Description: Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015)
Ref | Expression | ||
---|---|---|---|
Hypothesis | submss.b | ⊢ 𝐵 = ( Base ‘ 𝑀 ) | |
Assertion | submss | ⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → 𝑆 ⊆ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submss.b | ⊢ 𝐵 = ( Base ‘ 𝑀 ) | |
2 | submrcl | ⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → 𝑀 ∈ Mnd ) | |
3 | eqid | ⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) | |
4 | eqid | ⊢ ( 𝑀 ↾s 𝑆 ) = ( 𝑀 ↾s 𝑆 ) | |
5 | 1 3 4 | issubm2 | ⊢ ( 𝑀 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ( 0g ‘ 𝑀 ) ∈ 𝑆 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mnd ) ) ) |
6 | 2 5 | syl | ⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ( 0g ‘ 𝑀 ) ∈ 𝑆 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mnd ) ) ) |
7 | 6 | ibi | ⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝑆 ⊆ 𝐵 ∧ ( 0g ‘ 𝑀 ) ∈ 𝑆 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mnd ) ) |
8 | 7 | simp1d | ⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → 𝑆 ⊆ 𝐵 ) |