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Theorem fldgenfldext 33719
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 = (𝐸s 𝐹)
fldgenfldext.l 𝐿 = (𝐸s (𝐸 fldGen (𝐹𝐴)))
fldgenfldext.e (𝜑𝐸 ∈ Field)
fldgenfldext.f (𝜑𝐹 ∈ (SubDRing‘𝐸))
fldgenfldext.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
fldgenfldext (𝜑𝐿/FldExt𝐾)

Proof of Theorem fldgenfldext
StepHypRef Expression
1 fldgenfldext.l . . 3 𝐿 = (𝐸s (𝐸 fldGen (𝐹𝐴)))
2 fldgenfldext.b . . . 4 𝐵 = (Base‘𝐸)
3 fldgenfldext.e . . . 4 (𝜑𝐸 ∈ Field)
4 fldgenfldext.f . . . . . 6 (𝜑𝐹 ∈ (SubDRing‘𝐸))
52sdrgss 20795 . . . . . 6 (𝐹 ∈ (SubDRing‘𝐸) → 𝐹𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐹𝐵)
7 fldgenfldext.1 . . . . 5 (𝜑𝐴𝐵)
86, 7unssd 4191 . . . 4 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
92, 3, 8fldgenfld 33323 . . 3 (𝜑 → (𝐸s (𝐸 fldGen (𝐹𝐴))) ∈ Field)
101, 9eqeltrid 2844 . 2 (𝜑𝐿 ∈ Field)
11 fldgenfldext.k . . 3 𝐾 = (𝐸s 𝐹)
12 fldsdrgfld 20800 . . . 4 ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸s 𝐹) ∈ Field)
133, 4, 12syl2anc 584 . . 3 (𝜑 → (𝐸s 𝐹) ∈ Field)
1411, 13eqeltrid 2844 . 2 (𝜑𝐾 ∈ Field)
151oveq1i 7442 . . . . . 6 (𝐿s 𝐹) = ((𝐸s (𝐸 fldGen (𝐹𝐴))) ↾s 𝐹)
16 ovexd 7467 . . . . . . 7 (𝜑 → (𝐸 fldGen (𝐹𝐴)) ∈ V)
17 ressress 17294 . . . . . . 7 (((𝐸 fldGen (𝐹𝐴)) ∈ V ∧ 𝐹 ∈ (SubDRing‘𝐸)) → ((𝐸s (𝐸 fldGen (𝐹𝐴))) ↾s 𝐹) = (𝐸s ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹)))
1816, 4, 17syl2anc 584 . . . . . 6 (𝜑 → ((𝐸s (𝐸 fldGen (𝐹𝐴))) ↾s 𝐹) = (𝐸s ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹)))
1915, 18eqtrid 2788 . . . . 5 (𝜑 → (𝐿s 𝐹) = (𝐸s ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹)))
203flddrngd 20742 . . . . . . . . 9 (𝜑𝐸 ∈ DivRing)
212, 20, 8fldgenssid 33316 . . . . . . . 8 (𝜑 → (𝐹𝐴) ⊆ (𝐸 fldGen (𝐹𝐴)))
2221unssad 4192 . . . . . . 7 (𝜑𝐹 ⊆ (𝐸 fldGen (𝐹𝐴)))
23 sseqin2 4222 . . . . . . 7 (𝐹 ⊆ (𝐸 fldGen (𝐹𝐴)) ↔ ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹) = 𝐹)
2422, 23sylib 218 . . . . . 6 (𝜑 → ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹) = 𝐹)
2524oveq2d 7448 . . . . 5 (𝜑 → (𝐸s ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹)) = (𝐸s 𝐹))
2619, 25eqtrd 2776 . . . 4 (𝜑 → (𝐿s 𝐹) = (𝐸s 𝐹))
2711, 2ressbas2 17284 . . . . . 6 (𝐹𝐵𝐹 = (Base‘𝐾))
286, 27syl 17 . . . . 5 (𝜑𝐹 = (Base‘𝐾))
2928oveq2d 7448 . . . 4 (𝜑 → (𝐿s 𝐹) = (𝐿s (Base‘𝐾)))
3026, 29eqtr3d 2778 . . 3 (𝜑 → (𝐸s 𝐹) = (𝐿s (Base‘𝐾)))
3111, 30eqtrid 2788 . 2 (𝜑𝐾 = (𝐿s (Base‘𝐾)))
3210fldcrngd 20743 . . . . 5 (𝜑𝐿 ∈ CRing)
3332crngringd 20244 . . . 4 (𝜑𝐿 ∈ Ring)
3414fldcrngd 20743 . . . . . . 7 (𝜑𝐾 ∈ CRing)
3534crngringd 20244 . . . . . 6 (𝜑𝐾 ∈ Ring)
