Step |
Hyp |
Ref |
Expression |
1 |
|
qrng.q |
⊢ 𝑄 = ( ℂfld ↾s ℚ ) |
2 |
|
qsubdrg |
⊢ ( ℚ ∈ ( SubRing ‘ ℂfld ) ∧ ( ℂfld ↾s ℚ ) ∈ DivRing ) |
3 |
2
|
simpli |
⊢ ℚ ∈ ( SubRing ‘ ℂfld ) |
4 |
|
subrgsubg |
⊢ ( ℚ ∈ ( SubRing ‘ ℂfld ) → ℚ ∈ ( SubGrp ‘ ℂfld ) ) |
5 |
3 4
|
ax-mp |
⊢ ℚ ∈ ( SubGrp ‘ ℂfld ) |
6 |
|
eqid |
⊢ ( invg ‘ ℂfld ) = ( invg ‘ ℂfld ) |
7 |
|
eqid |
⊢ ( invg ‘ 𝑄 ) = ( invg ‘ 𝑄 ) |
8 |
1 6 7
|
subginv |
⊢ ( ( ℚ ∈ ( SubGrp ‘ ℂfld ) ∧ 𝑋 ∈ ℚ ) → ( ( invg ‘ ℂfld ) ‘ 𝑋 ) = ( ( invg ‘ 𝑄 ) ‘ 𝑋 ) ) |
9 |
5 8
|
mpan |
⊢ ( 𝑋 ∈ ℚ → ( ( invg ‘ ℂfld ) ‘ 𝑋 ) = ( ( invg ‘ 𝑄 ) ‘ 𝑋 ) ) |
10 |
|
qcn |
⊢ ( 𝑋 ∈ ℚ → 𝑋 ∈ ℂ ) |
11 |
|
cnfldneg |
⊢ ( 𝑋 ∈ ℂ → ( ( invg ‘ ℂfld ) ‘ 𝑋 ) = - 𝑋 ) |
12 |
10 11
|
syl |
⊢ ( 𝑋 ∈ ℚ → ( ( invg ‘ ℂfld ) ‘ 𝑋 ) = - 𝑋 ) |
13 |
9 12
|
eqtr3d |
⊢ ( 𝑋 ∈ ℚ → ( ( invg ‘ 𝑄 ) ‘ 𝑋 ) = - 𝑋 ) |