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Metamath Proof Explorer

Theorem mvrladdi

Description: Move the left term in a sum on the RHS to the LHS. (Contributed by David A. Wheeler, 11-Oct-2018)

Ref Expression
Hypotheses mvrraddi.1 ⊢ 𝐵 ∈ ℂ
mvrraddi.2 ⊢ 𝐶 ∈ ℂ
mvrraddi.3 ⊢ 𝐴 = ( 𝐵 + 𝐶 )
Assertion mvrladdi ( 𝐴 − 𝐵 ) = 𝐶

Proof

Step Hyp Ref Expression
1 mvrraddi.1 ⊢ 𝐵 ∈ ℂ
2 mvrraddi.2 ⊢ 𝐶 ∈ ℂ
3 mvrraddi.3 ⊢ 𝐴 = ( 𝐵 + 𝐶 )
4 1 2 3 comraddi ⊢ 𝐴 = ( 𝐶 + 𝐵 )
5 4 oveq1i ⊢ ( 𝐴 − 𝐵 ) = ( ( 𝐶 + 𝐵 ) − 𝐵 )
6 2 1 pncan3oi ⊢ ( ( 𝐶 + 𝐵 ) − 𝐵 ) = 𝐶
7 5 6 eqtri ⊢ ( 𝐴 − 𝐵 ) = 𝐶