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 = ( Base ‘ 𝐺 )
	grpsubadd.p	⊢ + = ( +g ‘ 𝐺 )
	grpsubadd.m	⊢ − = ( -g ‘ 𝐺 )
Assertion	grpnpncan	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑍 ) ) = ( 𝑋 − 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		grpsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubadd.p	⊢ + = ( +g ‘ 𝐺 )
3		grpsubadd.m	⊢ − = ( -g ‘ 𝐺 )
4		simpl	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐺 ∈ Grp )
5	1 3	grpsubcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) ∈ 𝐵 )
6	5	3adant3r3	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 − 𝑌 ) ∈ 𝐵 )
7		simpr2	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
8		simpr3	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
9	1 2 3	grpaddsubass	⊢ ( ( 𝐺 ∈ Grp ∧ ( ( 𝑋 − 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 − 𝑌 ) + 𝑌 ) − 𝑍 ) = ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑍 ) ) )
10	4 6 7 8 9	syl13anc	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 − 𝑌 ) + 𝑌 ) − 𝑍 ) = ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑍 ) ) )
11	1 2 3	grpnpcan	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 − 𝑌 ) + 𝑌 ) = 𝑋 )
12	11	3adant3r3	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + 𝑌 ) = 𝑋 )
13	12	oveq1d	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 − 𝑌 ) + 𝑌 ) − 𝑍 ) = ( 𝑋 − 𝑍 ) )
14	10 13	eqtr3d	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑍 ) ) = ( 𝑋 − 𝑍 ) )