Theorem nvmval

Description: Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvmval.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	nvmval.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	nvmval.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
	nvmval.3	⊢ 𝑀 = ( −𝑣 ‘ 𝑈 )
Assertion	nvmval	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝑀 𝐵 ) = ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nvmval.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvmval.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		nvmval.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4		nvmval.3	⊢ 𝑀 = ( −𝑣 ‘ 𝑈 )
5	2	nvgrp	⊢ ( 𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp )
6	1 2	bafval	⊢ 𝑋 = ran 𝐺
7		eqid	⊢ ( inv ‘ 𝐺 ) = ( inv ‘ 𝐺 )
8		eqid	⊢ ( /𝑔 ‘ 𝐺 ) = ( /𝑔 ‘ 𝐺 )
9	6 7 8	grpodivval	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( /𝑔 ‘ 𝐺 ) 𝐵 ) = ( 𝐴 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝐵 ) ) )
10	5 9	syl3an1	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( /𝑔 ‘ 𝐺 ) 𝐵 ) = ( 𝐴 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝐵 ) ) )
11	1 2 4 8	nvm	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝑀 𝐵 ) = ( 𝐴 ( /𝑔 ‘ 𝐺 ) 𝐵 ) )
12	1 2 3 7	nvinv	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ) → ( - 1 𝑆 𝐵 ) = ( ( inv ‘ 𝐺 ) ‘ 𝐵 ) )
13	12	3adant2	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( - 1 𝑆 𝐵 ) = ( ( inv ‘ 𝐺 ) ‘ 𝐵 ) )
14	13	oveq2d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) = ( 𝐴 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝐵 ) ) )
15	10 11 14	3eqtr4d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝑀 𝐵 ) = ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) )