Theorem fldgenfldext 34067

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 20907	. . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
7		fldgenfldext.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
8	6, 7	unssd 4144	. . . 4 ⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ 𝐵)
9	2, 3, 8	fldgenfld 33650	. . 3 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ∈ Field)
10	1, 9	eqeltrid 2866	. 2 ⊢ (𝜑 → 𝐿 ∈ Field)
11		fldgenfldext.k	. . 3 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
12		fldsdrgfld 20912	. . . 4 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
13	3, 4, 12	syl2anc 595	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
14	11, 13	eqeltrid 2866	. 2 ⊢ (𝜑 → 𝐾 ∈ Field)
15	1	oveq1i 7422	. . . . . 6 ⊢ (𝐿 ↾s 𝐹) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹)
16		ovexd 7447	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ∈ V)
17		ressress 17313	. . . . . . 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 595	. . . . . 6 ⊢ (𝜑 → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)))
19	15, 18	eqtrid 2809	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)))
20	3	flddrngd 20852	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ DivRing)
21	2, 20, 8	fldgenssid 33643	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)))
22	21	unssad 4145	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)))
23		sseqin2 4175	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)) ↔ ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹) = 𝐹)
24	22, 23	sylib 221	. . . . . 6 ⊢ (𝜑 → ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹) = 𝐹)
25	24	oveq2d 7428	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)) = (𝐸 ↾s 𝐹))
26	19, 25	eqtrd 2797	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s 𝐹))
27	11, 2	ressbas2 17304	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘𝐾))
28	6, 27	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
29	28	oveq2d 7428	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
30	26, 29	eqtr3d 2799	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
31	11, 30	eqtrid 2809	. 2 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
32	10	fldcrngd 20853	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ CRing)
33	32	crngringd 20334	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Ring)
34	14	fldcrngd 20853	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ CRing)
35	34	crngringd 20334	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Ring)
36	11, 35	eqeltrrid 2867	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
37	26, 36	eqeltrd 2862	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) ∈ Ring)
38	2, 20, 8	fldgenssv 33645	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵)
39	1, 2	ressbas2 17304	. . . . . . 7 ⊢ ((𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵 → (𝐸 fldGen (𝐹 ∪ 𝐴)) = (Base‘𝐿))
40	38, 39	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) = (Base‘𝐿))
41	22, 40	sseqtrd 3972	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
42	20	drngringd 20846	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Ring)
43		sdrgsubrg 20905	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
44		eqid 2762	. . . . . . . . . 10 ⊢ (1r‘𝐸) = (1r‘𝐸)
45	44	subrg1cl 20690	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
46	4, 43, 45	3syl 19	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
47	22, 46	sseldd 3937	. . . . . . 7 ⊢ (𝜑 → (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴)))
48	1, 2, 44	ress1r 33561	. . . . . . 7 ⊢ ((𝐸 ∈ Ring ∧ (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴)) ∧ (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵) → (1r‘𝐸) = (1r‘𝐿))
49	42, 47, 38, 48	syl3anc 1397	. . . . . 6 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐿))
50	49, 46	eqeltrrd 2863	. . . . 5 ⊢ (𝜑 → (1r‘𝐿) ∈ 𝐹)
51	41, 50	jca 520	. . . 4 ⊢ (𝜑 → (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹))
52		eqid 2762	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
53		eqid 2762	. . . . 5 ⊢ (1r‘𝐿) = (1r‘𝐿)
54	52, 53	issubrg 20681	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿 ↾s 𝐹) ∈ Ring) ∧ (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹)))
55	33, 37, 51, 54	syl21anbrc 1362	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
56	28, 55	eqeltrrd 2863	. 2 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
57		brfldext 34044	. . 3 ⊢ ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))))
58	57	biimpar 482	. 2 ⊢ (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾)
59	10, 14, 31, 56, 58	syl22anc 851	1 ⊢ (𝜑 → 𝐿/FldExt𝐾)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  Basecbs 17275   ↾_s cress 17296  1_rcur 20269  Ringcrg 20321  SubRingcsubrg 20679  Fieldcfield 20839  SubDRingcsdrg 20900   fldGen cfldgen 33640  /_FldExtcfldext 34037

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8220  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-er 8692  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-plusg 17329  df-mulr 17330  df-0g 17500  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-grp 19009  df-minusg 19010  df-subg 19195  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-cring 20324  df-oppr 20426  df-dvdsr 20446  df-unit 20447  df-invr 20477  df-dvr 20490  df-subrng 20656  df-subrg 20680  df-drng 20840  df-field 20841  df-sdrg 20901  df-fldgen 33641  df-fldext 34040

This theorem is used by:  fldextrspundgle  34077  fldextrspundglemul  34078  fldextrspundgdvdslem  34079  fldextrspundgdvds  34080  fldext2rspun  34081  rtelextdg2  34126  constrextdg2lem  34147  constrext2chnlem  34149