Theorem fldgensdrg 33658

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 33656	. 2 ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
5	2	drngringd 20864	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
6		eqid 2765	. . . . . 6 ⊢ (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) = (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
7		sseq2 3964	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝑥))
8	7	elrab 3652	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ↔ (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆 ⊆ 𝑥))
9	8	bilani 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → (𝑥 ∈ (SubDRing‘𝐹) ∧ 𝑆 ⊆ 𝑥))
10	9	simpld 500	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) → 𝑥 ∈ (SubDRing‘𝐹))
11		issdrg 20920	. . . . . . . . . 10 ⊢ (𝑥 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑥 ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s 𝑥) ∈ DivRing))
12	11	simp2bi 1164	. . . . . . . . 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 418	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} → 𝑥 ∈ (SubRing‘𝐹)))
15	14	ssrdv 3944	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ (SubRing‘𝐹))
16		sseq2 3964	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝐵))
17	1	sdrgid 20924	. . . . . . . . 9 ⊢ (𝐹 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝐹))
18	2, 17	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐹))
19	16, 18, 3	elrabd 3654	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
20	19	ne0d 4295	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ≠ ∅)
21	11	simp3bi 1165	. . . . . . 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 20935	. . . . 5 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ DivRing)
24	23	drngringd 20864	. . . 4 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ Ring)
25		intss1 4930	. . . . 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 20920	. . . . . . . . . 10 ⊢ (𝑎 ∈ (SubDRing‘𝐹) ↔ (𝐹 ∈ DivRing ∧ 𝑎 ∈ (SubRing‘𝐹) ∧ (𝐹 ↾s 𝑎) ∈ DivRing))
28	27	simp2bi 1164	. . . . . . . . 9 ⊢ (𝑎 ∈ (SubDRing‘𝐹) → 𝑎 ∈ (SubRing‘𝐹))
29		eqid 2765	. . . . . . . . . 10 ⊢ (1r‘𝐹) = (1r‘𝐹)
30	29	subrg1cl 20708	. . . . . . . . 9 ⊢ (𝑎 ∈ (SubRing‘𝐹) → (1r‘𝐹) ∈ 𝑎)
31	28, 30	syl 18	. . . . . . . 8 ⊢ (𝑎 ∈ (SubDRing‘𝐹) → (1r‘𝐹) ∈ 𝑎)
32	31	ad2antlr 740	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐹)) ∧ 𝑆 ⊆ 𝑎) → (1r‘𝐹) ∈ 𝑎)
33	32	ex 418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (SubDRing‘𝐹)) → (𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
34	33	ralrimiva 3159	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
35		fvex 6898	. . . . . 6 ⊢ (1r‘𝐹) ∈ V
36	35	elintrab 4927	. . . . 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 20699	. . . . 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 852	. . 3 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubRing‘𝐹))
41		issdrg 20920	. . 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 1362	. 2 ⊢ (𝜑 → ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ∈ (SubDRing‘𝐹))
43	4, 42	eqeltrd 2865	1 ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418   ⊆ wss 3906  ∩ cint 4914  ‘cfv 6540  (class class class)co 7416  Basecbs 17286   ↾_s cress 17307  1_rcur 20286  Ringcrg 20338  SubRingcsubrg 20697  DivRingcdr 20856  SubDRingcsdrg 20918   fldGen cfldgen 33654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-sets 17241  df-slot 17259  df-ndx 17271  df-base 17287  df-ress 17308  df-plusg 17340  df-mulr 17341  df-0g 17511  df-mgm 18715  df-sgrp 18798  df-mnd 18814  df-grp 19026  df-minusg 19027  df-subg 19212  df-cmn 19875  df-abl 19876  df-mgp 20240  df-rng 20254  df-ur 20287  df-ring 20340  df-oppr 20444  df-dvdsr 20464  df-unit 20465  df-invr 20495  df-dvr 20508  df-subrng 20674  df-subrg 20698  df-drng 20858  df-sdrg 20919  df-fldgen 33655

This theorem is used by:  fldgenfld  33664  1fldgenq  33666  fldextrspunlem2  34090  fldextrspundgdvdslem  34093  fldextrspundgdvds  34094  algextdeglem2  34131  algextdeglem4  34133  algextdeglem5  34134  constrextdg2lem  34161  constrext2chnlem  34163