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Theorem fldgensdrg 31907
Description: A generated subfield is a sub-division-ring. (Contributed by Thierry Arnoux, 11-Jan-2025.)
Hypotheses
Ref Expression
fldgenval.1 𝐵 = (Base‘𝐹)
fldgenval.2 (𝜑𝐹 ∈ DivRing)
fldgenval.3 (𝜑𝑆𝐵)
Assertion
Ref Expression
fldgensdrg (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))

Proof of Theorem fldgensdrg
Dummy variables 𝑎 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fldgenval.1 . . 3 𝐵 = (Base‘𝐹)
2 fldgenval.2 . . 3 (𝜑𝐹 ∈ DivRing)
3 fldgenval.3 . . 3 (𝜑𝑆𝐵)
41, 2, 3fldgenval 31905 . 2 (𝜑 → (𝐹 fldGen 𝑆) = {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
52drngringd 20146 . . . 4 (𝜑𝐹 ∈ Ring)
6 eqid 2738 . . . . . 6 (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) = (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
7 sseq2 3969 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑆𝑎𝑆𝑥))
87elrab 3644 . . . . . . . . . . . 12 (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ↔ (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆𝑥))
98biimpi 215 . . . . . . . . . . 11 (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} → (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆𝑥))
109adantl 483 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) → (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆𝑥))
1110simpld 496 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) → 𝑥 ∈ (SubDRing‘𝐹))
12 issdrg 20214 . . . . . . . . . 10 (𝑥 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑥 ∈ (SubRing‘𝐹) ∧ (𝐹s 𝑥) ∈ DivRing))
1312simp2bi 1147 . . . . . . . . 9 (𝑥 ∈ (SubDRing‘𝐹) → 𝑥 ∈ (SubRing‘𝐹))
1411, 13syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) → 𝑥 ∈ (SubRing‘𝐹))
1514ex 414 . . . . . . 7 (𝜑 → (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} → 𝑥 ∈ (SubRing‘𝐹)))
1615ssrdv 3949 . . . . . 6 (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ (SubRing‘𝐹))
17 sseq2 3969 . . . . . . . 8 (𝑎 = 𝐵 → (𝑆𝑎𝑆𝐵))
181sdrgid 20215 . . . . . . . . 9 (𝐹 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝐹))
192, 18syl 17 . . . . . . . 8 (𝜑𝐵 ∈ (SubDRing‘𝐹))
2017, 19, 3elrabd 3646 . . . . . . 7 (𝜑𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
2120ne0d 4294 . . . . . 6 (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ≠ ∅)
2212simp3bi 1148 . . . . . . 7 (𝑥 ∈ (SubDRing‘𝐹) → (𝐹s 𝑥) ∈ DivRing)
2311, 22syl 17 . . . . . 6 ((𝜑𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) → (𝐹s 𝑥) ∈ DivRing)
246, 2, 16, 21, 23subdrgint 20223 . . . . 5 (𝜑 → (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ DivRing)
2524drngringd 20146 . . . 4 (𝜑 → (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ Ring)
26 intss1 4923 . . . . 5 (𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ 𝐵)
2720, 26syl 17 . . . 4 (𝜑 {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ 𝐵)
28 issdrg 20214 . . . . . . . . . 10 (𝑎 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑎 ∈ (SubRing‘𝐹) ∧ (𝐹s 𝑎) ∈ DivRing))
2928simp2bi 1147 . . . . . . . . 9 (𝑎 ∈ (SubDRing‘𝐹) → 𝑎 ∈ (SubRing‘𝐹))
30 eqid 2738 . . . . . . . . . 10 (1r𝐹) = (1r𝐹)
3130subrg1cl 20183 . . . . . . . . 9 (𝑎 ∈ (SubRing‘𝐹) → (1r𝐹) ∈ 𝑎)
3229, 31syl 17 . . . . . . . 8 (𝑎 ∈ (SubDRing‘𝐹) → (1r𝐹) ∈ 𝑎)
3332ad2antlr 726 . . . . . . 7 (((𝜑𝑎 ∈ (SubDRing‘𝐹)) ∧ 𝑆𝑎) → (1r𝐹) ∈ 𝑎)
3433ex 414 . . . . . 6 ((𝜑𝑎 ∈ (SubDRing‘𝐹)) → (𝑆𝑎 → (1r𝐹) ∈ 𝑎))
3534ralrimiva 3142 . . . . 5 (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆𝑎 → (1r𝐹) ∈ 𝑎))
36 fvex 6853 . . . . . 6 (1r𝐹) ∈ V
3736elintrab 4920 . . . . 5 ((1r𝐹) ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆𝑎 → (1r𝐹) ∈ 𝑎))
3835, 37sylibr 233 . . . 4 (𝜑 → (1r𝐹) ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
391, 30issubrg 20175 . . . . 5 ( {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ∈ (SubRing‘𝐹) ↔ ((𝐹 ∈ Ring ∧ (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ Ring) ∧ ( {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ 𝐵 ∧ (1r𝐹) ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})))
4039biimpri 227 . . . 4 (((𝐹 ∈ Ring ∧ (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ Ring) ∧ ( {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ 𝐵 ∧ (1r𝐹) ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})) → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ∈ (SubRing‘𝐹))
415, 25, 27, 38, 40syl22anc 838 . . 3 (𝜑 {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ∈ (SubRing‘𝐹))
42 issdrg 20214 . . 3 ( {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ∈ (SubRing‘𝐹) ∧ (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ DivRing))
432, 41, 24, 42syl3anbrc 1344 . 2 (𝜑 {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ∈ (SubDRing‘𝐹))
444, 43eqeltrd 2839 1 (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3063  {crab 3406  wss 3909   cint 4906  cfv 6494  (class class class)co 7352  Basecbs 17043  s cress 17072  1rcur 19872  Ringcrg 19918  DivRingcdr 20138  SubRingcsubrg 20171  SubDRingcsdrg 20212   fldGen cfldgen 31903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-cnex 11066  ax-resscn 11067  ax-1cn 11068  ax-icn 11069  ax-addcl 11070  ax-addrcl 11071  ax-mulcl 11072  ax-mulrcl 11073  ax-mulcom 11074  ax-addass 11075  ax-mulass 11076  ax-distr 11077  ax-i2m1 11078  ax-1ne0 11079  ax-1rid 11080  ax-rnegex 11081  ax-rrecex 11082  ax-cnre 11083  ax-pre-lttri 11084  ax-pre-lttrn 11085  ax-pre-ltadd 11086  ax-pre-mulgt0 11087
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-iin 4956  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7796  df-1st 7914  df-2nd 7915  df-tpos 8150  df-frecs 8205  df-wrecs 8236  df-recs 8310  df-rdg 8349  df-er 8607  df-en 8843  df-dom 8844  df-sdom 8845  df-pnf 11150  df-mnf 11151  df-xr 11152  df-ltxr 11153  df-le 11154  df-sub 11346  df-neg 11347  df-nn 12113  df-2 12175  df-3 12176  df-sets 16996  df-slot 17014  df-ndx 17026  df-base 17044  df-ress 17073  df-plusg 17106  df-mulr 17107  df-0g 17283  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-grp 18711  df-minusg 18712  df-subg 18884  df-mgp 19856  df-ur 19873  df-ring 19920  df-oppr 20002  df-dvdsr 20023  df-unit 20024  df-invr 20054  df-dvr 20065  df-drng 20140  df-subrg 20173  df-sdrg 20213  df-fldgen 31904
This theorem is referenced by:  fldgenfld  31911  1fldgenq  31913
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