Theorem fldgensdrg 33574

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 33572	. 2 ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
5	2	drngringd 20817	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
6		eqid 2769	. . . . . 6 ⊢ (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) = (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
7		sseq2 3971	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝑥))
8	7	elrab 3659	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ↔ (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆 ⊆ 𝑥))
9	8	bilani 509	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆 ⊆ 𝑥))
10	9	simpld 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → 𝑥 ∈ (SubDRing‘𝐹))
11		issdrg 20865	. . . . . . . . . 10 ⊢ (𝑥 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑥 ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s 𝑥) ∈ DivRing))
12	11	simp2bi 1162	. . . . . . . . 9 ⊢ (𝑥 ∈ (SubDRing‘𝐹) → 𝑥 ∈ (SubRing‘𝐹))
13	10, 12	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → 𝑥 ∈ (SubRing‘𝐹))
14	13	ex 417	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} → 𝑥 ∈ (SubRing‘𝐹)))
15	14	ssrdv 3951	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ (SubRing‘𝐹))
16		sseq2 3971	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝐵))
17	1	sdrgid 20869	. . . . . . . . 9 ⊢ (𝐹 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝐹))
18	2, 17	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐹))
19	16, 18, 3	elrabd 3661	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
20	19	ne0d 4303	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ≠ ∅)
21	11	simp3bi 1163	. . . . . . 7 ⊢ (𝑥 ∈ (SubDRing‘𝐹) → (𝐹 ↾s 𝑥) ∈ DivRing)
22	10, 21	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → (𝐹 ↾s 𝑥) ∈ DivRing)
23	6, 2, 15, 20, 22	subdrgint 20880	. . . . 5 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ DivRing)
24	23	drngringd 20817	. . . 4 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ Ring)
25		intss1 4929	. . . . 5 ⊢ (𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ 𝐵)
26	19, 25	syl 18	. . . 4 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ 𝐵)
27		issdrg 20865	. . . . . . . . . 10 ⊢ (𝑎 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑎 ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s 𝑎) ∈ DivRing))
28	27	simp2bi 1162	. . . . . . . . 9 ⊢ (𝑎 ∈ (SubDRing‘𝐹) → 𝑎 ∈ (SubRing‘𝐹))
29		eqid 2769	. . . . . . . . . 10 ⊢ (1r‘𝐹) = (1r‘𝐹)
30	29	subrg1cl 20661	. . . . . . . . 9 ⊢ (𝑎 ∈ (SubRing‘𝐹) → (1r‘𝐹) ∈ 𝑎)
31	28, 30	syl 18	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐹) → (1r‘𝐹) ∈ 𝑎)
32	31	ad2antlr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐹)) ∧ 𝑆 ⊆ 𝑎) → (1r‘𝐹) ∈ 𝑎)
33	32	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐹)) → (𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
34	33	ralrimiva 3163	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
35		fvex 6892	. . . . . 6 ⊢ (1r‘𝐹) ∈ V
36	35	elintrab 4926	. . . . 5 ⊢ ((1r‘𝐹) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ↔ ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
37	34, 36	sylibr 237	. . . 4 ⊢ (𝜑 → (1r‘𝐹) ∈ ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
38	1, 29	issubrg 20652	. . . . 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})))
39	38	biimpri 231	. . . 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))
40	5, 24, 26, 37, 39	syl22anc 851	. . 3 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubRing‘𝐹))
41		issdrg 20865	. . 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))
42	2, 40, 23, 41	syl3anbrc 1360	. 2 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubDRing‘𝐹))
43	4, 42	eqeltrd 2869	1 ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   ⊆ wss 3913  ∩ cint 4913  ‘cfv 6533  (class class class)co 7408  Basecbs 17265   ↾_s cress 17286  1_rcur 20259  Ringcrg 20311  SubRingcsubrg 20650  DivRingcdr 20809  SubDRingcsdrg 20863   fldGen cfldgen 33570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-mulr 17320  df-0g 17490  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-grp 18999  df-minusg 19000  df-subg 19185  df-cmn 19848  df-abl 19849  df-mgp 20213  df-rng 20227  df-ur 20260  df-ring 20313  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-invr 20466  df-dvr 20479  df-subrng 20627  df-subrg 20651  df-drng 20811  df-sdrg 20864  df-fldgen 33571

This theorem is referenced by:  fldgenfld  33580  1fldgenq  33582  fldextrspunlem2  34008  fldextrspundgdvdslem  34011  fldextrspundgdvds  34012  algextdeglem2  34049  algextdeglem4  34051  algextdeglem5  34052  constrextdg2lem  34079  constrext2chnlem  34081