Theorem lspsnsub

Description: Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015)

	Ref	Expression
Hypotheses	lspsnsub.v	$$⊢V={\mathrm{Base}}_W$$
	lspsnsub.s	$$⊢\underset{\dot{}}{-}={-}_W$$
	lspsnsub.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
	lspsnsub.w	$$⊢\phi \to W\in \mathrm{LMod}$$
	lspsnsub.x	$$⊢\phi \to X\in V$$
	lspsnsub.y	$$⊢\phi \to Y\in V$$
Assertion	lspsnsub	$$⊢\phi \to N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)=N\left(\left\{Y\underset{\dot{}}{-}X\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		lspsnsub.v	$$⊢V={\mathrm{Base}}_W$$
2		lspsnsub.s	$$⊢\underset{\dot{}}{-}={-}_W$$
3		lspsnsub.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
4		lspsnsub.w	$$⊢\phi \to W\in \mathrm{LMod}$$
5		lspsnsub.x	$$⊢\phi \to X\in V$$
6		lspsnsub.y	$$⊢\phi \to Y\in V$$
7	1 2	lmodvsubcl	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{-}Y\in V$$
8	4 5 6 7	syl3anc	$$⊢\phi \to X\underset{\dot{}}{-}Y\in V$$
9		eqid	$$⊢{inv}_g\left(W\right)={inv}_g\left(W\right)$$
10	1 9 3	lspsnneg	$$⊢\left(W\in \mathrm{LMod}\wedge X\underset{\dot{}}{-}Y\in V\right)\to N\left(\left\{{inv}_g\left(W\right)\left(X\underset{\dot{}}{-}Y\right)\right\}\right)=N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)$$
11	4 8 10	syl2anc	$$⊢\phi \to N\left(\left\{{inv}_g\left(W\right)\left(X\underset{\dot{}}{-}Y\right)\right\}\right)=N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)$$
12		lmodgrp	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Grp}$$
13	4 12	syl	$$⊢\phi \to W\in \mathrm{Grp}$$
14	1 2 9	grpinvsub	$$⊢\left(W\in \mathrm{Grp}\wedge X\in V\wedge Y\in V\right)\to {inv}_g\left(W\right)\left(X\underset{\dot{}}{-}Y\right)=Y\underset{\dot{}}{-}X$$
15	13 5 6 14	syl3anc	$$⊢\phi \to {inv}_g\left(W\right)\left(X\underset{\dot{}}{-}Y\right)=Y\underset{\dot{}}{-}X$$
16	15	sneqd	$$⊢\phi \to \left\{{inv}_g\left(W\right)\left(X\underset{\dot{}}{-}Y\right)\right\}=\left\{Y\underset{\dot{}}{-}X\right\}$$
17	16	fveq2d	$$⊢\phi \to N\left(\left\{{inv}_g\left(W\right)\left(X\underset{\dot{}}{-}Y\right)\right\}\right)=N\left(\left\{Y\underset{\dot{}}{-}X\right\}\right)$$
18	11 17	eqtr3d	$$⊢\phi \to N\left(\left\{X\underset{\dot{}}{-}Y\right\}\right)=N\left(\left\{Y\underset{\dot{}}{-}X\right\}\right)$$