Theorem lsubrotld

Description: Rotate the variables left in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 21-Aug-2024)

	Ref	Expression
Hypotheses	lsubrotld.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	lsubrotld.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	lsubrotld.1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 𝐶 )
Assertion	lsubrotld	⊢ ( 𝜑 → ( 𝐵 + 𝐶 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lsubrotld.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		lsubrotld.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		lsubrotld.1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 𝐶 )
4	1 2	subcld	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ∈ ℂ )
5	3 4	eqeltrrd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6	2 5	addcld	⊢ ( 𝜑 → ( 𝐵 + 𝐶 ) ∈ ℂ )
7	2 5	pncan2d	⊢ ( 𝜑 → ( ( 𝐵 + 𝐶 ) − 𝐵 ) = 𝐶 )
8	7 3	eqtr4d	⊢ ( 𝜑 → ( ( 𝐵 + 𝐶 ) − 𝐵 ) = ( 𝐴 − 𝐵 ) )
9	6 1 2 8	subcan2d	⊢ ( 𝜑 → ( 𝐵 + 𝐶 ) = 𝐴 )