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Theorem fldgensdrg 33296
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 33294 . 2 (𝜑 → (𝐹 fldGen 𝑆) = {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
52drngringd 20754 . . . 4 (𝜑𝐹 ∈ Ring)
6 eqid 2735 . . . . . 6 (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) = (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
7 sseq2 4022 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑆𝑎𝑆𝑥))
87elrab 3695 . . . . . . . . . . . 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 20806 . . . . . . . . . 10 (𝑥 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑥 ∈ (SubRing‘𝐹) ∧ (𝐹s 𝑥) ∈ DivRing))
1312simp2bi 1145 . . . . . . . . 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 4001 . . . . . 6 (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ (SubRing‘𝐹))
17 sseq2 4022 . . . . . . . 8 (𝑎 = 𝐵 → (𝑆𝑎𝑆𝐵))
181sdrgid 20810 . . . . . . . . 9 (𝐹 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝐹))
192, 18syl 17 . . . . . . . 8 (𝜑𝐵 ∈ (SubDRing‘𝐹))
2017, 19, 3elrabd 3697 . . . . . . 7 (𝜑𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
2120ne0d 4348 . . . . . 6 (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ≠ ∅)
2212simp3bi 1146 . . . . . . 7 (𝑥 ∈ (SubDRing‘𝐹) → (𝐹s 𝑥) ∈ DivRing)
2311, 22syl 17 . . . . . 6 ((𝜑𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) → (𝐹s 𝑥) ∈ DivRing)
246, 2, 16, 21, 23subdrgint 20821 . . . . 5 (𝜑 → (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ DivRing)
2524drngringd 20754 . . . 4 (𝜑 → (𝐹s {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎}) ∈ Ring)
26 intss1 4968 . . . . 5 (𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ 𝐵)
2720, 26syl 17 . . . 4 (𝜑 {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ⊆ 𝐵)
28 issdrg 20806 . . . . . . . . . 10 (𝑎 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑎 ∈ (SubRing‘𝐹) ∧ (𝐹s 𝑎) ∈ DivRing))
2928simp2bi 1145 . . . . . . . . 9 (𝑎 ∈ (SubDRing‘𝐹) → 𝑎 ∈ (SubRing‘𝐹))
30 eqid 2735 . . . . . . . . . 10 (1r𝐹) = (1r𝐹)
3130subrg1cl 20597 . . . . . . . . 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 3144 . . . . 5 (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆𝑎 → (1r𝐹) ∈ 𝑎))
36 fvex 6920 . . . . . 6 (1r𝐹) ∈ V
3736elintrab 4965 . . . . 5 ((1r𝐹) ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆𝑎 → (1r𝐹) ∈ 𝑎))
3835, 37sylibr 234 . . . 4 (𝜑 → (1r𝐹) ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆𝑎})
391, 30issubrg 20588 . . . . 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 20806 . . 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 1342 . 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 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  wss 3963   cint 4951  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  1rcur 20199  Ringcrg 20251  SubRingcsubrg 20586  DivRingcdr 20746  SubDRingcsdrg 20804   fldGen cfldgen 33292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-subrng 20563  df-subrg 20587  df-drng 20748  df-sdrg 20805  df-fldgen 33293
This theorem is referenced by:  fldgenfld  33302  1fldgenq  33304  algextdeglem2  33724  algextdeglem4  33726  algextdeglem5  33727
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