Metamath Proof Explorer

Theorem ablsubsub4

Description: Law for double subtraction. (Contributed by NM, 7-Apr-2015)

Ref Expression
Hypotheses ablsubadd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
ablsubadd.p ⊢ + = ( +g ‘ 𝐺 )
ablsubadd.m ⊢ − = ( -g ‘ 𝐺 )
ablsubsub.g ⊢ ( 𝜑 → 𝐺 ∈ Abel )
ablsubsub.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
ablsubsub.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
ablsubsub.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion ablsubsub4 ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( 𝑋 − ( 𝑌 + 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 ablsubadd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 ablsubadd.p ⊢ + = ( +g ‘ 𝐺 )
3 ablsubadd.m ⊢ − = ( -g ‘ 𝐺 )
4 ablsubsub.g ⊢ ( 𝜑 → 𝐺 ∈ Abel )
5 ablsubsub.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 ablsubsub.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 ablsubsub.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8 ablgrp ⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
9 4 8 syl ⊢ ( 𝜑 → 𝐺 ∈ Grp )
10 1 3 grpsubcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) ∈ 𝐵 )
11 9 5 6 10 syl3anc ⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝐵 )
12 eqid ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
13 1 2 12 3 grpsubval ⊢ ( ( ( 𝑋 − 𝑌 ) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( ( 𝑋 − 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
14 11 7 13 syl2anc ⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( ( 𝑋 − 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
15 1 12 grpinvcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ∈ 𝐵 )
16 9 7 15 syl2anc ⊢ ( 𝜑 → ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ∈ 𝐵 )
17 1 2 3 4 5 6 16 ablsubsub ⊢ ( 𝜑 → ( 𝑋 − ( 𝑌 − ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) ) = ( ( 𝑋 − 𝑌 ) + ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) )
18 1 2 3 12 9 6 7 grpsubinv ⊢ ( 𝜑 → ( 𝑌 − ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) = ( 𝑌 + 𝑍 ) )
19 18 oveq2d ⊢ ( 𝜑 → ( 𝑋 − ( 𝑌 − ( ( invg ‘ 𝐺 ) ‘ 𝑍 ) ) ) = ( 𝑋 − ( 𝑌 + 𝑍 ) ) )
20 14 17 19 3eqtr2d ⊢ ( 𝜑 → ( ( 𝑋 − 𝑌 ) − 𝑍 ) = ( 𝑋 − ( 𝑌 + 𝑍 ) ) )