 6, 62 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 64 | | 3anass 1095 |
. . . . . 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 919 |
. . . 4
⊢ (𝜑 → ((𝑗 ∈ (𝐾...𝐿) ∨ 𝑗 ∈ (𝑀...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝐿)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))) |
| 67 | | elun 4153 |
. . . 4
⊢ (𝑗 ∈ ((𝐾...𝐿) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ (𝐾...𝐿) ∨ 𝑗 ∈ (𝑀...𝑁))) |
| 68 | | andi 1010 |
. . . 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 13552 |
. . . . 5
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 71 | 15, 6, 70 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 72 | | 3anass 1095 |
. . . 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)) |