Theorem matbas

Description: The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015)

	Ref	Expression
Hypotheses	matbas.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	matbas.g	⊢ 𝐺 = ( 𝑅 freeLMod ( 𝑁 × 𝑁 ) )
Assertion	matbas	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		matbas.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
2		matbas.g	⊢ 𝐺 = ( 𝑅 freeLMod ( 𝑁 × 𝑁 ) )
3		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		basendxnmulrndx	⊢ ( Base ‘ ndx ) ≠ ( .r ‘ ndx )
5	3 4	setsnid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ ( 𝐺 sSet ⟨ ( .r ‘ ndx ) , ( 𝑅 maMul ⟨ 𝑁 , 𝑁 , 𝑁 ⟩ ) ⟩ ) )
6		eqid	⊢ ( 𝑅 maMul ⟨ 𝑁 , 𝑁 , 𝑁 ⟩ ) = ( 𝑅 maMul ⟨ 𝑁 , 𝑁 , 𝑁 ⟩ )
7	1 2 6	matval	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐴 = ( 𝐺 sSet ⟨ ( .r ‘ ndx ) , ( 𝑅 maMul ⟨ 𝑁 , 𝑁 , 𝑁 ⟩ ) ⟩ ) )
8	7	fveq2d	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( Base ‘ 𝐴 ) = ( Base ‘ ( 𝐺 sSet ⟨ ( .r ‘ ndx ) , ( 𝑅 maMul ⟨ 𝑁 , 𝑁 , 𝑁 ⟩ ) ⟩ ) ) )
9	5 8	eqtr4id	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐴 ) )