Theorem nvinv

Description: Minus 1 times a vector is the underlying group's inverse element. Equation 2 of Kreyszig p. 51. (Contributed by NM, 15-Feb-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvinv.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	nvinv.2	$$⊢G={+}_v\left(U\right)$$
	nvinv.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
	nvinv.5	$$⊢M=\mathrm{inv}\left(G\right)$$
Assertion	nvinv	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to -1SA=M\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		nvinv.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		nvinv.2	$$⊢G={+}_v\left(U\right)$$
3		nvinv.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
4		nvinv.5	$$⊢M=\mathrm{inv}\left(G\right)$$
5		eqid	$$⊢1^{st}\left(U\right)=1^{st}\left(U\right)$$
6	5	nvvc	$$⊢U\in \mathrm{NrmCVec}\to 1^{st}\left(U\right)\in {CVec}_{\mathrm{OLD}}$$
7	2	vafval	$$⊢G=1^{st}\left(1^{st}\left(U\right)\right)$$
8	3	smfval	$$⊢S=2^{nd}\left(1^{st}\left(U\right)\right)$$
9	1 2	bafval	$$⊢X=\mathrm{ran}G$$
10	7 8 9 4	vcm	$$⊢\left(1^{st}\left(U\right)\in {CVec}_{\mathrm{OLD}}\wedge A\in X\right)\to -1SA=M\left(A\right)$$
11	6 10	sylan	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\right)\to -1SA=M\left(A\right)$$