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 |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ( 𝐺 Σg 𝑊 ) ∈ dom 𝑁 ) |
5 |
1 2 3
|
psgnval |
⊢ ( ( 𝐺 Σg 𝑊 ) ∈ dom 𝑁 → ( 𝑁 ‘ ( 𝐺 Σg 𝑊 ) ) = ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) |
6 |
4 5
|
syl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ( 𝑁 ‘ ( 𝐺 Σg 𝑊 ) ) = ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) |
7 |
|
simpr |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → 𝑊 ∈ Word 𝑇 ) |
8 |
|
eqidd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑊 ) ) |
9 |
|
eqidd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ 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 |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) |
18 |
|
ovexd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ∈ V ) |
19 |
1 2 3
|
psgneu |
⊢ ( ( 𝐺 Σg 𝑊 ) ∈ dom 𝑁 → ∃! 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) |
20 |
4 19
|
syl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ∃! 𝑠 ∃ 𝑤 ∈ 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 |
⊢ ( ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ 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 |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ( ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ↔ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) ) |
26 |
17 25
|
mpbid |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) |
27 |
6 26
|
eqtrd |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇 ) → ( 𝑁 ‘ ( 𝐺 Σg 𝑊 ) ) = ( - 1 ↑ ( ♯ ‘ 𝑊 ) ) ) |