# Metamath Proof Explorer

## Theorem issubmgm

Description: Expand definition of a submagma. (Contributed by AV, 25-Feb-2020)

Ref Expression
Hypotheses issubmgm.b 𝐵 = ( Base ‘ 𝑀 )
issubmgm.p + = ( +g𝑀 )
Assertion issubmgm ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆𝐵 ∧ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )

### Proof

Step Hyp Ref Expression
1 issubmgm.b 𝐵 = ( Base ‘ 𝑀 )
2 issubmgm.p + = ( +g𝑀 )
3 fveq2 ( 𝑚 = 𝑀 → ( Base ‘ 𝑚 ) = ( Base ‘ 𝑀 ) )
4 3 pweqd ( 𝑚 = 𝑀 → 𝒫 ( Base ‘ 𝑚 ) = 𝒫 ( Base ‘ 𝑀 ) )
5 fveq2 ( 𝑚 = 𝑀 → ( +g𝑚 ) = ( +g𝑀 ) )
6 5 oveqd ( 𝑚 = 𝑀 → ( 𝑥 ( +g𝑚 ) 𝑦 ) = ( 𝑥 ( +g𝑀 ) 𝑦 ) )
7 6 eleq1d ( 𝑚 = 𝑀 → ( ( 𝑥 ( +g𝑚 ) 𝑦 ) ∈ 𝑡 ↔ ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 ) )
8 7 2ralbidv ( 𝑚 = 𝑀 → ( ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑚 ) 𝑦 ) ∈ 𝑡 ↔ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 ) )
9 4 8 rabeqbidv ( 𝑚 = 𝑀 → { 𝑡 ∈ 𝒫 ( Base ‘ 𝑚 ) ∣ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑚 ) 𝑦 ) ∈ 𝑡 } = { 𝑡 ∈ 𝒫 ( Base ‘ 𝑀 ) ∣ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 } )
10 df-submgm SubMgm = ( 𝑚 ∈ Mgm ↦ { 𝑡 ∈ 𝒫 ( Base ‘ 𝑚 ) ∣ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑚 ) 𝑦 ) ∈ 𝑡 } )
11 fvex ( Base ‘ 𝑀 ) ∈ V
12 11 pwex 𝒫 ( Base ‘ 𝑀 ) ∈ V
13 12 rabex { 𝑡 ∈ 𝒫 ( Base ‘ 𝑀 ) ∣ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 } ∈ V
14 9 10 13 fvmpt ( 𝑀 ∈ Mgm → ( SubMgm ‘ 𝑀 ) = { 𝑡 ∈ 𝒫 ( Base ‘ 𝑀 ) ∣ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 } )
15 14 eleq2d ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ 𝑆 ∈ { 𝑡 ∈ 𝒫 ( Base ‘ 𝑀 ) ∣ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 } ) )
16 11 elpw2 ( 𝑆 ∈ 𝒫 ( Base ‘ 𝑀 ) ↔ 𝑆 ⊆ ( Base ‘ 𝑀 ) )
17 16 anbi1i ( ( 𝑆 ∈ 𝒫 ( Base ‘ 𝑀 ) ∧ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 ) )
18 eleq2 ( 𝑡 = 𝑆 → ( ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 ↔ ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 ) )
19 18 raleqbi1dv ( 𝑡 = 𝑆 → ( ∀ 𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 ↔ ∀ 𝑦𝑆 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 ) )
20 19 raleqbi1dv ( 𝑡 = 𝑆 → ( ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 ↔ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 ) )
21 20 elrab ( 𝑆 ∈ { 𝑡 ∈ 𝒫 ( Base ‘ 𝑀 ) ∣ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 } ↔ ( 𝑆 ∈ 𝒫 ( Base ‘ 𝑀 ) ∧ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 ) )
22 1 sseq2i ( 𝑆𝐵𝑆 ⊆ ( Base ‘ 𝑀 ) )
23 2 oveqi ( 𝑥 + 𝑦 ) = ( 𝑥 ( +g𝑀 ) 𝑦 )
24 23 eleq1i ( ( 𝑥 + 𝑦 ) ∈ 𝑆 ↔ ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 )
25 24 2ralbii ( ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ↔ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 )
26 22 25 anbi12i ( ( 𝑆𝐵 ∧ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑆 ) )
27 17 21 26 3bitr4i ( 𝑆 ∈ { 𝑡 ∈ 𝒫 ( Base ‘ 𝑀 ) ∣ ∀ 𝑥𝑡𝑦𝑡 ( 𝑥 ( +g𝑀 ) 𝑦 ) ∈ 𝑡 } ↔ ( 𝑆𝐵 ∧ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) )
28 15 27 syl6bb ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆𝐵 ∧ ∀ 𝑥𝑆𝑦𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )