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 ) ) )