Theorem grppncan

Description: Cancellation law for subtraction ( pncan analog). (Contributed by NM, 16-Apr-2014)

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

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		simp1	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to G\in \mathrm{Grp}$$
5		simp2	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to X\in B$$
6		simp3	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to Y\in B$$
7	1 2 3	grpaddsubass	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Y\in B\right)\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{-}Y=X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{-}Y\right)$$
8	4 5 6 6 7	syl13anc	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{-}Y=X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{-}Y\right)$$
9		eqid	$$⊢0_G=0_G$$
10	1 9 3	grpsubid	$$⊢\left(G\in \mathrm{Grp}\wedge Y\in B\right)\to Y\underset{\dot{}}{-}Y=0_G$$
11	10	oveq2d	$$⊢\left(G\in \mathrm{Grp}\wedge Y\in B\right)\to X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{-}Y\right)=X\underset{\dot{}}{+}{0}_G$$
12	11	3adant2	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{-}Y\right)=X\underset{\dot{}}{+}{0}_G$$
13	1 2 9	grprid	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\right)\to X\underset{\dot{}}{+}{0}_G=X$$
14	13	3adant3	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}{0}_G=X$$
15	8 12 14	3eqtrd	$$⊢\left(G\in \mathrm{Grp}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{-}Y=X$$