Theorem cnfldle 14539

Description: The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂ_fld ↾_s ℝ) 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 14529. (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 10060	. . . 4 ⊢ ℝ* ∈ V
2	1, 1	xpex 4834	. . 3 ⊢ (ℝ* × ℝ*) ∈ V
3		lerelxr 8217	. . 3 ⊢ ≤ ⊆ (ℝ* × ℝ*)
4	2, 3	ssexi 4222	. 2 ⊢ ≤ ∈ V
5		cnfldstr 14530	. . 3 ⊢ ℂfld Struct ⟨1, ;13⟩
6		pleslid 13243	. . 3 ⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
7		snsstp2 3819	. . . 4 ⊢ {⟨(le‘ndx), ≤ ⟩} ⊆ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}
8		ssun1 3367	. . . . 5 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ⊆ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})
9		ssun2 3368	. . . . . 6 ⊢ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ⊆ (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
10		df-cnfld 14529	. . . . . 6 ⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
11	9, 10	sseqtrri 3259	. . . . 5 ⊢ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ⊆ ℂfld
12	8, 11	sstri 3233	. . . 4 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ⊆ ℂfld
13	7, 12	sstri 3233	. . 3 ⊢ {⟨(le‘ndx), ≤ ⟩} ⊆ ℂfld
14	5, 6, 13	strslfv 13085	. 2 ⊢ ( ≤ ∈ V → ≤ = (le‘ℂfld))
15	4, 14	ax-mp 5	1 ⊢ ≤ = (le‘ℂfld)

Syntax hints:   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∪ cun 3195  {csn 3666  {ctp 3668  ⟨cop 3669   × cxp 4717   ∘ ccom 4723  ‘cfv 5318  (class class class)co 6007   ∈ cmpo 6009  ℂcc 8005  1c1 8008   + caddc 8010   · cmul 8012  ℝ^*cxr 8188   ≤ cle 8190   − cmin 8325  3c3 9170  ;cdc 9586  ∗ccj 11358  abscabs 11516  ndxcnx 13037  Basecbs 13040  +_gcplusg 13118  ._rcmulr 13119  *_𝑟cstv 13120  TopSetcts 13124  lecple 13125  distcds 13127  UnifSetcunif 13128  MetOpencmopn 14513  metUnifcmetu 14514  ℂ_fldccnfld 14528

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-z 9455  df-dec 9587  df-uz 9731  df-rp 9858  df-fz 10213  df-cj 11361  df-abs 11518  df-struct 13042  df-ndx 13043  df-slot 13044  df-base 13046  df-plusg 13131  df-mulr 13132  df-starv 13133  df-tset 13137  df-ple 13138  df-ds 13140  df-unif 13141  df-topgen 13301  df-bl 14518  df-mopn 14519  df-fg 14521  df-metu 14522  df-cnfld 14529

This theorem is referenced by: (None)