Metamath Proof Explorer


Theorem psgnvalii

Description: Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015)

Ref Expression
Hypotheses psgnval.g 𝐺 = ( SymGrp ‘ 𝐷 )
psgnval.t 𝑇 = ran ( pmTrsp ‘ 𝐷 )
psgnval.n 𝑁 = ( pmSgn ‘ 𝐷 )
Assertion psgnvalii ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( 𝑁 ‘ ( 𝐺 Σg 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) )

Proof

Step Hyp Ref Expression
1 psgnval.g 𝐺 = ( SymGrp ‘ 𝐷 )
2 psgnval.t 𝑇 = ran ( pmTrsp ‘ 𝐷 )
3 psgnval.n 𝑁 = ( pmSgn ‘ 𝐷 )
4 1 2 3 psgneldm2i ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( 𝐺 Σg 𝑊 ) ∈ dom 𝑁 )
5 1 2 3 psgnval ( ( 𝐺 Σg 𝑊 ) ∈ dom 𝑁 → ( 𝑁 ‘ ( 𝐺 Σg 𝑊 ) ) = ( ℩ 𝑠𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) )
6 4 5 syl ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( 𝑁 ‘ ( 𝐺 Σg 𝑊 ) ) = ( ℩ 𝑠𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) )
7 simpr ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → 𝑊 ∈ Word 𝑇 )
8 eqidd ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑊 ) )
9 eqidd ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) )
10 oveq2 ( 𝑤 = 𝑊 → ( 𝐺 Σg 𝑤 ) = ( 𝐺 Σg 𝑊 ) )
11 10 eqeq2d ( 𝑤 = 𝑊 → ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ↔ ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑊 ) ) )
12 fveq2 ( 𝑤 = 𝑊 → ( ♯ ‘ 𝑤 ) = ( ♯ ‘ 𝑊 ) )
13 12 oveq2d ( 𝑤 = 𝑊 → ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) )
14 13 eqeq2d ( 𝑤 = 𝑊 → ( ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ↔ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) )
15 11 14 anbi12d ( 𝑤 = 𝑊 → ( ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ↔ ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑊 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) ) )
16 15 rspcev ( ( 𝑊 ∈ Word 𝑇 ∧ ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑊 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) ) → ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) )
17 7 8 9 16 syl12anc ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) )
18 ovexd ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ∈ V )
19 1 2 3 psgneu ( ( 𝐺 Σg 𝑊 ) ∈ dom 𝑁 → ∃! 𝑠𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) )
20 4 19 syl ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ∃! 𝑠𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) )
21 eqeq1 ( 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) → ( 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ↔ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) )
22 21 anbi2d ( 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) → ( ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ↔ ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) )
23 22 rexbidv ( 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) → ( ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ↔ ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) )
24 23 adantl ( ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) → ( ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ↔ ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) )
25 18 20 24 iota2d ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ↔ ( ℩ 𝑠𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) )
26 17 25 mpbid ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( ℩ 𝑠𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) )
27 6 26 eqtrd ( ( 𝐷𝑉𝑊 ∈ Word 𝑇 ) → ( 𝑁 ‘ ( 𝐺 Σg 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) )