| Step | Hyp | Ref
| Expression |
| 1 | | simprl 771 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℤ) |
| 2 | 1 | zred 12722 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℝ) |
| 3 | | fzuntd.m |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 4 | 3 | zred 12722 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 5 | 4 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑀 ∈ ℝ) |
| 6 | | fzuntd.n |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 7 | 6 | zred 12722 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℝ) |
| 8 | 7 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑁 ∈ ℝ) |
| 9 | | simprr 773 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑀) |
| 10 | | fzuntd.mn |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ≤ 𝑁) |
| 11 | 10 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑀 ≤ 𝑁) |
| 12 | 2, 5, 8, 9, 11 | letrd 11418 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑁) |
| 13 | 12 | expr 456 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 → 𝑗 ≤ 𝑁)) |
| 14 | 13 | anim2d 612 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 15 | | fzuntd.k |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ ℤ) |
| 16 | 15 | zred 12722 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ ℝ) |
| 17 | 16 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ∈ ℝ) |
| 18 | 4 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑀 ∈ ℝ) |
| 19 | | simprl 771 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑗 ∈ ℤ) |
| 20 | 19 | zred 12722 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑗 ∈ ℝ) |
| 21 | | fzuntd.km |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ≤ 𝑀) |
| 22 | 21 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑀) |
| 23 | | simprr 773 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑀 ≤ 𝑗) |
| 24 | 17, 18, 20, 22, 23 | letrd 11418 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑗) |
| 25 | 24 | expr 456 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑀 ≤ 𝑗 → 𝐾 ≤ 𝑗)) |
| 26 | 25 | anim1d 611 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 27 | 14, 26 | jaod 860 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 28 | | orc 868 |
. . . . . . . . 9
⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗)) |
| 29 | | orc 868 |
. . . . . . . . 9
⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)) |
| 30 | 28, 29 | jca 511 |
. . . . . . . 8
⊢ (𝐾 ≤ 𝑗 → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))) |
| 31 | 30 | ad2antrl 728 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))) |
| 32 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ) |
| 33 | 32 | zred 12722 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ) |
| 34 | 4 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑀 ∈ ℝ) |
| 35 | 33, 34 | letrid 11413 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗)) |
| 36 | 35 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗)) |
| 37 | | simprr 773 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → 𝑗 ≤ 𝑁) |
| 38 | 37 | olcd 875 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁)) |
| 39 | 36, 38 | jca 511 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁))) |
| 40 | | orddi 1012 |
. . . . . . 7
⊢ (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)) ∧ ((𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁)))) |
| 41 | 31, 39, 40 | sylanbrc 583 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 42 | 41 | ex 412 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
| 43 | 27, 42 | impbid 212 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 44 | 43 | pm5.32da 579 |
. . 3
⊢ (𝜑 → ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
| 45 | | elfz1 13552 |
. . . . . . 7
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀))) |
| 46 | 15, 3, 45 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀))) |
| 47 | | 3anass 1095 |
. . . . . 6
⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀))) |
| 48 | 46, 47 | bitrdi 287 |
. . . . 5
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))) |
| 49 | | elfz1 13552 |
. . . . . . 7
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 50 | 3, 6, 49 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 51 | | 3anass 1095 |
. . . . . 6
⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 52 | 50, 51 | bitrdi 287 |
. . . . 5
⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
| 53 | 48, 52 | orbi12d 919 |
. . . 4
⊢ (𝜑 → ((𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))) |
| 54 | | elun 4153 |
. . . 4
⊢ (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁))) |
| 55 | | andi 1010 |
. . . 4
⊢ ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
| 56 | 53, 54, 55 | 3bitr4g 314 |
. . 3
⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))) |
| 57 | | elfz1 13552 |
. . . . 5
⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 58 | 15, 6, 57 | syl2anc 584 |
. . . 4
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 59 | | 3anass 1095 |
. . . 4
⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) |
| 60 | 58, 59 | bitrdi 287 |
. . 3
⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))) |
| 61 | 44, 56, 60 | 3bitr4d 311 |
. 2
⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ 𝑗 ∈ (𝐾...𝑁))) |
| 62 | 61 | eqrdv 2735 |
1
⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁)) |