Theorem scmatel

Description: An N x N scalar matrix over (a ring) R . (Contributed by AV, 18-Dec-2019)

	Ref	Expression
Hypotheses	scmatval.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	scmatval.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
	scmatval.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	scmatval.1	⊢ 1 = ( 1r ‘ 𝐴 )
	scmatval.t	⊢ · = ( ·𝑠 ‘ 𝐴 )
	scmatval.s	⊢ 𝑆 = ( 𝑁 ScMat 𝑅 )
Assertion	scmatel	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝑆 ↔ ( 𝑀 ∈ 𝐵 ∧ ∃ 𝑐 ∈ 𝐾 𝑀 = ( 𝑐 · 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		scmatval.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
2		scmatval.a	⊢ 𝐴 = ( 𝑁 Mat 𝑅 )
3		scmatval.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
4		scmatval.1	⊢ 1 = ( 1r ‘ 𝐴 )
5		scmatval.t	⊢ · = ( ·𝑠 ‘ 𝐴 )
6		scmatval.s	⊢ 𝑆 = ( 𝑁 ScMat 𝑅 )
7	1 2 3 4 5 6	scmatval	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝑆 = { 𝑚 ∈ 𝐵 ∣ ∃ 𝑐 ∈ 𝐾 𝑚 = ( 𝑐 · 1 ) } )
8	7	eleq2d	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝑆 ↔ 𝑀 ∈ { 𝑚 ∈ 𝐵 ∣ ∃ 𝑐 ∈ 𝐾 𝑚 = ( 𝑐 · 1 ) } ) )
9		eqeq1	⊢ ( 𝑚 = 𝑀 → ( 𝑚 = ( 𝑐 · 1 ) ↔ 𝑀 = ( 𝑐 · 1 ) ) )
10	9	rexbidv	⊢ ( 𝑚 = 𝑀 → ( ∃ 𝑐 ∈ 𝐾 𝑚 = ( 𝑐 · 1 ) ↔ ∃ 𝑐 ∈ 𝐾 𝑀 = ( 𝑐 · 1 ) ) )
11	10	elrab	⊢ ( 𝑀 ∈ { 𝑚 ∈ 𝐵 ∣ ∃ 𝑐 ∈ 𝐾 𝑚 = ( 𝑐 · 1 ) } ↔ ( 𝑀 ∈ 𝐵 ∧ ∃ 𝑐 ∈ 𝐾 𝑀 = ( 𝑐 · 1 ) ) )
12	8 11	bitrdi	⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝑆 ↔ ( 𝑀 ∈ 𝐵 ∧ ∃ 𝑐 ∈ 𝐾 𝑀 = ( 𝑐 · 1 ) ) ) )