Theorem lcoval

Description: The value of a linear combination. (Contributed by AV, 5-Apr-2019) (Revised by AV, 28-Jul-2019)

	Ref	Expression
Hypotheses	lcoop.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
	lcoop.s	⊢ 𝑆 = ( Scalar ‘ 𝑀 )
	lcoop.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
Assertion	lcoval	⊢ ( ( 𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐶 ∈ ( 𝑀 LinCo 𝑉 ) ↔ ( 𝐶 ∈ 𝐵 ∧ ∃ 𝑠 ∈ ( 𝑅 ↑m 𝑉 ) ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝐶 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lcoop.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
2		lcoop.s	⊢ 𝑆 = ( Scalar ‘ 𝑀 )
3		lcoop.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
4	1 2 3	lcoop	⊢ ( ( 𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝑀 LinCo 𝑉 ) = { 𝑐 ∈ 𝐵 ∣ ∃ 𝑠 ∈ ( 𝑅 ↑m 𝑉 ) ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝑐 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) } )
5	4	eleq2d	⊢ ( ( 𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐶 ∈ ( 𝑀 LinCo 𝑉 ) ↔ 𝐶 ∈ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑠 ∈ ( 𝑅 ↑m 𝑉 ) ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝑐 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) } ) )
6		eqeq1	⊢ ( 𝑐 = 𝐶 → ( 𝑐 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ↔ 𝐶 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) )
7	6	anbi2d	⊢ ( 𝑐 = 𝐶 → ( ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝑐 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) ↔ ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝐶 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) ) )
8	7	rexbidv	⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑠 ∈ ( 𝑅 ↑m 𝑉 ) ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝑐 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) ↔ ∃ 𝑠 ∈ ( 𝑅 ↑m 𝑉 ) ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝐶 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) ) )
9	8	elrab	⊢ ( 𝐶 ∈ { 𝑐 ∈ 𝐵 ∣ ∃ 𝑠 ∈ ( 𝑅 ↑m 𝑉 ) ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝑐 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) } ↔ ( 𝐶 ∈ 𝐵 ∧ ∃ 𝑠 ∈ ( 𝑅 ↑m 𝑉 ) ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝐶 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) ) )
10	5 9	bitrdi	⊢ ( ( 𝑀 ∈ 𝑋 ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐶 ∈ ( 𝑀 LinCo 𝑉 ) ↔ ( 𝐶 ∈ 𝐵 ∧ ∃ 𝑠 ∈ ( 𝑅 ↑m 𝑉 ) ( 𝑠 finSupp ( 0g ‘ 𝑆 ) ∧ 𝐶 = ( 𝑠 ( linC ‘ 𝑀 ) 𝑉 ) ) ) ) )