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

Metamath Proof Explorer

Theorem ngplcan

Description: Cancel left addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015)

Ref Expression
Hypotheses ngprcan.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
ngprcan.p ⊢ + = ( +g ‘ 𝐺 )
ngprcan.d ⊢ 𝐷 = ( dist ‘ 𝐺 )
Assertion ngplcan ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 + 𝐴 ) 𝐷 ( 𝐶 + 𝐵 ) ) = ( 𝐴 𝐷 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ngprcan.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
2 ngprcan.p ⊢ + = ( +g ‘ 𝐺 )
3 ngprcan.d ⊢ 𝐷 = ( dist ‘ 𝐺 )
4 simplr ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐺 ∈ Abel )
5 simpr3 ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐶 ∈ 𝑋 )
6 simpr1 ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐴 ∈ 𝑋 )
7 1 2 ablcom ⊢ ( ( 𝐺 ∈ Abel ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐶 + 𝐴 ) = ( 𝐴 + 𝐶 ) )
8 4 5 6 7 syl3anc ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐶 + 𝐴 ) = ( 𝐴 + 𝐶 ) )
9 simpr2 ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → 𝐵 ∈ 𝑋 )
10 1 2 ablcom ⊢ ( ( 𝐺 ∈ Abel ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐶 + 𝐵 ) = ( 𝐵 + 𝐶 ) )
11 4 5 9 10 syl3anc ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐶 + 𝐵 ) = ( 𝐵 + 𝐶 ) )
12 8 11 oveq12d ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 + 𝐴 ) 𝐷 ( 𝐶 + 𝐵 ) ) = ( ( 𝐴 + 𝐶 ) 𝐷 ( 𝐵 + 𝐶 ) ) )
13 1 2 3 ngprcan ⊢ ( ( 𝐺 ∈ NrmGrp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 + 𝐶 ) 𝐷 ( 𝐵 + 𝐶 ) ) = ( 𝐴 𝐷 𝐵 ) )
14 13 adantlr ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 + 𝐶 ) 𝐷 ( 𝐵 + 𝐶 ) ) = ( 𝐴 𝐷 𝐵 ) )
15 12 14 eqtrd ⊢ ( ( ( 𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 + 𝐴 ) 𝐷 ( 𝐶 + 𝐵 ) ) = ( 𝐴 𝐷 𝐵 ) )