Theorem cnfldstr

Description: The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015) (Revised by Thierry Arnoux, 17-Dec-2017) Revise df-cnfld . (Revised by GG, 31-Mar-2025)

	Ref	Expression
Assertion	cnfldstr	$$⊢{ℂ}_{\mathrm{fld}}\mathrm{Struct}⟨1,13⟩$$

Proof

Step	Hyp	Ref	Expression
1		df-cnfld	$$⊢{ℂ}_{\mathrm{fld}}=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼u+v\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼uv\right)⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},*⟩\right\}\right)\cup \left(\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\cup \left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}\right)$$
2		eqid	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼u+v\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼uv\right)⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},*⟩\right\}=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼u+v\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼uv\right)⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},*⟩\right\}$$
3	2	srngstr	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼u+v\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼uv\right)⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},*⟩\right\}\right)\mathrm{Struct}⟨1,4⟩$$
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	$$⊢\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\mathrm{Struct}⟨9,12⟩$$
17		3nn	$$⊢3\in \mathbb{N}$$
18	9 17	decnncl	$$⊢13\in \mathbb{N}$$
19		unifndx	$$⊢\mathrm{UnifSet}\left(\mathrm{ndx}\right)=13$$
20	18 19	strle1	$$⊢\left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}\mathrm{Struct}⟨13,13⟩$$
21		2nn0	$$⊢2\in {\mathbb{N}}_0$$
22		2lt3	$$⊢2<3$$
23	9 21 17 22	declt	$$⊢12<13$$
24	16 20 23	strleun	$$⊢\left(\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\cup \left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}\right)\mathrm{Struct}⟨9,13⟩$$
25		4lt9	$$⊢4<9$$
26	3 24 25	strleun	$$⊢\left(\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼u+v\right)⟩,⟨{\cdot }_{\mathrm{ndx}},\left(u\in ℂ,v\in ℂ⟼uv\right)⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},*⟩\right\}\right)\cup \left(\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\cup \left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}\right)\right)\mathrm{Struct}⟨1,13⟩$$
27	1 26	eqbrtri	$$⊢{ℂ}_{\mathrm{fld}}\mathrm{Struct}⟨1,13⟩$$