Step |
Hyp |
Ref |
Expression |
1 |
|
zlmlem2.1 |
⊢ 𝑊 = ( ℤMod ‘ 𝐺 ) |
2 |
|
zlm1.1 |
⊢ 1 = ( 1r ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
4 |
3
|
a1i |
⊢ ( ⊤ → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) ) |
5 |
1 3
|
zlmbas |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝑊 ) |
6 |
5
|
a1i |
⊢ ( ⊤ → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑊 ) ) |
7 |
|
eqid |
⊢ ( .r ‘ 𝐺 ) = ( .r ‘ 𝐺 ) |
8 |
1 7
|
zlmmulr |
⊢ ( .r ‘ 𝐺 ) = ( .r ‘ 𝑊 ) |
9 |
8
|
a1i |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( .r ‘ 𝐺 ) = ( .r ‘ 𝑊 ) ) |
10 |
9
|
oveqd |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( .r ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝑊 ) 𝑦 ) ) |
11 |
4 6 10
|
rngidpropd |
⊢ ( ⊤ → ( 1r ‘ 𝐺 ) = ( 1r ‘ 𝑊 ) ) |
12 |
11
|
mptru |
⊢ ( 1r ‘ 𝐺 ) = ( 1r ‘ 𝑊 ) |
13 |
2 12
|
eqtri |
⊢ 1 = ( 1r ‘ 𝑊 ) |