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	⊢ (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 33314	. 2 ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
5	2	drngringd 20737	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
6		eqid 2737	. . . . . 6 ⊢ (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) = (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
7		sseq2 4010	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝑥))
8	7	elrab 3692	. . . . . . . . . . . 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 20789	. . . . . . . . . 10 ⊢ (𝑥 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑥 ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s 𝑥) ∈ DivRing))
13	12	simp2bi 1147	. . . . . . . . 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 3989	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ (SubRing‘𝐹))
17		sseq2 4010	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝐵))
18	1	sdrgid 20793	. . . . . . . . 9 ⊢ (𝐹 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝐹))
19	2, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐹))
20	17, 19, 3	elrabd 3694	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
21	20	ne0d 4342	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ≠ ∅)
22	12	simp3bi 1148	. . . . . . 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 20804	. . . . 5 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ DivRing)
25	24	drngringd 20737	. . . 4 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ Ring)
26		intss1 4963	. . . . 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 20789	. . . . . . . . . 10 ⊢ (𝑎 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑎 ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s 𝑎) ∈ DivRing))
29	28	simp2bi 1147	. . . . . . . . 9 ⊢ (𝑎 ∈ (SubDRing‘𝐹) → 𝑎 ∈ (SubRing‘𝐹))
30		eqid 2737	. . . . . . . . . 10 ⊢ (1r‘𝐹) = (1r‘𝐹)
31	30	subrg1cl 20580	. . . . . . . . 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 3146	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
36		fvex 6919	. . . . . 6 ⊢ (1r‘𝐹) ∈ V
37	36	elintrab 4960	. . . . 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 20571	. . . . 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 839	. . 3 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubRing‘𝐹))
42		issdrg 20789	. . 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 2841	1 ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))

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  1_rcur 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