Step |
Hyp |
Ref |
Expression |
1 |
|
divsval |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → ( 𝐴 /su 𝐵 ) = ( ℩ 𝑦 ∈ No ( 𝐵 ·s 𝑦 ) = 𝐴 ) ) |
2 |
1
|
adantr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) ∧ ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s ) → ( 𝐴 /su 𝐵 ) = ( ℩ 𝑦 ∈ No ( 𝐵 ·s 𝑦 ) = 𝐴 ) ) |
3 |
|
3anrot |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) ↔ ( 𝐵 ∈ No ∧ 𝐵 ≠ 0s ∧ 𝐴 ∈ No ) ) |
4 |
|
noreceuw |
⊢ ( ( ( 𝐵 ∈ No ∧ 𝐵 ≠ 0s ∧ 𝐴 ∈ No ) ∧ ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s ) → ∃! 𝑦 ∈ No ( 𝐵 ·s 𝑦 ) = 𝐴 ) |
5 |
3 4
|
sylanb |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) ∧ ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s ) → ∃! 𝑦 ∈ No ( 𝐵 ·s 𝑦 ) = 𝐴 ) |
6 |
|
riotacl |
⊢ ( ∃! 𝑦 ∈ No ( 𝐵 ·s 𝑦 ) = 𝐴 → ( ℩ 𝑦 ∈ No ( 𝐵 ·s 𝑦 ) = 𝐴 ) ∈ No ) |
7 |
5 6
|
syl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) ∧ ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s ) → ( ℩ 𝑦 ∈ No ( 𝐵 ·s 𝑦 ) = 𝐴 ) ∈ No ) |
8 |
2 7
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) ∧ ∃ 𝑥 ∈ No ( 𝐵 ·s 𝑥 ) = 1s ) → ( 𝐴 /su 𝐵 ) ∈ No ) |