Description: Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020)
Ref | Expression | ||
---|---|---|---|
Hypothesis | submgmss.b | ⊢ 𝐵 = ( Base ‘ 𝑀 ) | |
Assertion | submgmss | ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑆 ⊆ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submgmss.b | ⊢ 𝐵 = ( Base ‘ 𝑀 ) | |
2 | submgmrcl | ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑀 ∈ Mgm ) | |
3 | eqid | ⊢ ( 𝑀 ↾s 𝑆 ) = ( 𝑀 ↾s 𝑆 ) | |
4 | 1 3 | issubmgm2 | ⊢ ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mgm ) ) ) |
5 | 2 4 | syl | ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mgm ) ) ) |
6 | 5 | ibi | ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ⊆ 𝐵 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mgm ) ) |
7 | 6 | simpld | ⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑆 ⊆ 𝐵 ) |