Theorem addsubsd

Description: Law for surreal addition and subtraction. (Contributed by Scott Fenton, 4-Mar-2025)

	Ref	Expression
Hypotheses	addsubsassd.1	$$⊢\phi \to A\in \mathrm{No}$$
	addsubsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
	addsubsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	addsubsd	$$⊢\phi \to \left(A{+}_{s}B\right){-}_{s}C=\left(A{-}_{s}C\right){+}_{s}B$$

Proof

Step	Hyp	Ref	Expression
1		addsubsassd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsubsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsubsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
4	3	negscld	$$⊢\phi \to {+}_s\left(C\right)\in \mathrm{No}$$
5	1 2 4	adds32d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_s{+}_s\left(C\right)=\left(A{+}_s{+}_s\left(C\right)\right){+}_{s}B$$
6	1 2	addscld	$$⊢\phi \to A{+}_{s}B\in \mathrm{No}$$
7	6 3	subsvald	$$⊢\phi \to \left(A{+}_{s}B\right){-}_{s}C=\left(A{+}_{s}B\right){+}_s{+}_s\left(C\right)$$
8	1 3	subsvald	$$⊢\phi \to A{-}_{s}C=A{+}_s{+}_s\left(C\right)$$
9	8	oveq1d	$$⊢\phi \to \left(A{-}_{s}C\right){+}_{s}B=\left(A{+}_s{+}_s\left(C\right)\right){+}_{s}B$$
10	5 7 9	3eqtr4d	$$⊢\phi \to \left(A{+}_{s}B\right){-}_{s}C=\left(A{-}_{s}C\right){+}_{s}B$$