Theorem odrngstr

Description: Functionality of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015) (Proof shortened by AV, 15-Sep-2021)

	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	odrngstr	$$⊢W\mathrm{Struct}⟨1,12⟩$$

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		eqid	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
3	2	rngstr	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\mathrm{Struct}⟨1,3⟩$$
4		9nn	$$⊢9\in \mathbb{N}$$
5		tsetndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)=9$$
6		9lt10	$$⊢9<10$$
7		10nn	$$⊢10\in \mathbb{N}$$
8		plendx	$$⊢{\le }_{\mathrm{ndx}}=10$$
9		1nn0	$$⊢1\in {\mathbb{N}}_0$$
10		0nn0	$$⊢0\in {\mathbb{N}}_0$$
11		2nn	$$⊢2\in \mathbb{N}$$
12		2pos	$$⊢0<2$$
13	9 10 11 12	declt	$$⊢10<12$$
14	9 11	decnncl	$$⊢12\in \mathbb{N}$$
15		dsndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)=12$$
16	4 5 6 7 8 13 14 15	strle3	$$⊢\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}\mathrm{Struct}⟨9,12⟩$$
17		3lt9	$$⊢3<9$$
18	3 16 17	strleun	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),J⟩,⟨{\le }_{\mathrm{ndx}},\underset{\dot{}}{\le }⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}\right)\mathrm{Struct}⟨1,12⟩$$
19	1 18	eqbrtri	$$⊢W\mathrm{Struct}⟨1,12⟩$$