Theorem grpnpncan0

Description: Cancellation law for group subtraction ( npncan2 analog). (Contributed by AV, 24-Nov-2019)

	Ref	Expression
Hypotheses	grpsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	grpsubadd.p	⊢ + = ( +g ‘ 𝐺 )
	grpsubadd.m	⊢ − = ( -g ‘ 𝐺 )
	grpnpncan0.0	⊢ 0 = ( 0g ‘ 𝐺 )
Assertion	grpnpncan0	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑋 ) ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		grpsubadd.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		grpsubadd.p	⊢ + = ( +g ‘ 𝐺 )
3		grpsubadd.m	⊢ − = ( -g ‘ 𝐺 )
4		grpnpncan0.0	⊢ 0 = ( 0g ‘ 𝐺 )
5		simpl	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐺 ∈ Grp )
6		simprl	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
7		simprr	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
8	1 2 3	grpnpncan	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑋 ) ) = ( 𝑋 − 𝑋 ) )
9	5 6 7 6 8	syl13anc	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑋 ) ) = ( 𝑋 − 𝑋 ) )
10	1 4 3	grpsubid	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 𝑋 ) = 0 )
11	10	adantrr	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 − 𝑋 ) = 0 )
12	9 11	eqtrd	⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 − 𝑌 ) + ( 𝑌 − 𝑋 ) ) = 0 )