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Metamath Proof Explorer

Theorem symgcl

Description: The group operation of the symmetric group on A is closed, i.e. a magma. (Contributed by Mario Carneiro, 12-Jan-2015) (Revised by Mario Carneiro, 28-Jan-2015)

Ref Expression
Hypotheses symgov.1 ⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
symgov.2 ⊢ 𝐵 = ( Base ‘ 𝐺 )
symgov.3 ⊢ + = ( +g ‘ 𝐺 )
Assertion symgcl ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 symgov.1 ⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2 symgov.2 ⊢ 𝐵 = ( Base ‘ 𝐺 )
3 symgov.3 ⊢ + = ( +g ‘ 𝐺 )
4 1 2 3 symgov ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑋 ∘ 𝑌 ) )
5 1 2 symgbasf1o ⊢ ( 𝑋 ∈ 𝐵 → 𝑋 : 𝐴 –1-1-onto→ 𝐴 )
6 1 2 symgbasf1o ⊢ ( 𝑌 ∈ 𝐵 → 𝑌 : 𝐴 –1-1-onto→ 𝐴 )
7 f1oco ⊢ ( ( 𝑋 : 𝐴 –1-1-onto→ 𝐴 ∧ 𝑌 : 𝐴 –1-1-onto→ 𝐴 ) → ( 𝑋 ∘ 𝑌 ) : 𝐴 –1-1-onto→ 𝐴 )
8 5 6 7 syl2an ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∘ 𝑌 ) : 𝐴 –1-1-onto→ 𝐴 )
9 coexg ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∘ 𝑌 ) ∈ V )
10 1 2 elsymgbas2 ⊢ ( ( 𝑋 ∘ 𝑌 ) ∈ V → ( ( 𝑋 ∘ 𝑌 ) ∈ 𝐵 ↔ ( 𝑋 ∘ 𝑌 ) : 𝐴 –1-1-onto→ 𝐴 ) )
11 9 10 syl ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∘ 𝑌 ) ∈ 𝐵 ↔ ( 𝑋 ∘ 𝑌 ) : 𝐴 –1-1-onto→ 𝐴 ) )
12 8 11 mpbird ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∘ 𝑌 ) ∈ 𝐵 )
13 4 12 eqeltrd ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )