Theorem nvgf

Description: Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvgf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	nvgf.2	$$⊢G={+}_v\left(U\right)$$
Assertion	nvgf	$$⊢U\in \mathrm{NrmCVec}\to G:X\times X⟶X$$

Proof

Step	Hyp	Ref	Expression
1		nvgf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		nvgf.2	$$⊢G={+}_v\left(U\right)$$
3	2	nvgrp	$$⊢U\in \mathrm{NrmCVec}\to G\in \mathrm{GrpOp}$$
4	1 2	bafval	$$⊢X=\mathrm{ran}G$$
5	4	grpofo	$$⊢G\in \mathrm{GrpOp}\to G:X\times X\underset{onto}{⟶}X$$
6		fof	$$⊢G:X\times X\underset{onto}{⟶}X\to G:X\times X⟶X$$
7	3 5 6	3syl	$$⊢U\in \mathrm{NrmCVec}\to G:X\times X⟶X$$