Theorem addresr

Description: Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996) (New usage is discouraged.)

	Ref	Expression
Assertion	addresr	$$⊢\left(A\in 𝑹\wedge B\in 𝑹\right)\to ⟨A,0_{𝑹}⟩+⟨B,0_{𝑹}⟩=⟨A{+}_{𝑹}B,0_{𝑹}⟩$$

Proof

Step	Hyp	Ref	Expression
1		0r	$$⊢0_{𝑹}\in 𝑹$$
2		addcnsr	$$⊢\left(\left(A\in 𝑹\wedge 0_{𝑹}\in 𝑹\right)\wedge \left(B\in 𝑹\wedge 0_{𝑹}\in 𝑹\right)\right)\to ⟨A,0_{𝑹}⟩+⟨B,0_{𝑹}⟩=⟨A{+}_{𝑹}B,0_{𝑹}{+}_{𝑹}{0}_{𝑹}⟩$$
3	2	an4s	$$⊢\left(\left(A\in 𝑹\wedge B\in 𝑹\right)\wedge \left(0_{𝑹}\in 𝑹\wedge 0_{𝑹}\in 𝑹\right)\right)\to ⟨A,0_{𝑹}⟩+⟨B,0_{𝑹}⟩=⟨A{+}_{𝑹}B,0_{𝑹}{+}_{𝑹}{0}_{𝑹}⟩$$
4	1 1 3	mpanr12	$$⊢\left(A\in 𝑹\wedge B\in 𝑹\right)\to ⟨A,0_{𝑹}⟩+⟨B,0_{𝑹}⟩=⟨A{+}_{𝑹}B,0_{𝑹}{+}_{𝑹}{0}_{𝑹}⟩$$
5		0idsr	$$⊢0_{𝑹}\in 𝑹\to 0_{𝑹}{+}_{𝑹}{0}_{𝑹}=0_{𝑹}$$
6	1 5	ax-mp	$$⊢0_{𝑹}{+}_{𝑹}{0}_{𝑹}=0_{𝑹}$$
7	6	opeq2i	$$⊢⟨A{+}_{𝑹}B,0_{𝑹}{+}_{𝑹}{0}_{𝑹}⟩=⟨A{+}_{𝑹}B,0_{𝑹}⟩$$
8	4 7	eqtrdi	$$⊢\left(A\in 𝑹\wedge B\in 𝑹\right)\to ⟨A,0_{𝑹}⟩+⟨B,0_{𝑹}⟩=⟨A{+}_{𝑹}B,0_{𝑹}⟩$$