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Metamath Proof Explorer

Theorem symgextsymg

Description: The extension of a permutation is an element of the extended symmetric group. (Contributed by AV, 9-Mar-2019)

Ref Expression
Hypotheses symgext.s ⊢ 𝑆 = ( Base ‘ ( SymGrp ‘ ( 𝑁 ∖ { 𝐾 } ) ) )
symgext.e ⊢ 𝐸 = ( 𝑥 ∈ 𝑁 ↦ if ( 𝑥 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑥 ) ) )
Assertion symgextsymg ( ( 𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → 𝐸 ∈ ( Base ‘ ( SymGrp ‘ 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 symgext.s ⊢ 𝑆 = ( Base ‘ ( SymGrp ‘ ( 𝑁 ∖ { 𝐾 } ) ) )
2 symgext.e ⊢ 𝐸 = ( 𝑥 ∈ 𝑁 ↦ if ( 𝑥 = 𝐾 , 𝐾 , ( 𝑍 ‘ 𝑥 ) ) )
3 1 2 symgextf1o ⊢ ( ( 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → 𝐸 : 𝑁 –1-1-onto→ 𝑁 )
4 3 3adant1 ⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → 𝐸 : 𝑁 –1-1-onto→ 𝑁 )
5 eqid ⊢ ( SymGrp ‘ 𝑁 ) = ( SymGrp ‘ 𝑁 )
6 eqid ⊢ ( Base ‘ ( SymGrp ‘ 𝑁 ) ) = ( Base ‘ ( SymGrp ‘ 𝑁 ) )
7 5 6 elsymgbas ⊢ ( 𝑁 ∈ 𝑉 → ( 𝐸 ∈ ( Base ‘ ( SymGrp ‘ 𝑁 ) ) ↔ 𝐸 : 𝑁 –1-1-onto→ 𝑁 ) )
8 7 3ad2ant1 ⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → ( 𝐸 ∈ ( Base ‘ ( SymGrp ‘ 𝑁 ) ) ↔ 𝐸 : 𝑁 –1-1-onto→ 𝑁 ) )
9 4 8 mpbird ⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆 ) → 𝐸 ∈ ( Base ‘ ( SymGrp ‘ 𝑁 ) ) )