Step |
Hyp |
Ref |
Expression |
1 |
|
quorem.1 |
⊢ 𝑄 = ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) |
2 |
|
quorem.2 |
⊢ 𝑅 = ( 𝐴 − ( 𝐵 · 𝑄 ) ) |
3 |
|
fldivnn0 |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℕ0 ) |
4 |
1 3
|
eqeltrid |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → 𝑄 ∈ ℕ0 ) |
5 |
|
nn0z |
⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ ) |
6 |
1 2
|
quoremz |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℕ ) → ( ( 𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0 ) ∧ ( 𝑅 < 𝐵 ∧ 𝐴 = ( ( 𝐵 · 𝑄 ) + 𝑅 ) ) ) ) |
7 |
5 6
|
sylan |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0 ) ∧ ( 𝑅 < 𝐵 ∧ 𝐴 = ( ( 𝐵 · 𝑄 ) + 𝑅 ) ) ) ) |
8 |
|
simpl |
⊢ ( ( 𝑄 ∈ ℕ0 ∧ 𝑄 ∈ ℤ ) → 𝑄 ∈ ℕ0 ) |
9 |
8
|
anim1i |
⊢ ( ( ( 𝑄 ∈ ℕ0 ∧ 𝑄 ∈ ℤ ) ∧ 𝑅 ∈ ℕ0 ) → ( 𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ) ) |
10 |
9
|
anasss |
⊢ ( ( 𝑄 ∈ ℕ0 ∧ ( 𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0 ) ) → ( 𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ) ) |
11 |
10
|
anim1i |
⊢ ( ( ( 𝑄 ∈ ℕ0 ∧ ( 𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0 ) ) ∧ ( 𝑅 < 𝐵 ∧ 𝐴 = ( ( 𝐵 · 𝑄 ) + 𝑅 ) ) ) → ( ( 𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ) ∧ ( 𝑅 < 𝐵 ∧ 𝐴 = ( ( 𝐵 · 𝑄 ) + 𝑅 ) ) ) ) |
12 |
11
|
anasss |
⊢ ( ( 𝑄 ∈ ℕ0 ∧ ( ( 𝑄 ∈ ℤ ∧ 𝑅 ∈ ℕ0 ) ∧ ( 𝑅 < 𝐵 ∧ 𝐴 = ( ( 𝐵 · 𝑄 ) + 𝑅 ) ) ) ) → ( ( 𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ) ∧ ( 𝑅 < 𝐵 ∧ 𝐴 = ( ( 𝐵 · 𝑄 ) + 𝑅 ) ) ) ) |
13 |
4 7 12
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 𝑄 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ) ∧ ( 𝑅 < 𝐵 ∧ 𝐴 = ( ( 𝐵 · 𝑄 ) + 𝑅 ) ) ) ) |