Theorem mvrrsubd

Description: Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd .

EDITORIAL: Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 21-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		mvrrsubd.a	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
2		mvrrsubd.b	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
3		mvrrsubd.1	⊢ ( 𝜑 → 𝐴 = ( 𝐵 − 𝐶 ) )
4	1 2	subcld	⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) ∈ ℂ )
5	3 4	eqeltrd	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
6	5 2	addcld	⊢ ( 𝜑 → ( 𝐴 + 𝐶 ) ∈ ℂ )
7	5 2	pncand	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) − 𝐶 ) = 𝐴 )
8	7 3	eqtrd	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) − 𝐶 ) = ( 𝐵 − 𝐶 ) )
9	6 1 2 8	subcan2d	⊢ ( 𝜑 → ( 𝐴 + 𝐶 ) = 𝐵 )