Theorem addsubsassd

Description: Associative-type law for surreal addition and subtraction. (Contributed by Scott Fenton, 6-Feb-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	addsubsassd	$$⊢\phi \to \left(A{+}_{s}B\right){-}_{s}C=A{+}_s\left(B{-}_{s}C\right)$$

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	addsassd	$$⊢\phi \to \left(A{+}_{s}B\right){+}_s{+}_s\left(C\right)=A{+}_s\left(B{+}_s{+}_s\left(C\right)\right)$$
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	2 3	subsvald	$$⊢\phi \to B{-}_{s}C=B{+}_s{+}_s\left(C\right)$$
9	8	oveq2d	$$⊢\phi \to A{+}_s\left(B{-}_{s}C\right)=A{+}_s\left(B{+}_s{+}_s\left(C\right)\right)$$
10	5 7 9	3eqtr4d	$$⊢\phi \to \left(A{+}_{s}B\right){-}_{s}C=A{+}_s\left(B{-}_{s}C\right)$$