Metamath Proof Explorer

Theorem zrhpsgninv

Description: The embedded sign of a permutation equals the embedded sign of the inverse of the permutation. (Contributed by SO, 9-Jul-2018)

Ref Expression
Hypotheses zrhpsgninv.p ⊢ 𝑃 = ( Base ‘ ( SymGrp ‘ 𝑁 ) )
zrhpsgninv.y ⊢ 𝑌 = ( ℤRHom ‘ 𝑅 )
zrhpsgninv.s ⊢ 𝑆 = ( pmSgn ‘ 𝑁 )
Assertion zrhpsgninv ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( ( 𝑌 ∘ 𝑆 ) ‘ ◡ 𝐹 ) = ( ( 𝑌 ∘ 𝑆 ) ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 zrhpsgninv.p ⊢ 𝑃 = ( Base ‘ ( SymGrp ‘ 𝑁 ) )
2 zrhpsgninv.y ⊢ 𝑌 = ( ℤRHom ‘ 𝑅 )
3 zrhpsgninv.s ⊢ 𝑆 = ( pmSgn ‘ 𝑁 )
4 eqid ⊢ ( SymGrp ‘ 𝑁 ) = ( SymGrp ‘ 𝑁 )
5 4 3 1 psgninv ⊢ ( ( 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( 𝑆 ‘ ◡ 𝐹 ) = ( 𝑆 ‘ 𝐹 ) )
6 5 3adant1 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( 𝑆 ‘ ◡ 𝐹 ) = ( 𝑆 ‘ 𝐹 ) )
7 6 fveq2d ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( 𝑌 ‘ ( 𝑆 ‘ ◡ 𝐹 ) ) = ( 𝑌 ‘ ( 𝑆 ‘ 𝐹 ) ) )
8 eqid ⊢ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) = ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } )
9 4 3 8 psgnghm2 ⊢ ( 𝑁 ∈ Fin → 𝑆 ∈ ( ( SymGrp ‘ 𝑁 ) GrpHom ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) )
10 eqid ⊢ ( Base ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) = ( Base ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) )
11 1 10 ghmf ⊢ ( 𝑆 ∈ ( ( SymGrp ‘ 𝑁 ) GrpHom ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) → 𝑆 : 𝑃 ⟶ ( Base ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) )
12 9 11 syl ⊢ ( 𝑁 ∈ Fin → 𝑆 : 𝑃 ⟶ ( Base ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) )
13 12 3ad2ant2 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → 𝑆 : 𝑃 ⟶ ( Base ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) )
14 eqid ⊢ ( invg ‘ ( SymGrp ‘ 𝑁 ) ) = ( invg ‘ ( SymGrp ‘ 𝑁 ) )
15 4 1 14 symginv ⊢ ( 𝐹 ∈ 𝑃 → ( ( invg ‘ ( SymGrp ‘ 𝑁 ) ) ‘ 𝐹 ) = ◡ 𝐹 )
16 15 3ad2ant3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( ( invg ‘ ( SymGrp ‘ 𝑁 ) ) ‘ 𝐹 ) = ◡ 𝐹 )
17 4 symggrp ⊢ ( 𝑁 ∈ Fin → ( SymGrp ‘ 𝑁 ) ∈ Grp )
18 17 3ad2ant2 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( SymGrp ‘ 𝑁 ) ∈ Grp )
19 simp3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → 𝐹 ∈ 𝑃 )
20 1 14 grpinvcl ⊢ ( ( ( SymGrp ‘ 𝑁 ) ∈ Grp ∧ 𝐹 ∈ 𝑃 ) → ( ( invg ‘ ( SymGrp ‘ 𝑁 ) ) ‘ 𝐹 ) ∈ 𝑃 )
21 18 19 20 syl2anc ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( ( invg ‘ ( SymGrp ‘ 𝑁 ) ) ‘ 𝐹 ) ∈ 𝑃 )
22 16 21 eqeltrrd ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ◡ 𝐹 ∈ 𝑃 )
23 fvco3 ⊢ ( ( 𝑆 : 𝑃 ⟶ ( Base ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) ∧ ◡ 𝐹 ∈ 𝑃 ) → ( ( 𝑌 ∘ 𝑆 ) ‘ ◡ 𝐹 ) = ( 𝑌 ‘ ( 𝑆 ‘ ◡ 𝐹 ) ) )
24 13 22 23 syl2anc ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( ( 𝑌 ∘ 𝑆 ) ‘ ◡ 𝐹 ) = ( 𝑌 ‘ ( 𝑆 ‘ ◡ 𝐹 ) ) )
25 fvco3 ⊢ ( ( 𝑆 : 𝑃 ⟶ ( Base ‘ ( ( mulGrp ‘ ℂfld ) ↾s { 1 , - 1 } ) ) ∧ 𝐹 ∈ 𝑃 ) → ( ( 𝑌 ∘ 𝑆 ) ‘ 𝐹 ) = ( 𝑌 ‘ ( 𝑆 ‘ 𝐹 ) ) )
26 13 19 25 syl2anc ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( ( 𝑌 ∘ 𝑆 ) ‘ 𝐹 ) = ( 𝑌 ‘ ( 𝑆 ‘ 𝐹 ) ) )
27 7 24 26 3eqtr4d ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝐹 ∈ 𝑃 ) → ( ( 𝑌 ∘ 𝑆 ) ‘ ◡ 𝐹 ) = ( ( 𝑌 ∘ 𝑆 ) ‘ 𝐹 ) )