Proof of Theorem quoremz
| Step | Hyp | Ref
| Expression |
| 1 | | quorem.1 |
. . 3
⊢ 𝑄 = (⌊‘(𝐴 / 𝐵)) |
| 2 | | zre 12617 |
. . . . . 6
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℝ) |
| 3 | 2 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈
ℝ) |
| 4 | | nnre 12273 |
. . . . . 6
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℝ) |
| 5 | 4 | adantl 481 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈
ℝ) |
| 6 | | nnne0 12300 |
. . . . . 6
⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) |
| 7 | 6 | adantl 481 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ≠ 0) |
| 8 | 3, 5, 7 | redivcld 12095 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ∈ ℝ) |
| 9 | 8 | flcld 13838 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) →
(⌊‘(𝐴 / 𝐵)) ∈
ℤ) |
| 10 | 1, 9 | eqeltrid 2845 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑄 ∈
ℤ) |
| 11 | | quorem.2 |
. . 3
⊢ 𝑅 = (𝐴 − (𝐵 · 𝑄)) |
| 12 | 10 | zcnd 12723 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑄 ∈
ℂ) |
| 13 | | nncn 12274 |
. . . . . . . 8
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℂ) |
| 14 | 13 | adantl 481 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈
ℂ) |
| 15 | 12, 14, 7 | divcan3d 12048 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐵 · 𝑄) / 𝐵) = 𝑄) |
| 16 | | flle 13839 |
. . . . . . . 8
⊢ ((𝐴 / 𝐵) ∈ ℝ →
(⌊‘(𝐴 / 𝐵)) ≤ (𝐴 / 𝐵)) |
| 17 | 8, 16 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) →
(⌊‘(𝐴 / 𝐵)) ≤ (𝐴 / 𝐵)) |
| 18 | 1, 17 | eqbrtrid 5178 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑄 ≤ (𝐴 / 𝐵)) |
| 19 | 15, 18 | eqbrtrd 5165 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐵 · 𝑄) / 𝐵) ≤ (𝐴 / 𝐵)) |
| 20 | | nnz 12634 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℤ) |
| 21 | 20 | adantl 481 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈
ℤ) |
| 22 | 21, 10 | zmulcld 12728 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 · 𝑄) ∈ ℤ) |
| 23 | 22 | zred 12722 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 · 𝑄) ∈ ℝ) |
| 24 | | nngt0 12297 |
. . . . . . 7
⊢ (𝐵 ∈ ℕ → 0 <
𝐵) |
| 25 | 24 | adantl 481 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 0 <
𝐵) |
| 26 | | lediv1 12133 |
. . . . . 6
⊢ (((𝐵 · 𝑄) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → ((𝐵 · 𝑄) ≤ 𝐴 ↔ ((𝐵 · 𝑄) / 𝐵) ≤ (𝐴 / 𝐵))) |
| 27 | 23, 3, 5, 25, 26 | syl112anc 1376 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐵 · 𝑄) ≤ 𝐴 ↔ ((𝐵 · 𝑄) / 𝐵) ≤ (𝐴 / 𝐵))) |
| 28 | 19, 27 | mpbird 257 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 · 𝑄) ≤ 𝐴) |
| 29 | | simpl 482 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈
ℤ) |
| 30 | | znn0sub 12664 |
. . . . 5
⊢ (((𝐵 · 𝑄) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐵 · 𝑄) ≤ 𝐴 ↔ (𝐴 − (𝐵 · 𝑄)) ∈
ℕ0)) |
| 31 | 22, 29, 30 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐵 · 𝑄) ≤ 𝐴 ↔ (𝐴 − (𝐵 · 𝑄)) ∈
ℕ0)) |
| 32 | 28, 31 | mpbid 232 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐴 − (𝐵 · 𝑄)) ∈
ℕ0) |
| 33 | 11, 32 | eqeltrid 2845 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑅 ∈
ℕ0) |
| 34 | 1 | oveq2i 7442 |
. . . . . 6
⊢ ((𝐴 / 𝐵) − 𝑄) = ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) |
| 35 | | fraclt1 13842 |
. . . . . . 7
⊢ ((𝐴 / 𝐵) ∈ ℝ → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < 1) |
| 36 | 8, 35 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) − (⌊‘(𝐴 / 𝐵))) < 1) |
| 37 | 34, 36 | eqbrtrid 5178 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) − 𝑄) < 1) |
| 38 | 11 | oveq1i 7441 |
. . . . . 6
⊢ (𝑅 / 𝐵) = ((𝐴 − (𝐵 · 𝑄)) / 𝐵) |
| 39 | | zcn 12618 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℂ) |
| 40 | 39 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈
ℂ) |
| 41 | 22 | zcnd 12723 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 · 𝑄) ∈ ℂ) |
| 42 | 13, 6 | jca 511 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℕ → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
| 43 | 42 | adantl 481 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
| 44 | | divsubdir 11961 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 · 𝑄) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → ((𝐴 − (𝐵 · 𝑄)) / 𝐵) = ((𝐴 / 𝐵) − ((𝐵 · 𝑄) / 𝐵))) |
| 45 | 40, 41, 43, 44 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 − (𝐵 · 𝑄)) / 𝐵) = ((𝐴 / 𝐵) − ((𝐵 · 𝑄) / 𝐵))) |
| 46 | 15 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) − ((𝐵 · 𝑄) / 𝐵)) = ((𝐴 / 𝐵) − 𝑄)) |
| 47 | 45, 46 | eqtrd 2777 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐴 − (𝐵 · 𝑄)) / 𝐵) = ((𝐴 / 𝐵) − 𝑄)) |
| 48 | 38, 47 | eqtrid 2789 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝑅 / 𝐵) = ((𝐴 / 𝐵) − 𝑄)) |
| 49 | 13, 6 | dividd 12041 |
. . . . . 6
⊢ (𝐵 ∈ ℕ → (𝐵 / 𝐵) = 1) |
| 50 | 49 | adantl 481 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝐵 / 𝐵) = 1) |
| 51 | 37, 48, 50 | 3brtr4d 5175 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝑅 / 𝐵) < (𝐵 / 𝐵)) |
| 52 | 33 | nn0red 12588 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑅 ∈
ℝ) |
| 53 | | ltdiv1 12132 |
. . . . 5
⊢ ((𝑅 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 <
𝐵)) → (𝑅 < 𝐵 ↔ (𝑅 / 𝐵) < (𝐵 / 𝐵))) |
| 54 | 52, 5, 5, 25, 53 | syl112anc 1376 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝑅 < 𝐵 ↔ (𝑅 / 𝐵) < (𝐵 / 𝐵))) |
| 55 | 51, 54 | mpbird 257 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝑅 < 𝐵) |
| 56 | 11 | oveq2i 7442 |
. . . 4
⊢ ((𝐵 · 𝑄) + 𝑅) = ((𝐵 · 𝑄) + (𝐴 − (𝐵 · 𝑄))) |
| 57 | 41, 40 | pncan3d 11623 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝐵 · 𝑄) + (𝐴 − (𝐵 · 𝑄))) = 𝐴) |
| 58 | 56, 57 | eqtr2id 2790 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → 𝐴 = ((𝐵 · 𝑄) + 𝑅)) |
| 59 | 55, 58 | jca 511 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → (𝑅 < 𝐵 ∧ 𝐴 = ((𝐵 · 𝑄) + 𝑅))) |
| 60 | 10, 33, 59 | jca31 514 |
1
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ) → ((𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0)
∧ (𝑅 < 𝐵 ∧ 𝐴 = ((𝐵 · 𝑄) + 𝑅)))) |