Theorem odrngds

Description: The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Hypothesis	odrngstr.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}$$
Assertion	odrngds	$$⊢D\in V\to D=\mathrm{dist}\left(W\right)$$

Proof

Step	Hyp	Ref	Expression
1		odrngstr.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}$$
2	1	odrngstr	$$⊢W\mathrm{Struct}⟨1,12⟩$$
3		dsid	$$⊢\mathrm{dist}=\mathrm{Slot}\mathrm{dist}\left(\mathrm{ndx}\right)$$
4		snsstp3	$$⊢\left\{⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}\subseteq \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}$$
5		ssun2	$$⊢\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}$$
6	5 1	sseqtrri	$$⊢\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}\subseteq W$$
7	4 6	sstri	$$⊢\left\{⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}\subseteq W$$
8	2 3 7	strfv	$$⊢D\in V\to D=\mathrm{dist}\left(W\right)$$