lspsnsub

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

	Ref	Expression
Hypotheses	lspsnsub.v	$⊢ V = {Base}_{W}$
	lspsnsub.s	$⊢ \underset{˙}{-} = -_{W}$
	lspsnsub.n	$⊢ N = LSpan (W)$
	lspsnsub.w	$⊢ φ \to W \in LMod$
	lspsnsub.x	$⊢ φ \to X \in V$
	lspsnsub.y	$⊢ φ \to Y \in V$
Assertion	lspsnsub	$⊢ φ \to N (\{X \underset{˙}{-} Y\}) = N (\{Y \underset{˙}{-} X\})$

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

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