Theorem grpnpncan

Description: Cancellation law for group subtraction. ( npncan analog.) (Contributed by NM, 15-Feb-2008) (Revised by Mario Carneiro, 2-Dec-2014)

	Ref	Expression
Hypotheses	grpsubadd.b	$$⊢B={\mathrm{Base}}_G$$
	grpsubadd.p	$$⊢\underset{\dot{}}{+}={+}_G$$
	grpsubadd.m	$$⊢\underset{\dot{}}{-}={-}_G$$
Assertion	grpnpncan	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}\left(Y\underset{\dot{}}{-}Z\right)=X\underset{\dot{}}{-}Z$$

Proof

Step	Hyp	Ref	Expression
1		grpsubadd.b	$$⊢B={\mathrm{Base}}_G$$
2		grpsubadd.p	$$⊢\underset{\dot{}}{+}={+}_G$$
3		grpsubadd.m	$$⊢\underset{\dot{}}{-}={-}_G$$
4		simpl	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to G\in \mathrm{Grp}$$
5	1 3	grpsubcl	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{-}Y\in B$$
6	5	3adant3r3	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to X\underset{\dot{}}{-}Y\in B$$
7		simpr2	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to Y\in B$$
8		simpr3	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to Z\in B$$
9	1 2 3	grpaddsubass	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\underset{\dot{}}{-}Y\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}Y\right)\underset{\dot{}}{-}Z=\left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}\left(Y\underset{\dot{}}{-}Z\right)$$
10	4 6 7 8 9	syl13anc	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}Y\right)\underset{\dot{}}{-}Z=\left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}\left(Y\underset{\dot{}}{-}Z\right)$$
11	1 2 3	grpnpcan	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}Y=X$$
12	11	3adant3r3	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}Y=X$$
13	12	oveq1d	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}Y\right)\underset{\dot{}}{-}Z=X\underset{\dot{}}{-}Z$$
14	10 13	eqtr3d	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{-}Y\right)\underset{\dot{}}{+}\left(Y\underset{\dot{}}{-}Z\right)=X\underset{\dot{}}{-}Z$$