Theorem nvnegneg

Description: Double negative of a vector. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvnegneg.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvnegneg.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
Assertion	nvnegneg	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( - 1 𝑆 ( - 1 𝑆 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nvnegneg.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvnegneg.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
3		neg1cn	⊢ - 1 ∈ ℂ
4	1 2	nvscl	⊢ ( ( 𝑈 ∈ NrmCVec ∧ - 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋 ) → ( - 1 𝑆 𝐴 ) ∈ 𝑋 )
5	3 4	mp3an2	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( - 1 𝑆 𝐴 ) ∈ 𝑋 )
6		eqid	⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 )
7		eqid	⊢ ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) = ( inv ‘ ( +𝑣 ‘ 𝑈 ) )
8	1 6 2 7	nvinv	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( - 1 𝑆 𝐴 ) ∈ 𝑋 ) → ( - 1 𝑆 ( - 1 𝑆 𝐴 ) ) = ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ ( - 1 𝑆 𝐴 ) ) )
9	5 8	syldan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( - 1 𝑆 ( - 1 𝑆 𝐴 ) ) = ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ ( - 1 𝑆 𝐴 ) ) )
10	1 6 2 7	nvinv	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( - 1 𝑆 𝐴 ) = ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ 𝐴 ) )
11	10	fveq2d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ ( - 1 𝑆 𝐴 ) ) = ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ 𝐴 ) ) )
12	6	nvgrp	⊢ ( 𝑈 ∈ NrmCVec → ( +𝑣 ‘ 𝑈 ) ∈ GrpOp )
13	1 6	bafval	⊢ 𝑋 = ran ( +𝑣 ‘ 𝑈 )
14	13 7	grpo2inv	⊢ ( ( ( +𝑣 ‘ 𝑈 ) ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ 𝐴 ) ) = 𝐴 )
15	12 14	sylan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ ( ( inv ‘ ( +𝑣 ‘ 𝑈 ) ) ‘ 𝐴 ) ) = 𝐴 )
16	9 11 15	3eqtrd	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( - 1 𝑆 ( - 1 𝑆 𝐴 ) ) = 𝐴 )