Theorem fldgenfldext 33818

Description: A subfield 𝐹 extended with a set 𝐴 forms a field extension. (Contributed by Thierry Arnoux, 22-Jun-2025.)

Hypotheses

Ref	Expression
fldgenfldext.b	⊢ 𝐵 = (Base‘𝐸)
fldgenfldext.k	⊢ 𝐾 = (𝐸 ↾s 𝐹)
fldgenfldext.l	⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴)))
fldgenfldext.e	⊢ (𝜑 → 𝐸 ∈ Field)
fldgenfldext.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
fldgenfldext.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
fldgenfldext	⊢ (𝜑 → 𝐿/FldExt𝐾)

Proof of Theorem fldgenfldext

Step	Hyp	Ref	Expression
1		fldgenfldext.l	. . 3 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴)))
2		fldgenfldext.b	. . . 4 ⊢ 𝐵 = (Base‘𝐸)
3		fldgenfldext.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Field)
4		fldgenfldext.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	2	sdrgss 20728	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
7		fldgenfldext.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
8	6, 7	unssd 4133	. . . 4 ⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ 𝐵)
9	2, 3, 8	fldgenfld 33386	. . 3 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ∈ Field)
10	1, 9	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝐿 ∈ Field)
11		fldgenfldext.k	. . 3 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
12		fldsdrgfld 20733	. . . 4 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
13	3, 4, 12	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
14	11, 13	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝐾 ∈ Field)
15	1	oveq1i 7368	. . . . . 6 ⊢ (𝐿 ↾s 𝐹) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹)
16		ovexd 7393	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ∈ V)
17		ressress 17175	. . . . . . 7 ⊢ (((𝐸 fldGen (𝐹 ∪ 𝐴)) ∈ V ∧ 𝐹 ∈ (SubDRing‘𝐸)) → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)))
18	16, 4, 17	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)))
19	15, 18	eqtrid 2784	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)))
20	3	flddrngd 20676	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ DivRing)
21	2, 20, 8	fldgenssid 33379	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)))
22	21	unssad 4134	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)))
23		sseqin2 4164	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)) ↔ ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹) = 𝐹)
24	22, 23	sylib 218	. . . . . 6 ⊢ (𝜑 → ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹) = 𝐹)
25	24	oveq2d 7374	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)) = (𝐸 ↾s 𝐹))
26	19, 25	eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s 𝐹))
27	11, 2	ressbas2 17166	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘𝐾))
28	6, 27	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
29	28	oveq2d 7374	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
30	26, 29	eqtr3d 2774	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
31	11, 30	eqtrid 2784	. 2 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
32	10	fldcrngd 20677	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ CRing)
33	32	crngringd 20185	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Ring)
34	14	fldcrngd 20677	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ CRing)
35	34	crngringd 20185	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Ring)
36	11, 35	eqeltrrid 2842	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
37	26, 36	eqeltrd 2837	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) ∈ Ring)
38	2, 20, 8	fldgenssv 33381	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵)
39	1, 2	ressbas2 17166	. . . . . . 7 ⊢ ((𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵 → (𝐸 fldGen (𝐹 ∪ 𝐴)) = (Base‘𝐿))
40	38, 39	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) = (Base‘𝐿))
41	22, 40	sseqtrd 3959	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
42	20	drngringd 20672	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Ring)
43		sdrgsubrg 20726	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
44		eqid 2737	. . . . . . . . . 10 ⊢ (1r‘𝐸) = (1r‘𝐸)
45	44	subrg1cl 20515	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
46	4, 43, 45	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
47	22, 46	sseldd 3923	. . . . . . 7 ⊢ (𝜑 → (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴)))
48	1, 2, 44	ress1r 33299	. . . . . . 7 ⊢ ((𝐸 ∈ Ring ∧ (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴)) ∧ (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵) → (1r‘𝐸) = (1r‘𝐿))
49	42, 47, 38, 48	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐿))
50	49, 46	eqeltrrd 2838	. . . . 5 ⊢ (𝜑 → (1r‘𝐿) ∈ 𝐹)
51	41, 50	jca 511	. . . 4 ⊢ (𝜑 → (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹))
52		eqid 2737	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
53		eqid 2737	. . . . 5 ⊢ (1r‘𝐿) = (1r‘𝐿)
54	52, 53	issubrg 20506	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿 ↾s 𝐹) ∈ Ring) ∧ (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹)))
55	33, 37, 51, 54	syl21anbrc 1346	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
56	28, 55	eqeltrrd 2838	. 2 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
57		brfldext 33795	. . 3 ⊢ ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))))
58	57	biimpar 477	. 2 ⊢ (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾)
59	10, 14, 31, 56, 58	syl22anc 839	1 ⊢ (𝜑 → 𝐿/FldExt𝐾)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890   class class class wbr 5086  ‘cfv 6490  (class class class)co 7358  Basecbs 17137   ↾_s cress 17158  1_rcur 20120  Ringcrg 20172  SubRingcsubrg 20504  Fieldcfield 20665  SubDRingcsdrg 20721   fldGen cfldgen 33376  /_FldExtcfldext 33788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17138  df-ress 17159  df-plusg 17191  df-mulr 17192  df-0g 17362  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18870  df-minusg 18871  df-subg 19057  df-cmn 19715  df-abl 19716  df-mgp 20080  df-rng 20092  df-ur 20121  df-ring 20174  df-cring 20175  df-oppr 20275  df-dvdsr 20295  df-unit 20296  df-invr 20326  df-dvr 20339  df-subrng 20481  df-subrg 20505  df-drng 20666  df-field 20667  df-sdrg 20722  df-fldgen 33377  df-fldext 33791

This theorem is referenced by:  fldextrspundgle  33828  fldextrspundglemul  33829  fldextrspundgdvdslem  33830  fldextrspundgdvds  33831  fldext2rspun  33832  rtelextdg2  33877  constrextdg2lem  33898  constrext2chnlem  33900