3611, 35eqeltrrid 2845 . . . . 5 (𝜑 → (𝐸s 𝐹) ∈ Ring)
3726, 36eqeltrd 2840 . . . 4 (𝜑 → (𝐿s 𝐹) ∈ Ring)
382, 20, 8fldgenssv 33318 . . . . . . 7 (𝜑 → (𝐸 fldGen (𝐹𝐴)) ⊆ 𝐵)
391, 2ressbas2 17284 . . . . . . 7 ((𝐸 fldGen (𝐹𝐴)) ⊆ 𝐵 → (𝐸 fldGen (𝐹𝐴)) = (Base‘𝐿))
4038, 39syl 17 . . . . . 6 (𝜑 → (𝐸 fldGen (𝐹𝐴)) = (Base‘𝐿))
4122, 40sseqtrd 4019 . . . . 5 (𝜑𝐹 ⊆ (Base‘𝐿))
4220drngringd 20738 . . . . . . 7 (𝜑𝐸 ∈ Ring)
43 sdrgsubrg 20793 . . . . . . . . 9 (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
44 eqid 2736 . . . . . . . . . 10 (1r𝐸) = (1r𝐸)
4544subrg1cl 20581 . . . . . . . . 9 (𝐹 ∈ (SubRing‘𝐸) → (1r𝐸) ∈ 𝐹)
464, 43, 453syl 18 . . . . . . . 8 (𝜑 → (1r𝐸) ∈ 𝐹)
4722, 46sseldd 3983 . . . . . . 7 (𝜑 → (1r𝐸) ∈ (𝐸 fldGen (𝐹𝐴)))
481, 2, 44ress1r 33239 . . . . . . 7 ((𝐸 ∈ Ring ∧ (1r𝐸) ∈ (𝐸 fldGen (𝐹𝐴)) ∧ (𝐸 fldGen (𝐹𝐴)) ⊆ 𝐵) → (1r𝐸) = (1r𝐿))
4942, 47, 38, 48syl3anc 1372 . . . . . 6 (𝜑 → (1r𝐸) = (1r𝐿))
5049, 46eqeltrrd 2841 . . . . 5 (𝜑 → (1r𝐿) ∈ 𝐹)
5141, 50jca 511 . . . 4 (𝜑 → (𝐹 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝐹))
52 eqid 2736 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
53 eqid 2736 . . . . 5 (1r𝐿) = (1r𝐿)
5452, 53issubrg 20572 . . . 4 (𝐹 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝐹) ∈ Ring) ∧ (𝐹 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝐹)))
5533, 37, 51, 54syl21anbrc 1344 . . 3 (𝜑𝐹 ∈ (SubRing‘𝐿))
5628, 55eqeltrrd 2841 . 2 (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
57 brfldext 33699 . . 3 ((𝐿 ∈ Field ∧ 𝐾 ∈ Field) → (𝐿/FldExt𝐾 ↔ (𝐾 = (𝐿s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))))
5857biimpar 477 . 2 (((𝐿 ∈ Field ∧ 𝐾 ∈ Field) ∧ (𝐾 = (𝐿s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐿))) → 𝐿/FldExt𝐾)
5910, 14, 31, 56, 58syl22anc 838 1 (𝜑𝐿/FldExt𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  cin 3949  wss 3950   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  s cress 17275  1rcur 20179  Ringcrg 20231  SubRingcsubrg 20570  Fieldcfield 20731  SubDRingcsdrg 20788   fldGen cfldgen 33313  /FldExtcfldext 33690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-subg 19142  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-cring 20234  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-dvr 20402  df-subrng 20547  df-subrg 20571  df-drng 20732  df-field 20733  df-sdrg 20789  df-fldgen 33314  df-fldext 33694
This theorem is referenced by:  fldextrspundgle  33729  fldextrspundglemul  33730  fldextrspundgdvdslem  33731  fldextrspundgdvds  33732  fldext2rspun  33733  rtelextdg2  33769  constrextdg2lem  33790
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