Theorem fldgensdrg 33266

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	⊢ (𝜑 → 𝐹 ∈ DivRing)
fldgenval.3	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

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.

Step	Hyp	Ref	Expression
1		fldgenval.1	. . 3 ⊢ 𝐵 = (Base‘𝐹)
2		fldgenval.2	. . 3 ⊢ (𝜑 → 𝐹 ∈ DivRing)
3		fldgenval.3	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
4	1, 2, 3	fldgenval 33264	. 2 ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
5	2	drngringd 20640	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
6		eqid 2729	. . . . . 6 ⊢ (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) = (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
7		sseq2 3964	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝑥))
8	7	elrab 3650	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ↔ (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆 ⊆ 𝑥))
9	8	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} → (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆 ⊆ 𝑥))
10	9	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆 ⊆ 𝑥))
11	10	simpld 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → 𝑥 ∈ (SubDRing‘𝐹))
12		issdrg 20691	. . . . . . . . . 10 ⊢ (𝑥 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑥 ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s 𝑥) ∈ DivRing))
13	12	simp2bi 1146	. . . . . . . . 9 ⊢ (𝑥 ∈ (SubDRing‘𝐹) → 𝑥 ∈ (SubRing‘𝐹))
14	11, 13	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → 𝑥 ∈ (SubRing‘𝐹))
15	14	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} → 𝑥 ∈ (SubRing‘𝐹)))
16	15	ssrdv 3943	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ (SubRing‘𝐹))
17		sseq2 3964	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝐵))
18	1	sdrgid 20695	. . . . . . . . 9 ⊢ (𝐹 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝐹))
19	2, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐹))
20	17, 19, 3	elrabd 3652	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
21	20	ne0d 4295	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ≠ ∅)
22	12	simp3bi 1147	. . . . . . 7 ⊢ (𝑥 ∈ (SubDRing‘𝐹) → (𝐹 ↾s 𝑥) ∈ DivRing)
23	11, 22	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → (𝐹 ↾s 𝑥) ∈ DivRing)
24	6, 2, 16, 21, 23	subdrgint 20706	. . . . 5 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ DivRing)
25	24	drngringd 20640	. . . 4 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ Ring)
26		intss1 4916	. . . . 5 ⊢ (𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ 𝐵)
27	20, 26	syl 17	. . . 4 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ 𝐵)
28		issdrg 20691	. . . . . . . . . 10 ⊢ (𝑎 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑎 ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s 𝑎) ∈ DivRing))
29	28	simp2bi 1146	. . . . . . . . 9 ⊢ (𝑎 ∈ (SubDRing‘𝐹) → 𝑎 ∈ (SubRing‘𝐹))
30		eqid 2729	. . . . . . . . . 10 ⊢ (1r‘𝐹) = (1r‘𝐹)
31	30	subrg1cl 20483	. . . . . . . . 9 ⊢ (𝑎 ∈ (SubRing‘𝐹) → (1r‘𝐹) ∈ 𝑎)
32	29, 31	syl 17	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐹) → (1r‘𝐹) ∈ 𝑎)
33	32	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐹)) ∧ 𝑆 ⊆ 𝑎) → (1r‘𝐹) ∈ 𝑎)
34	33	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐹)) → (𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
35	34	ralrimiva 3121	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
36		fvex 6839	. . . . . 6 ⊢ (1r‘𝐹) ∈ V
37	36	elintrab 4913	. . . . 5 ⊢ ((1r‘𝐹) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
38	35, 37	sylibr 234	. . . 4 ⊢ (𝜑 → (1r‘𝐹) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
39	1, 30	issubrg 20474	. . . . 5 ⊢ (∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubRing‘𝐹) ↔ ((𝐹 ∈ Ring ∧ (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ Ring) ∧ (∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ 𝐵 ∧ (1r‘𝐹) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})))
40	39	biimpri 228	. . . 4 ⊢ (((𝐹 ∈ Ring ∧ (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ Ring) ∧ (∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ 𝐵 ∧ (1r‘𝐹) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})) → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubRing‘𝐹))
41	5, 25, 27, 38, 40	syl22anc 838	. . 3 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubRing‘𝐹))
42		issdrg 20691	. . 3 ⊢ (∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ DivRing))
43	2, 41, 24, 42	syl3anbrc 1344	. 2 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubDRing‘𝐹))
44	4, 43	eqeltrd 2828	1 ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396   ⊆ wss 3905  ∩ cint 4899  ‘cfv 6486  (class class class)co 7353  Basecbs 17138   ↾_s cress 17159  1_rcur 20084  Ringcrg 20136  SubRingcsubrg 20472  DivRingcdr 20632  SubDRingcsdrg 20689   fldGen cfldgen 33262

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-mulr 17193  df-0g 17363  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-grp 18833  df-minusg 18834  df-subg 19020  df-cmn 19679  df-abl 19680  df-mgp 20044  df-rng 20056  df-ur 20085  df-ring 20138  df-oppr 20240  df-dvdsr 20260  df-unit 20261  df-invr 20291  df-dvr 20304  df-subrng 20449  df-subrg 20473  df-drng 20634  df-sdrg 20690  df-fldgen 33263

This theorem is referenced by:  fldgenfld  33272  1fldgenq  33274  fldextrspunlem2  33651  fldextrspundgdvdslem  33654  fldextrspundgdvds  33655  algextdeglem2  33687  algextdeglem4  33689  algextdeglem5  33690  constrextdg2lem  33717  constrext2chnlem  33719