Theorem nvmfval

Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 11-Sep-2007) (Revised by Mario Carneiro, 23-Dec-2013) (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	nvmfval	⊢ ( 𝑈 ∈ NrmCVec → 𝑀 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) )

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	2 4	vsfval	⊢ 𝑀 = ( /𝑔 ‘ 𝐺 )
9	6 7 8	grpodivfval	⊢ ( 𝐺 ∈ GrpOp → 𝑀 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ) )
10	5 9	syl	⊢ ( 𝑈 ∈ NrmCVec → 𝑀 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ) )
11	1 2 3 7	nvinv	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋 ) → ( - 1 𝑆 𝑦 ) = ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) )
12	11	3adant2	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( - 1 𝑆 𝑦 ) = ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) )
13	12	oveq2d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) → ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) = ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) )
14	13	mpoeq3dva	⊢ ( 𝑈 ∈ NrmCVec → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( ( inv ‘ 𝐺 ) ‘ 𝑦 ) ) ) )
15	10 14	eqtr4d	⊢ ( 𝑈 ∈ NrmCVec → 𝑀 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ ( 𝑥 𝐺 ( - 1 𝑆 𝑦 ) ) ) )