Theorem fldgenfldext 33806

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 20730	. . . . . 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 4145	. . . 4 ⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ 𝐵)
9	2, 3, 8	fldgenfld 33383	. . 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 20735	. . . 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 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 17178	. . . . . . 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 20678	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ DivRing)
21	2, 20, 8	fldgenssid 33376	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∪ 𝐴) ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)))
22	21	unssad 4146	. . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ 𝐴)))
23		sseqin2 4176	. . . . . . 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 7376	. . . . 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 17169	. . . . . 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 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 20679	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ CRing)
33	32	crngringd 20185	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Ring)
34	14	fldcrngd 20679	. . . . . . 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 33378	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵)
39	1, 2	ressbas2 17169	. . . . . . 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 3971	. . . . 5 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
42	20	drngringd 20674	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Ring)
43		sdrgsubrg 20728	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
44		eqid 2737	. . . . . . . . . 10 ⊢ (1r‘𝐸) = (1r‘𝐸)
45	44	subrg1cl 20517	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubRing‘𝐸) → (1r‘𝐸) ∈ 𝐹)
46	4, 43, 45	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝐸) ∈ 𝐹)
47	22, 46	sseldd 3935	. . . . . . 7 ⊢ (𝜑 → (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴)))
48	1, 2, 44	ress1r 33296	. . . . . . 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 20508	. . . 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 33783	. . 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 3441   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5099  ‘cfv 6493  (class class class)co 7360  Basecbs 17140   ↾_s cress 17161  1_rcur 20120  Ringcrg 20172  SubRingcsubrg 20506  Fieldcfield 20667  SubDRingcsdrg 20723   fldGen cfldgen 33373  /_FldExtcfldext 33776

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  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 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-3 12213  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-mulr 17195  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  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 20277  df-dvdsr 20297  df-unit 20298  df-invr 20328  df-dvr 20341  df-subrng 20483  df-subrg 20507  df-drng 20668  df-field 20669  df-sdrg 20724  df-fldgen 33374  df-fldext 33779

This theorem is referenced by:  fldextrspundgle  33816  fldextrspundglemul  33817  fldextrspundgdvdslem  33818  fldextrspundgdvds  33819  fldext2rspun  33820  rtelextdg2  33865  constrextdg2lem  33886  constrext2chnlem  33888