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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 ⊢ + = ( +g ‘ 𝑀 )
Assertion issubmgm ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )

Proof

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