Theorem srads

Description: Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by AV, 29-Oct-2024)

	Ref	Expression
Hypotheses	srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
	srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
Assertion	srads	$$⊢\phi \to \mathrm{dist}\left(W\right)=\mathrm{dist}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
2		srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
3		dsid	$$⊢\mathrm{dist}=\mathrm{Slot}\mathrm{dist}\left(\mathrm{ndx}\right)$$
4		slotsdnscsi	$$⊢\left(\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)\wedge \mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}\wedge \mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\right)$$
5	4	simp1i	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)$$
6	5	necomi	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne \mathrm{dist}\left(\mathrm{ndx}\right)$$
7	4	simp2i	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
8	7	necomi	$$⊢{\cdot }_{\mathrm{ndx}}\ne \mathrm{dist}\left(\mathrm{ndx}\right)$$
9	4	simp3i	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)$$
10	9	necomi	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne \mathrm{dist}\left(\mathrm{ndx}\right)$$
11	1 2 3 6 8 10	sralem	$$⊢\phi \to \mathrm{dist}\left(W\right)=\mathrm{dist}\left(A\right)$$