Step |
Hyp |
Ref |
Expression |
1 |
|
psgnval.g |
⊢ 𝐺 = ( SymGrp ‘ 𝐷 ) |
2 |
|
psgnval.t |
⊢ 𝑇 = ran ( pmTrsp ‘ 𝐷 ) |
3 |
|
psgnval.n |
⊢ 𝑁 = ( pmSgn ‘ 𝐷 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
5 |
|
eqid |
⊢ { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } |
6 |
1 4 5 3
|
psgnfn |
⊢ 𝑁 Fn { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } |
7 |
6
|
fndmi |
⊢ dom 𝑁 = { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } |
8 |
|
eqid |
⊢ ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) = ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) |
9 |
2 1 4 8
|
symggen |
⊢ ( 𝐷 ∈ 𝑉 → ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ 𝑇 ) = { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } ) |
10 |
1
|
symggrp |
⊢ ( 𝐷 ∈ 𝑉 → 𝐺 ∈ Grp ) |
11 |
10
|
grpmndd |
⊢ ( 𝐷 ∈ 𝑉 → 𝐺 ∈ Mnd ) |
12 |
2 1 4
|
symgtrf |
⊢ 𝑇 ⊆ ( Base ‘ 𝐺 ) |
13 |
4 8
|
gsumwspan |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑇 ⊆ ( Base ‘ 𝐺 ) ) → ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ 𝑇 ) = ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) ) |
14 |
11 12 13
|
sylancl |
⊢ ( 𝐷 ∈ 𝑉 → ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ 𝑇 ) = ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) ) |
15 |
9 14
|
eqtr3d |
⊢ ( 𝐷 ∈ 𝑉 → { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) ) |
16 |
7 15
|
eqtrid |
⊢ ( 𝐷 ∈ 𝑉 → dom 𝑁 = ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) ) |
17 |
16
|
eleq2d |
⊢ ( 𝐷 ∈ 𝑉 → ( 𝑃 ∈ dom 𝑁 ↔ 𝑃 ∈ ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) ) ) |
18 |
|
eqid |
⊢ ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) = ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) |
19 |
|
ovex |
⊢ ( 𝐺 Σg 𝑤 ) ∈ V |
20 |
18 19
|
elrnmpti |
⊢ ( 𝑃 ∈ ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) ↔ ∃ 𝑤 ∈ Word 𝑇 𝑃 = ( 𝐺 Σg 𝑤 ) ) |
21 |
17 20
|
bitrdi |
⊢ ( 𝐷 ∈ 𝑉 → ( 𝑃 ∈ dom 𝑁 ↔ ∃ 𝑤 ∈ Word 𝑇 𝑃 = ( 𝐺 Σg 𝑤 ) ) ) |