Theorem cnfldle 14764

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 14754. (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 10195	. . . 4 ⊢ ℝ* ∈ V
2	1, 1	xpex 4868	. . 3 ⊢ (ℝ* × ℝ*) ∈ V
3		lerelxr 8341	. . 3 ⊢ ≤ ⊆ (ℝ* × ℝ*)
4	2, 3	ssexi 4250	. 2 ⊢ ≤ ∈ V
5		cnfldstr 14755	. . 3 ⊢ ℂfld Struct ⟨1, ;13⟩
6		pleslid 13436	. . 3 ⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
7		snsstp2 3847	. . . 4 ⊢ {⟨(le‘ndx), ≤ ⟩} ⊆ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩}
8		ssun1 3384	. . . . 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 3385	. . . . . 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 14754	. . . . . 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 3275	. . . . 5 ⊢ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) ⊆ ℂfld
12	8, 11	sstri 3249	. . . 4 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ⊆ ℂfld
13	7, 12	sstri 3249	. . 3 ⊢ {⟨(le‘ndx), ≤ ⟩} ⊆ ℂfld
14	5, 6, 13	strslfv 13278	. 2 ⊢ ( ≤ ∈ V → ≤ = (le‘ℂfld))
15	4, 14	ax-mp 5	1 ⊢ ≤ = (le‘ℂfld)

Syntax hints:   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∪ cun 3211  {csn 3691  {ctp 3693  ⟨cop 3694   × cxp 4749   ∘ ccom 4755  ‘cfv 5354  (class class class)co 6052   ∈ cmpo 6054  ℂcc 8130  1c1 8133   + caddc 8135   · cmul 8137  ℝ^*cxr 8312   ≤ cle 8314   − cmin 8449  3c3 9294  ;cdc 9715  ∗ccj 11532  abscabs 11690  ndxcnx 13230  Basecbs 13233  +_gcplusg 13311  ._rcmulr 13312  *_𝑟cstv 13313  TopSetcts 13317  lecple 13318  distcds 13320  UnifSetcunif 13321  MetOpencmopn 14738  metUnifcmetu 14739  ℂ_fldccnfld 14753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-z 9583  df-dec 9716  df-uz 9860  df-rp 9993  df-fz 10349  df-cj 11535  df-abs 11692  df-struct 13235  df-ndx 13236  df-slot 13237  df-base 13239  df-plusg 13324  df-mulr 13325  df-starv 13326  df-tset 13330  df-ple 13331  df-ds 13333  df-unif 13334  df-topgen 13494  df-bl 14743  df-mopn 14744  df-fg 14746  df-metu 14747  df-cnfld 14754

This theorem is referenced by: (None)