Theorem setsmsds

Description: The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015) (Proof shortened by AV, 11-Nov-2024)

	Ref	Expression
Hypotheses	setsms.x	$$⊢\phi \to X={\mathrm{Base}}_M$$
	setsms.d	$$⊢\phi \to D={\mathrm{dist}\left(M\right)\upharpoonright }_{\left(X\times X\right)}$$
	setsms.k	$$⊢\phi \to K=M\mathrm{sSet}⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(D\right)⟩$$
Assertion	setsmsds	$$⊢\phi \to \mathrm{dist}\left(M\right)=\mathrm{dist}\left(K\right)$$

Proof

Step	Hyp	Ref	Expression
1		setsms.x	$$⊢\phi \to X={\mathrm{Base}}_M$$
2		setsms.d	$$⊢\phi \to D={\mathrm{dist}\left(M\right)\upharpoonright }_{\left(X\times X\right)}$$
3		setsms.k	$$⊢\phi \to K=M\mathrm{sSet}⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(D\right)⟩$$
4		dsid	$$⊢\mathrm{dist}=\mathrm{Slot}\mathrm{dist}\left(\mathrm{ndx}\right)$$
5		dsndxntsetndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{TopSet}\left(\mathrm{ndx}\right)$$
6	4 5	setsnid	$$⊢\mathrm{dist}\left(M\right)=\mathrm{dist}\left(M\mathrm{sSet}⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(D\right)⟩\right)$$
7	3	fveq2d	$$⊢\phi \to \mathrm{dist}\left(K\right)=\mathrm{dist}\left(M\mathrm{sSet}⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(D\right)⟩\right)$$
8	6 7	eqtr4id	$$⊢\phi \to \mathrm{dist}\left(M\right)=\mathrm{dist}\left(K\right)$$