Theorem fldgenfldext 34024

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 20875	. . . . . 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 33607	. . 3 ⊢ (𝜑 → (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ∈ Field)
10	1, 9	eqeltrid 2865	. 2 ⊢ (𝜑 → 𝐿 ∈ Field)
11		fldgenfldext.k	. . 3 ⊢ 𝐾 = (𝐸 ↾s 𝐹)
12		fldsdrgfld 20880	. . . 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 2865	. 2 ⊢ (𝜑 → 𝐾 ∈ Field)
15	1	oveq1i 7420	. . . . . 6 ⊢ (𝐿 ↾s 𝐹) = ((𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴))) ↾s 𝐹)
16		ovexd 7445	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ∈ V)
17		ressress 17306	. . . . . . 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 2808	. . . . 5 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)))
20	3	flddrngd 20826	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ DivRing)
21	2, 20, 8	fldgenssid 33600	. . . . . . . 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 7426	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s ((𝐸 fldGen (𝐹 ∪ 𝐴)) ∩ 𝐹)) = (𝐸 ↾s 𝐹))
26	19, 25	eqtrd 2796	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐸 ↾s 𝐹))
27	11, 2	ressbas2 17297	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘𝐾))
28	6, 27	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘𝐾))
29	28	oveq2d 7426	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
30	26, 29	eqtr3d 2798	. . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (𝐿 ↾s (Base‘𝐾)))
31	11, 30	eqtrid 2808	. 2 ⊢ (𝜑 → 𝐾 = (𝐿 ↾s (Base‘𝐾)))
32	10	fldcrngd 20827	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ CRing)
33	32	crngringd 20327	. . . 4 ⊢ (𝜑 → 𝐿 ∈ Ring)
34	14	fldcrngd 20827	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ CRing)
35	34	crngringd 20327	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Ring)
36	11, 35	eqeltrrid 2866	. . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Ring)
37	26, 36	eqeltrd 2861	. . . 4 ⊢ (𝜑 → (𝐿 ↾s 𝐹) ∈ Ring)
38	2, 20, 8	fldgenssv 33602	. . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵)
39	1, 2	ressbas2 17297	. . . . . . 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 20820	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Ring)
43		sdrgsubrg 20873	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
44		eqid 2761	. . . . . . . . . 10 ⊢ (1r‘𝐸) = (1r‘𝐸)
45	44	subrg1cl 20664	. . . . . . . . 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 33518	. . . . . . 7 ⊢ ((𝐸 ∈ Ring ∧ (1r‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ 𝐴)) ∧ (𝐸 fldGen (𝐹 ∪ 𝐴)) ⊆ 𝐵) → (1r‘𝐸) = (1r‘𝐿))
49	42, 47, 38, 48	syl3anc 1396	. . . . . 6 ⊢ (𝜑 → (1r‘𝐸) = (1r‘𝐿))
50	49, 46	eqeltrrd 2862	. . . . 5 ⊢ (𝜑 → (1r‘𝐿) ∈ 𝐹)
51	41, 50	jca 520	. . . 4 ⊢ (𝜑 → (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹))
52		eqid 2761	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
53		eqid 2761	. . . . 5 ⊢ (1r‘𝐿) = (1r‘𝐿)
54	52, 53	issubrg 20655	. . . 4 ⊢ (𝐹 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿 ↾s 𝐹) ∈ Ring) ∧ (𝐹 ⊆ (Base‘𝐿) ∧ (1r‘𝐿) ∈ 𝐹)))
55	33, 37, 51, 54	syl21anbrc 1361	. . 3 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐿))
56	28, 55	eqeltrrd 2862	. 2 ⊢ (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
57		brfldext 34001	. . 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  Vcvv 3453   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904   class class class wbr 5108  ‘cfv 6536  (class class class)co 7410  Basecbs 17268   ↾_s cress 17289  1_rcur 20262  Ringcrg 20314  SubRingcsubrg 20653  Fieldcfield 20813  SubDRingcsdrg 20868   fldGen cfldgen 33597  /_FldExtcfldext 33994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  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 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-grp 19002  df-minusg 19003  df-subg 19188  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-cring 20317  df-oppr 20418  df-dvdsr 20438  df-unit 20439  df-invr 20469  df-dvr 20482  df-subrng 20630  df-subrg 20654  df-drng 20814  df-field 20815  df-sdrg 20869  df-fldgen 33598  df-fldext 33997

This theorem is referenced by:  fldextrspundgle  34034  fldextrspundglemul  34035  fldextrspundgdvdslem  34036  fldextrspundgdvds  34037  fldext2rspun  34038  rtelextdg2  34083  constrextdg2lem  34104  constrext2chnlem  34106