Theorem imasvalstr

Description: Structure product value is a structure. (Contributed by Stefan O'Rear, 3-Jan-2015) (Revised by Mario Carneiro, 30-Apr-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Hypothesis	imasvalstr.u	$$⊢U=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),O⟩,⟨{\le }_{\mathrm{ndx}},L⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}$$
Assertion	imasvalstr	$$⊢U\mathrm{Struct}⟨1,12⟩$$

Proof

Step	Hyp	Ref	Expression
1		imasvalstr.u	$$⊢U=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),O⟩,⟨{\le }_{\mathrm{ndx}},L⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}$$
2		eqid	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}$$
3	2	ipsstr	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}\right)\mathrm{Struct}⟨1,8⟩$$
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	$$⊢\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),O⟩,⟨{\le }_{\mathrm{ndx}},L⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}\mathrm{Struct}⟨9,12⟩$$
17		8lt9	$$⊢8<9$$
18	3 16 17	strleun	$$⊢\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),\underset{\dot{}}{,}⟩\right\}\right)\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),O⟩,⟨{\le }_{\mathrm{ndx}},L⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),D⟩\right\}\right)\mathrm{Struct}⟨1,12⟩$$
19	1 18	eqbrtri	$$⊢U\mathrm{Struct}⟨1,12⟩$$