Metamath Proof Explorer


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 + = ( +g𝐺 )
grpsubadd.m = ( -g𝐺 )
grpnpncan0.0 0 = ( 0g𝐺 )
Assertion grpnpncan0 ( ( 𝐺 ∈ Grp ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( ( 𝑋 𝑌 ) + ( 𝑌 𝑋 ) ) = 0 )

Proof

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