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Theorem fldgensdrg 33316
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 33314 . 2 (𝜑 → (𝐹 fldGen 𝑆) = {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
52drngringd 20737 . . . 4 (𝜑𝐹 ∈ Ring)
6 eqid 2737 . . . . . 6 (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) = (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
7 sseq2 4010 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑆𝑎𝑆𝑥))
87elrab 3692 . . . . . . . . . . . 12 (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ↔ (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆𝑥))
98biimpi 216 . . . . . . . . . . 11 (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} → (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆𝑥))
109adantl 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) → (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆𝑥))
1110simpld 494 . . . . . . . . 9 ((𝜑𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) → 𝑥 ∈ (SubDRing‘𝐹))
12 issdrg 20789 . . . . . . . . . 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 412 . . . . . . 7 (𝜑 → (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} → 𝑥 ∈ (SubRing‘𝐹)))
1615ssrdv 3989 . . . . . 6 (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ (SubRing‘𝐹))
17 sseq2 4010 . . . . . . . 8 (𝑎 = 𝐵 → (𝑆𝑎𝑆𝐵))
181sdrgid 20793 . . . . . . . . 9 (𝐹 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝐹))
192, 18syl 17 . . . . . . . 8 (𝜑𝐵 ∈ (SubDRing‘𝐹))
2017, 19, 3elrabd 3694 . . . . . . 7 (𝜑𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
2120ne0d 4342 . . . . . 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 20804 . . . . 5 (𝜑 → (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ DivRing)
2524drngringd 20737 . . . 4 (𝜑 → (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ Ring)
26 intss1 4963 . . . . 5 (𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ 𝐵)
2720, 26syl 17 . . . 4 (𝜑 {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ 𝐵)
28 issdrg 20789 . . . . . . . . . 10 (𝑎 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑎 ∈ (SubRing‘𝐹) ∧ (𝐹s 𝑎) ∈ DivRing))
2928simp2bi 1147 . . . . . . . . 9 (𝑎 ∈ (SubDRing‘𝐹) → 𝑎 ∈ (SubRing‘𝐹))
30 eqid 2737 . . . . . . . . . 10 (1r𝐹) = (1r𝐹)
3130subrg1cl 20580 . . . . . . . . 9 (𝑎 ∈ (SubRing‘𝐹) → (1r𝐹) ∈ 𝑎)
3229, 31syl 17 . . . . . . . 8 (𝑎 ∈ (SubDRing‘𝐹) → (1r𝐹) ∈ 𝑎)
3332ad2antlr 727 . . . . . . 7 (((𝜑𝑎 ∈ (SubDRing‘𝐹)) ∧ 𝑆𝑎) → (1r𝐹) ∈ 𝑎)
3433ex 412 . . . . . 6 ((𝜑𝑎 ∈ (SubDRing‘𝐹)) → (𝑆𝑎 → (1r𝐹) ∈ 𝑎))
3534ralrimiva 3146 . . . . 5 (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆𝑎 → (1r𝐹) ∈ 𝑎))
36 fvex 6919 . . . . . 6 (1r𝐹) ∈ V
3736elintrab 4960 . . . . 5 ((1r𝐹) ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆𝑎 → (1r𝐹) ∈ 𝑎))
3835, 37sylibr 234 . . . 4 (𝜑 → (1r𝐹) ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
391, 30issubrg 20571 . . . . 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 228 . . . 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 839 . . 3 (𝜑 {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ∈ (SubRing‘𝐹))
42 issdrg 20789 . . 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 2841 1 (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951   cint 4946  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  1rcur 20178  Ringcrg 20230  SubRingcsubrg 20569  DivRingcdr 20729  SubDRingcsdrg 20787   fldGen cfldgen 33312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-subrng 20546  df-subrg 20570  df-drng 20731  df-sdrg 20788  df-fldgen 33313
This theorem is referenced by:  fldgenfld  33322  1fldgenq  33324  fldextrspunlem2  33727  fldextrspundgdvdslem  33730  fldextrspundgdvds  33731  algextdeglem2  33759  algextdeglem4  33761  algextdeglem5  33762  constrextdg2lem  33789
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