This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ablpnpcan

Description: Cancellation law for mixed addition and subtraction. ( pnpcan analog.) (Contributed by NM, 29-May-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 )
ablpnpcan.g ⊢ ( 𝜑 → 𝐺 ∈ Abel )
ablpnpcan.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
ablpnpcan.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
ablpnpcan.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion ablpnpcan ( 𝜑 → ( ( 𝑋 + 𝑌 ) − ( 𝑋 + 𝑍 ) ) = ( 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 ablpnpcan.g ⊢ ( 𝜑 → 𝐺 ∈ Abel )
9 ablpnpcan.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
10 ablpnpcan.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
11 ablpnpcan.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
12 1 2 3 ablsub4 ⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) − ( 𝑋 + 𝑍 ) ) = ( ( 𝑋 − 𝑋 ) + ( 𝑌 − 𝑍 ) ) )
13 4 5 6 5 7 12 syl122anc ⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) − ( 𝑋 + 𝑍 ) ) = ( ( 𝑋 − 𝑋 ) + ( 𝑌 − 𝑍 ) ) )
14 ablgrp ⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp )
15 4 14 syl ⊢ ( 𝜑 → 𝐺 ∈ Grp )
16 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
17 1 16 3 grpsubid ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 𝑋 ) = ( 0g ‘ 𝐺 ) )
18 15 5 17 syl2anc ⊢ ( 𝜑 → ( 𝑋 − 𝑋 ) = ( 0g ‘ 𝐺 ) )
19 18 oveq1d ⊢ ( 𝜑 → ( ( 𝑋 − 𝑋 ) + ( 𝑌 − 𝑍 ) ) = ( ( 0g ‘ 𝐺 ) + ( 𝑌 − 𝑍 ) ) )
20 1 3 grpsubcl ⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 − 𝑍 ) ∈ 𝐵 )
21 15 6 7 20 syl3anc ⊢ ( 𝜑 → ( 𝑌 − 𝑍 ) ∈ 𝐵 )
22 1 2 16 grplid ⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑌 − 𝑍 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝐺 ) + ( 𝑌 − 𝑍 ) ) = ( 𝑌 − 𝑍 ) )
23 15 21 22 syl2anc ⊢ ( 𝜑 → ( ( 0g ‘ 𝐺 ) + ( 𝑌 − 𝑍 ) ) = ( 𝑌 − 𝑍 ) )
24 13 19 23 3eqtrd ⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) − ( 𝑋 + 𝑍 ) ) = ( 𝑌 − 𝑍 ) )