Theorem nsmndex1

Description: The base set B of the constructed monoid S is not a submonoid of the monoid M of endofunctions on set NN0 , although M e. Mnd and S e. Mnd and B C_ ( BaseM ) hold. (Contributed by AV, 17-Feb-2024)

	Ref	Expression
Hypotheses	smndex1ibas.m	⊢ 𝑀 = ( EndoFMnd ‘ ℕ0 )
	smndex1ibas.n	⊢ 𝑁 ∈ ℕ
	smndex1ibas.i	⊢ 𝐼 = ( 𝑥 ∈ ℕ0 ↦ ( 𝑥 mod 𝑁 ) )
	smndex1ibas.g	⊢ 𝐺 = ( 𝑛 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝑥 ∈ ℕ0 ↦ 𝑛 ) )
	smndex1mgm.b	⊢ 𝐵 = ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } )
	smndex1mgm.s	⊢ 𝑆 = ( 𝑀 ↾s 𝐵 )
Assertion	nsmndex1	⊢ 𝐵 ∉ ( SubMnd ‘ 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		smndex1ibas.m	⊢ 𝑀 = ( EndoFMnd ‘ ℕ0 )
2		smndex1ibas.n	⊢ 𝑁 ∈ ℕ
3		smndex1ibas.i	⊢ 𝐼 = ( 𝑥 ∈ ℕ0 ↦ ( 𝑥 mod 𝑁 ) )
4		smndex1ibas.g	⊢ 𝐺 = ( 𝑛 ∈ ( 0 ..^ 𝑁 ) ↦ ( 𝑥 ∈ ℕ0 ↦ 𝑛 ) )
5		smndex1mgm.b	⊢ 𝐵 = ( { 𝐼 } ∪ ∪ 𝑛 ∈ ( 0 ..^ 𝑁 ) { ( 𝐺 ‘ 𝑛 ) } )
6		smndex1mgm.s	⊢ 𝑆 = ( 𝑀 ↾s 𝐵 )
7	1 2 3 4 5 6	smndex1n0mnd	⊢ ( 0g ‘ 𝑀 ) ∉ 𝐵
8	7	neli	⊢ ¬ ( 0g ‘ 𝑀 ) ∈ 𝐵
9	8	intnan	⊢ ¬ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 )
10	9	intnan	⊢ ¬ ( ( 𝑀 ∈ Mnd ∧ ( 𝑀 ↾s 𝐵 ) ∈ Mnd ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ) )
11		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
12		eqid	⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 )
13	11 12	issubmndb	⊢ ( 𝐵 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( ( 𝑀 ∈ Mnd ∧ ( 𝑀 ↾s 𝐵 ) ∈ Mnd ) ∧ ( 𝐵 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝐵 ) ) )
14	10 13	mtbir	⊢ ¬ 𝐵 ∈ ( SubMnd ‘ 𝑀 )
15	14	nelir	⊢ 𝐵 ∉ ( SubMnd ‘ 𝑀 )