Metamath Proof Explorer


Theorem psgneldm2

Description: The finitary permutations are the span of the transpositions. (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 psgneldm2 ( 𝐷𝑉 → ( 𝑃 ∈ dom 𝑁 ↔ ∃ 𝑤 ∈ Word 𝑇 𝑃 = ( 𝐺 Σg 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 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 ( 𝐷𝑉 → ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ 𝑇 ) = { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } )
10 1 symggrp ( 𝐷𝑉𝐺 ∈ Grp )
11 10 grpmndd ( 𝐷𝑉𝐺 ∈ 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 ( 𝐷𝑉 → ( ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) ‘ 𝑇 ) = ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) )
15 9 14 eqtr3d ( 𝐷𝑉 → { 𝑝 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) )
16 7 15 eqtrid ( 𝐷𝑉 → dom 𝑁 = ran ( 𝑤 ∈ Word 𝑇 ↦ ( 𝐺 Σg 𝑤 ) ) )
17 16 eleq2d ( 𝐷𝑉 → ( 𝑃 ∈ dom 𝑁𝑃 ∈ 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 ( 𝐷𝑉 → ( 𝑃 ∈ dom 𝑁 ↔ ∃ 𝑤 ∈ Word 𝑇 𝑃 = ( 𝐺 Σg 𝑤 ) ) )