Step |
Hyp |
Ref |
Expression |
1 |
|
evpmsubg.s |
⊢ 𝑆 = ( SymGrp ‘ 𝐷 ) |
2 |
|
evpmsubg.a |
⊢ 𝐴 = ( pmEven ‘ 𝐷 ) |
3 |
2
|
evpmval |
⊢ ( 𝐷 ∈ Fin → 𝐴 = ( ◡ ( pmSgn ‘ 𝐷 ) “ { 1 } ) ) |
4 |
|
eqid |
⊢ ( pmSgn ‘ 𝐷 ) = ( pmSgn ‘ 𝐷 ) |
5 |
|
eqid |
⊢ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) = ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) |
6 |
1 4 5
|
psgnghm2 |
⊢ ( 𝐷 ∈ Fin → ( pmSgn ‘ 𝐷 ) ∈ ( 𝑆 GrpHom ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) ) |
7 |
5
|
cnmsgngrp |
⊢ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ∈ Grp |
8 |
5
|
cnmsgn0g |
⊢ 1 = ( 0g ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) |
9 |
8
|
0subg |
⊢ ( ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ∈ Grp → { 1 } ∈ ( SubGrp ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) ) |
10 |
7 9
|
ax-mp |
⊢ { 1 } ∈ ( SubGrp ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) |
11 |
|
ghmpreima |
⊢ ( ( ( pmSgn ‘ 𝐷 ) ∈ ( 𝑆 GrpHom ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) ∧ { 1 } ∈ ( SubGrp ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) ) → ( ◡ ( pmSgn ‘ 𝐷 ) “ { 1 } ) ∈ ( SubGrp ‘ 𝑆 ) ) |
12 |
6 10 11
|
sylancl |
⊢ ( 𝐷 ∈ Fin → ( ◡ ( pmSgn ‘ 𝐷 ) “ { 1 } ) ∈ ( SubGrp ‘ 𝑆 ) ) |
13 |
3 12
|
eqeltrd |
⊢ ( 𝐷 ∈ Fin → 𝐴 ∈ ( SubGrp ‘ 𝑆 ) ) |