Metamath Proof Explorer


Theorem ablsub32

Description: Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015)

Ref Expression
Hypotheses ablnncan.b 𝐵 = ( Base ‘ 𝐺 )
ablnncan.m = ( -g𝐺 )
ablnncan.g ( 𝜑𝐺 ∈ Abel )
ablnncan.x ( 𝜑𝑋𝐵 )
ablnncan.y ( 𝜑𝑌𝐵 )
ablsub32.z ( 𝜑𝑍𝐵 )
Assertion ablsub32 ( 𝜑 → ( ( 𝑋 𝑌 ) 𝑍 ) = ( ( 𝑋 𝑍 ) 𝑌 ) )

Proof

Step Hyp Ref Expression
1 ablnncan.b 𝐵 = ( Base ‘ 𝐺 )
2 ablnncan.m = ( -g𝐺 )
3 ablnncan.g ( 𝜑𝐺 ∈ Abel )
4 ablnncan.x ( 𝜑𝑋𝐵 )
5 ablnncan.y ( 𝜑𝑌𝐵 )
6 ablsub32.z ( 𝜑𝑍𝐵 )
7 eqid ( +g𝐺 ) = ( +g𝐺 )
8 1 7 ablcom ( ( 𝐺 ∈ Abel ∧ 𝑌𝐵𝑍𝐵 ) → ( 𝑌 ( +g𝐺 ) 𝑍 ) = ( 𝑍 ( +g𝐺 ) 𝑌 ) )
9 3 5 6 8 syl3anc ( 𝜑 → ( 𝑌 ( +g𝐺 ) 𝑍 ) = ( 𝑍 ( +g𝐺 ) 𝑌 ) )
10 9 oveq2d ( 𝜑 → ( 𝑋 ( 𝑌 ( +g𝐺 ) 𝑍 ) ) = ( 𝑋 ( 𝑍 ( +g𝐺 ) 𝑌 ) ) )
11 1 7 2 3 4 5 6 ablsubsub4 ( 𝜑 → ( ( 𝑋 𝑌 ) 𝑍 ) = ( 𝑋 ( 𝑌 ( +g𝐺 ) 𝑍 ) ) )
12 1 7 2 3 4 6 5 ablsubsub4 ( 𝜑 → ( ( 𝑋 𝑍 ) 𝑌 ) = ( 𝑋 ( 𝑍 ( +g𝐺 ) 𝑌 ) ) )
13 10 11 12 3eqtr4d ( 𝜑 → ( ( 𝑋 𝑌 ) 𝑍 ) = ( ( 𝑋 𝑍 ) 𝑌 ) )