Theorem ppncan

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005)

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

Proof

Step	Hyp	Ref	Expression
1		addcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
2	1	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
3	2	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − ( 𝐵 − 𝐶 ) ) = ( ( 𝐵 + 𝐴 ) − ( 𝐵 − 𝐶 ) ) )
4		addcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )
5	4	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )
6		subsub2	⊢ ( ( ( 𝐴 + 𝐵 ) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 + 𝐵 ) + ( 𝐶 − 𝐵 ) ) )
7	5 6	syld3an1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 + 𝐵 ) + ( 𝐶 − 𝐵 ) ) )
8		pnncan	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐴 ) − ( 𝐵 − 𝐶 ) ) = ( 𝐴 + 𝐶 ) )
9	8	3com12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐴 ) − ( 𝐵 − 𝐶 ) ) = ( 𝐴 + 𝐶 ) )
10	3 7 9	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) + ( 𝐶 − 𝐵 ) ) = ( 𝐴 + 𝐶 ) )