Theorem fldgenfldext 33864

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 20769	. . . . . 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 4124	. . . 4 ⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ 𝐵)
9	2, 3, 8	fldgenfld 33408	. . 3 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ∈ Field)
10	1, 9	eqeltrid 2845	. 2 ⊢ (𝜑 → 𝐿 ∈ Field)
11		fldgenfldext.k	. . 3 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
12		fldsdrgfld 20774	. . . 4 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field)
13	3, 4, 12	syl2anc 591	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field)
14	11, 13	eqeltrid 2845	. 2 ⊢ (𝜑 → 𝐾 ∈ Field)
15	1	oveq1i 7370	. . . . . 6 ⊢ (𝐿 ↾s 𝐹) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹)
16		ovexd 7395	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ∈ V)
17		ressress 17212	. . . . . . 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 591	. . . . . 6 ⊢ (𝜑 → ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)))
19	15, 18	eqtrid 2788	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)))
20	3	flddrngd 20717	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ DivRing)
21	2, 20, 8	fldgenssid 33401	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)))
22	21	unssad 4125	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)))
23		sseqin2 4155	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)) ↔ ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹) = 𝐹)
24	22, 23	sylib 220	. . . . . 6 ⊢ (𝜑 → ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹) = 𝐹)
25	24	oveq2d 7376	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)) = (𝐸 ↾s 𝐹))
26	19, 25	eqtrd 2776	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s 𝐹))
27	11, 2	ressbas2 17203	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘𝐾))
28	6, 27	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
29	28	oveq2d 7376	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
30	26, 29	eqtr3d 2778	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
31	11, 30	eqtrid 2788	. 2 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
32	10	fldcrngd 20718	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ CRing)
33	32	crngringd 20222	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Ring)
34	14	fldcrngd 20718	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ CRing)
35	34	crngringd 20222	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Ring)
36	11, 35	eqeltrrid 2846	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
37	26, 36	eqeltrd 2841	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) ∈ Ring)
38	2, 20, 8	fldgenssv 33403	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵)
39	1, 2	ressbas2 17203	. . . . . . 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 3953	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
42	20	drngringd 20713	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Ring)
43		sdrgsubrg 20767	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
44		eqid 2741	. . . . . . . . . 10 ⊢ (1r‘𝐸) = (1r‘𝐸)
45	44	subrg1cl 20556	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
46	4, 43, 45	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
47	22, 46	sseldd 3918	. . . . . . 7 ⊢ (𝜑 → (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴)))
48	1, 2, 44	ress1r 33318	. . . . . . 7 ⊢ ((𝐸 ∈ Ring ∧ (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴)) ∧ (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵) → (1r‘𝐸) = (1r‘𝐿))
49	42, 47, 38, 48	syl3anc 1380	. . . . . 6 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐿))
50	49, 46	eqeltrrd 2842	. . . . 5 ⊢ (𝜑 → (1r‘𝐿) ∈ 𝐹)
51	41, 50	jca 517	. . . 4 ⊢ (𝜑 → (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹))
52		eqid 2741	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
53		eqid 2741	. . . . 5 ⊢ (1r‘𝐿) = (1r‘𝐿)
54	52, 53	issubrg 20547	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿 ↾s 𝐹) ∈ Ring) ∧ (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹)))
55	33, 37, 51, 54	syl21anbrc 1352	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
56	28, 55	eqeltrrd 2842	. 2 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
57		brfldext 33841	. . 3 ⊢ ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))))
58	57	biimpar 479	. 2 ⊢ (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾)
59	10, 14, 31, 56, 58	syl22anc 845	1 ⊢ (𝜑 → 𝐿/FldExt𝐾)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433   ∪ cun 3883   ∩ cin 3884   ⊆ wss 3885   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360  Basecbs 17174   ↾_s cress 17195  1_rcur 20157  Ringcrg 20209  SubRingcsubrg 20545  Fieldcfield 20706  SubDRingcsdrg 20762   fldGen cfldgen 33398  /_FldExtcfldext 33834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-0g 17399  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-minusg 18908  df-subg 19094  df-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-ring 20211  df-cring 20212  df-oppr 20312  df-dvdsr 20332  df-unit 20333  df-invr 20363  df-dvr 20376  df-subrng 20522  df-subrg 20546  df-drng 20707  df-field 20708  df-sdrg 20763  df-fldgen 33399  df-fldext 33837

This theorem is referenced by:  fldextrspundgle  33874  fldextrspundglemul  33875  fldextrspundgdvdslem  33876  fldextrspundgdvds  33877  fldext2rspun  33878  rtelextdg2  33923  constrextdg2lem  33944  constrext2chnlem  33946