Theorem fldgensdrg 33923

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 33921	. 2 ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
5	2	drngringd 20994	. . . 4 ⊢ (𝜑 → 𝐹 ∈ Ring)
6		eqid 2760	. . . . . 6 ⊢ (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) = (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
7		sseq2 3934	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝑥))
8	7	elrab 3644	. . . . . . . . . . 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 21051	. . . . . . . . . 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 3914	. . . . . 6 ⊢ (𝜑 → {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎} ⊆ (SubRing‘𝐹))
16		sseq2 3934	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (𝑆 ⊆ 𝑎 ↔ 𝑆 ⊆ 𝐵))
17	1	sdrgid 21055	. . . . . . . . 9 ⊢ (𝐹 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝐹))
18	2, 17	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (SubDRing‘𝐹))
19	16, 18, 3	elrabd 3646	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
20	19	ne0d 4287	. . . . . 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 21066	. . . . 5 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ DivRing)
24	23	drngringd 20994	. . . 4 ⊢ (𝜑 → (𝐹 ↾s ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) ∈ Ring)
25		intss1 4922	. . . . 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 21051	. . . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (1r‘𝐹) = (1r‘𝐹)
30	29	subrg1cl 20838	. . . . . . . . 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 3154	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (SubDRing‘𝐹)(𝑆 ⊆ 𝑎 → (1r‘𝐹) ∈ 𝑎))
35		fvex 6899	. . . . . 6 ⊢ (1r‘𝐹) ∈ V
36	35	elintrab 4919	. . . . 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 20829	. . . . 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 21051	. . 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 2860	1 ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895  ∩ cint 4906  ‘cfv 6539  (class class class)co 7421  Basecbs 17393   ↾_s cress 17414  1_rcur 20413  Ringcrg 20465  SubRingcsubrg 20827  DivRingcdr 20986  SubDRingcsdrg 21049   fldGen cfldgen 33919

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-tpos 8247  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-3 12410  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-ress 17415  df-plusg 17447  df-mulr 17448  df-0g 17618  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-grp 19153  df-minusg 19154  df-subg 19339  df-cmn 20002  df-abl 20003  df-mgp 20367  df-rng 20381  df-ur 20414  df-ring 20467  df-oppr 20573  df-dvdsr 20593  df-unit 20594  df-invr 20624  df-dvr 20637  df-subrng 20804  df-subrg 20828  df-drng 20988  df-sdrg 21050  df-fldgen 33920

This theorem is used by:  fldgenfld  33929  1fldgenq  33931  fldextrspunlem2  34356  fldextrspundgdvdslem  34359  fldextrspundgdvds  34360  algextdeglem2  34397  algextdeglem4  34399  algextdeglem5  34400  constrextdg2lem  34427  constrext2chnlem  34429