Theorem cnfldle 14587

Description: The ordering of the field of complex numbers. Note that this is not actually an ordering on CC, but we put it in the structure anyway because restricting to RR does not affect this component, so that (ℂ_fld ↾_s RR ) is an ordered field even though ℂ_fld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14577. (Revised by GG, 31-Mar-2025.)

Assertion

Ref	Expression
cnfldle	|- <_ = ( le ` ℂfld )

Proof of Theorem cnfldle

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrex 10091	. . . 4 |- RR* e. _V
2	1, 1	xpex 4842	. . 3 |- ( RR* X. RR* ) e. _V
3		lerelxr 8242	. . 3 |- <_ C_ ( RR* X. RR* )
4	2, 3	ssexi 4227	. 2 |- <_ e. _V
5		cnfldstr 14578	. . 3 |- ℂfld Struct <. 1 , ; 1 3 >.
6		pleslid 13290	. . 3 |- ( le = Slot ( le ` ndx ) /\ ( le ` ndx ) e. NN )
7		snsstp2 3824	. . . 4 |- { <. ( le ` ndx ) , <_ >. } C_ { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. }
8		ssun1 3370	. . . . 5 |- { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } C_ ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } )
9		ssun2 3371	. . . . . 6 |- ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) C_ ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( u e. CC , v e. CC |-> ( u + v ) ) >. , <. ( .r ` ndx ) , ( u e. CC , v e. CC |-> ( u x. v ) ) >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
10		df-cnfld 14577	. . . . . 6 |- ℂfld = ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( u e. CC , v e. CC |-> ( u + v ) ) >. , <. ( .r ` ndx ) , ( u e. CC , v e. CC |-> ( u x. v ) ) >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
11	9, 10	sseqtrri 3262	. . . . 5 |- ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) C_ ℂfld
12	8, 11	sstri 3236	. . . 4 |- { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } C_ ℂfld
13	7, 12	sstri 3236	. . 3 |- { <. ( le ` ndx ) , <_ >. } C_ ℂfld
14	5, 6, 13	strslfv 13132	. 2 |- ( <_ e. _V -> <_ = ( le ` ℂfld ) )
15	4, 14	ax-mp 5	1 |- <_ = ( le ` ℂfld )

Syntax hints:   = wceq 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   {csn 3669   {ctp 3671   <.cop 3672   X. cxp 4723   o. ccom 4729   ` cfv 5326  (class class class)co 6018   e. cmpo 6020   CCcc 8030   1c1 8033   + caddc 8035   x. cmul 8037   RR*cxr 8213   <_ cle 8215   - cmin 8350   3c3 9195  ;cdc 9611   *ccj 11404   abscabs 11562   ndxcnx 13084   Basecbs 13087   +g cplusg 13165   .rcmulr 13166   *rcstv 13167  TopSetcts 13171   lecple 13172   distcds 13174   UnifSetcunif 13175   MetOpencmopn 14561  metUnifcmetu 14562  ℂ_fldccnfld 14576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-rp 9889  df-fz 10244  df-cj 11407  df-abs 11564  df-struct 13089  df-ndx 13090  df-slot 13091  df-base 13093  df-plusg 13178  df-mulr 13179  df-starv 13180  df-tset 13184  df-ple 13185  df-ds 13187  df-unif 13188  df-topgen 13348  df-bl 14566  df-mopn 14567  df-fg 14569  df-metu 14570  df-cnfld 14577

This theorem is referenced by: (None)