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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 ⊢ − = ( -g ‘ 𝐺 )
ablnncan.g ⊢ ( 𝜑 → 𝐺 ∈ Abel )
ablnncan.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
ablnncan.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
ablsub32.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion ablsub32 ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( ( 𝑋 − 𝑍 ) − 𝑌 ) )

Proof

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