Theorem negsubsdi2d

Description: Distribution of negative over subtraction. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	negsubsdi2d.1	$$⊢\phi \to A\in \mathrm{No}$$
	negsubsdi2d.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	negsubsdi2d	$$⊢\phi \to {+}_s\left(A{-}_{s}B\right)=B{-}_{s}A$$

Proof

Step	Hyp	Ref	Expression
1		negsubsdi2d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		negsubsdi2d.2	$$⊢\phi \to B\in \mathrm{No}$$
3	2	negscld	$$⊢\phi \to {+}_s\left(B\right)\in \mathrm{No}$$
4		negsdi	$$⊢\left(A\in \mathrm{No}\wedge {+}_s\left(B\right)\in \mathrm{No}\right)\to {+}_s\left(A{+}_s{+}_s\left(B\right)\right)={+}_s\left(A\right){+}_s{+}_s\left({+}_s\left(B\right)\right)$$
5	1 3 4	syl2anc	$$⊢\phi \to {+}_s\left(A{+}_s{+}_s\left(B\right)\right)={+}_s\left(A\right){+}_s{+}_s\left({+}_s\left(B\right)\right)$$
6		negnegs	$$⊢B\in \mathrm{No}\to {+}_s\left({+}_s\left(B\right)\right)=B$$
7	2 6	syl	$$⊢\phi \to {+}_s\left({+}_s\left(B\right)\right)=B$$
8	7	oveq2d	$$⊢\phi \to {+}_s\left(A\right){+}_s{+}_s\left({+}_s\left(B\right)\right)={+}_s\left(A\right){+}_{s}B$$
9	1	negscld	$$⊢\phi \to {+}_s\left(A\right)\in \mathrm{No}$$
10	9 2	addscomd	$$⊢\phi \to {+}_s\left(A\right){+}_{s}B=B{+}_s{+}_s\left(A\right)$$
11	5 8 10	3eqtrd	$$⊢\phi \to {+}_s\left(A{+}_s{+}_s\left(B\right)\right)=B{+}_s{+}_s\left(A\right)$$
12	1 2	subsvald	$$⊢\phi \to A{-}_{s}B=A{+}_s{+}_s\left(B\right)$$
13	12	fveq2d	$$⊢\phi \to {+}_s\left(A{-}_{s}B\right)={+}_s\left(A{+}_s{+}_s\left(B\right)\right)$$
14	2 1	subsvald	$$⊢\phi \to B{-}_{s}A=B{+}_s{+}_s\left(A\right)$$
15	11 13 14	3eqtr4d	$$⊢\phi \to {+}_s\left(A{-}_{s}B\right)=B{-}_{s}A$$