Theorem baerlem3

Description: An equality that holds when X , Y , Z are independent (non-colinear) vectors. Part (3) in Baer p. 45. TODO fix ref. (Contributed by NM, 9-Apr-2015)

	Ref	Expression
Hypotheses	baerlem3.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	baerlem3.m	⊢ − = ( -g ‘ 𝑊 )
	baerlem3.o	⊢ 0 = ( 0g ‘ 𝑊 )
	baerlem3.s	⊢ ⊕ = ( LSSum ‘ 𝑊 )
	baerlem3.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
	baerlem3.w	⊢ ( 𝜑 → 𝑊 ∈ LVec )
	baerlem3.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	baerlem3.c	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	baerlem3.d	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
	baerlem3.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	baerlem3.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	baerlem3	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 − 𝑍 ) } ) = ( ( ( 𝑁 ‘ { 𝑌 } ) ⊕ ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊕ ( 𝑁 ‘ { ( 𝑋 − 𝑍 ) } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		baerlem3.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		baerlem3.m	⊢ − = ( -g ‘ 𝑊 )
3		baerlem3.o	⊢ 0 = ( 0g ‘ 𝑊 )
4		baerlem3.s	⊢ ⊕ = ( LSSum ‘ 𝑊 )
5		baerlem3.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
6		baerlem3.w	⊢ ( 𝜑 → 𝑊 ∈ LVec )
7		baerlem3.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
8		baerlem3.c	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
9		baerlem3.d	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
10		baerlem3.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
11		baerlem3.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
12		eqid	⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
13		eqid	⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 )
14		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
15		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
16		eqid	⊢ ( +g ‘ ( Scalar ‘ 𝑊 ) ) = ( +g ‘ ( Scalar ‘ 𝑊 ) )
17		eqid	⊢ ( -g ‘ ( Scalar ‘ 𝑊 ) ) = ( -g ‘ ( Scalar ‘ 𝑊 ) )
18		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
19		eqid	⊢ ( invg ‘ ( Scalar ‘ 𝑊 ) ) = ( invg ‘ ( Scalar ‘ 𝑊 ) )
20	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19	baerlem3lem2	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 − 𝑍 ) } ) = ( ( ( 𝑁 ‘ { 𝑌 } ) ⊕ ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ⊕ ( 𝑁 ‘ { ( 𝑋 − 𝑍 ) } ) ) ) )