Theorem nppcan2

Description: Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005)

	Ref	Expression
Assertion	nppcan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − ( 𝐵 + 𝐶 ) ) + 𝐶 ) = ( 𝐴 − 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		addcl	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + 𝐶 ) ∈ ℂ )
2	1	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + 𝐶 ) ∈ ℂ )
3		subsub	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 + 𝐶 ) ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( ( 𝐵 + 𝐶 ) − 𝐶 ) ) = ( ( 𝐴 − ( 𝐵 + 𝐶 ) ) + 𝐶 ) )
4	2 3	syld3an2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( ( 𝐵 + 𝐶 ) − 𝐶 ) ) = ( ( 𝐴 − ( 𝐵 + 𝐶 ) ) + 𝐶 ) )
5		pncan	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐶 ) − 𝐶 ) = 𝐵 )
6	5	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐶 ) − 𝐶 ) = 𝐵 )
7	6	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( ( 𝐵 + 𝐶 ) − 𝐶 ) ) = ( 𝐴 − 𝐵 ) )
8	4 7	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − ( 𝐵 + 𝐶 ) ) + 𝐶 ) = ( 𝐴 − 𝐵 ) )