Theorem symg1hash

Description: The symmetric group on a singleton has cardinality 1 . (Contributed by AV, 9-Dec-2018)

	Ref	Expression
Hypotheses	symg1bas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
	symg1bas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
	symg1bas.0	⊢ 𝐴 = { 𝐼 }
Assertion	symg1hash	⊢ ( 𝐼 ∈ 𝑉 → ( ♯ ‘ 𝐵 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		symg1bas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2		symg1bas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
3		symg1bas.0	⊢ 𝐴 = { 𝐼 }
4		snfi	⊢ { 𝐼 } ∈ Fin
5	3 4	eqeltri	⊢ 𝐴 ∈ Fin
6	1 2	symghash	⊢ ( 𝐴 ∈ Fin → ( ♯ ‘ 𝐵 ) = ( ! ‘ ( ♯ ‘ 𝐴 ) ) )
7	5 6	ax-mp	⊢ ( ♯ ‘ 𝐵 ) = ( ! ‘ ( ♯ ‘ 𝐴 ) )
8	3	fveq2i	⊢ ( ♯ ‘ 𝐴 ) = ( ♯ ‘ { 𝐼 } )
9		hashsng	⊢ ( 𝐼 ∈ 𝑉 → ( ♯ ‘ { 𝐼 } ) = 1 )
10	8 9	eqtrid	⊢ ( 𝐼 ∈ 𝑉 → ( ♯ ‘ 𝐴 ) = 1 )
11	10	fveq2d	⊢ ( 𝐼 ∈ 𝑉 → ( ! ‘ ( ♯ ‘ 𝐴 ) ) = ( ! ‘ 1 ) )
12		fac1	⊢ ( ! ‘ 1 ) = 1
13	11 12	eqtrdi	⊢ ( 𝐼 ∈ 𝑉 → ( ! ‘ ( ♯ ‘ 𝐴 ) ) = 1 )
14	7 13	eqtrid	⊢ ( 𝐼 ∈ 𝑉 → ( ♯ ‘ 𝐵 ) = 1 )