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Theorem fldgenfldext 33693
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 20811 . . . . . 6 (𝐹 ∈ (SubDRing‘𝐸) → 𝐹𝐵)
64, 5syl 17 . . . . 5 (𝜑𝐹𝐵)
7 fldgenfldext.1 . . . . 5 (𝜑𝐴𝐵)
86, 7unssd 4202 . . . 4 (𝜑 → (𝐹𝐴) ⊆ 𝐵)
92, 3, 8fldgenfld 33302 . . 3 (𝜑 → (𝐸s (𝐸 fldGen (𝐹𝐴))) ∈ Field)
101, 9eqeltrid 2843 . 2 (𝜑𝐿 ∈ Field)
11 fldgenfldext.k . . 3 𝐾 = (𝐸s 𝐹)
12 fldsdrgfld 20816 . . . 4 ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸s 𝐹) ∈ Field)
133, 4, 12syl2anc 584 . . 3 (𝜑 → (𝐸s 𝐹) ∈ Field)
1411, 13eqeltrid 2843 . 2 (𝜑𝐾 ∈ Field)
151oveq1i 7441 . . . . . 6 (𝐿s 𝐹) = ((𝐸s (𝐸 fldGen (𝐹𝐴))) ↾s 𝐹)
16 ovexd 7466 . . . . . . 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 2787 . . . . 5 (𝜑 → (𝐿s 𝐹) = (𝐸s ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹)))
203flddrngd 20758 . . . . . . . . 9 (𝜑𝐸 ∈ DivRing)
212, 20, 8fldgenssid 33295 . . . . . . . 8 (𝜑 → (𝐹𝐴) ⊆ (𝐸 fldGen (𝐹𝐴)))
2221unssad 4203 . . . . . . 7 (𝜑𝐹 ⊆ (𝐸 fldGen (𝐹𝐴)))
23 sseqin2 4231 . . . . . . 7 (𝐹 ⊆ (𝐸 fldGen (𝐹𝐴)) ↔ ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹) = 𝐹)
2422, 23sylib 218 . . . . . 6 (𝜑 → ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹) = 𝐹)
2524oveq2d 7447 . . . . 5 (𝜑 → (𝐸s ((𝐸 fldGen (𝐹𝐴)) ∩ 𝐹)) = (𝐸s 𝐹))
2619, 25eqtrd 2775 . . . 4 (𝜑 → (𝐿s 𝐹) = (𝐸s 𝐹))
2711, 2ressbas2 17283 . . . . . 6 (𝐹𝐵𝐹 = (Base‘𝐾))
286, 27syl 17 . . . . 5 (𝜑𝐹 = (Base‘𝐾))
2928oveq2d 7447 . . . 4 (𝜑 → (𝐿s 𝐹) = (𝐿s (Base‘𝐾)))
3026, 29eqtr3d 2777 . . 3 (𝜑 → (𝐸s 𝐹) = (𝐿s (Base‘𝐾)))
3111, 30eqtrid 2787 . 2 (𝜑𝐾 = (𝐿s (Base‘𝐾)))
3210fldcrngd 20759 . . . . 5 (𝜑𝐿 ∈ CRing)
3332crngringd 20264 . . . 4 (𝜑𝐿 ∈ Ring)
3414fldcrngd 20759 . . . . . . 7 (𝜑𝐾 ∈ CRing)
3534crngringd 20264 . . . . . 6 (𝜑𝐾 ∈ Ring)
3611, 35eqeltrrid 2844 . . . . 5 (𝜑 → (𝐸s 𝐹) ∈ Ring)
3726, 36eqeltrd 2839 . . . 4 (𝜑 → (𝐿s 𝐹) ∈ Ring)
382, 20, 8fldgenssv 33297 . . . . . . 7 (𝜑 → (𝐸 fldGen (𝐹𝐴)) ⊆ 𝐵)
391, 2ressbas2 17283 . . . . . . 7 ((𝐸 fldGen (𝐹𝐴)) ⊆ 𝐵 → (𝐸 fldGen (𝐹𝐴)) = (Base‘𝐿))
4038, 39syl 17 . . . . . 6 (𝜑 → (𝐸 fldGen (𝐹𝐴)) = (Base‘𝐿))
4122, 40sseqtrd 4036 . . . . 5 (𝜑𝐹 ⊆ (Base‘𝐿))
4220drngringd 20754 . . . . . . 7 (𝜑𝐸 ∈ Ring)
43 sdrgsubrg 20809 . . . . . . . . 9 (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
44 eqid 2735 . . . . . . . . . 10 (1r𝐸) = (1r𝐸)
4544subrg1cl 20597 . . . . . . . . 9 (𝐹 ∈ (SubRing‘𝐸) → (1r𝐸) ∈ 𝐹)
464, 43, 453syl 18 . . . . . . . 8 (𝜑 → (1r𝐸) ∈ 𝐹)
4722, 46sseldd 3996 . . . . . . 7 (𝜑 → (1r𝐸) ∈ (𝐸 fldGen (𝐹𝐴)))
481, 2, 44ress1r 33224 . . . . . . 7 ((𝐸 ∈ Ring ∧ (1r𝐸) ∈ (𝐸 fldGen (𝐹𝐴)) ∧ (𝐸 fldGen (𝐹𝐴)) ⊆ 𝐵) → (1r𝐸) = (1r𝐿))
4942, 47, 38, 48syl3anc 1370 . . . . . 6 (𝜑 → (1r𝐸) = (1r𝐿))
5049, 46eqeltrrd 2840 . . . . 5 (𝜑 → (1r𝐿) ∈ 𝐹)
5141, 50jca 511 . . . 4 (𝜑 → (𝐹 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝐹))
52 eqid 2735 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
53 eqid 2735 . . . . 5 (1r𝐿) = (1r𝐿)
5452, 53issubrg 20588 . . . 4 (𝐹 ∈ (SubRing‘𝐿) ↔ ((𝐿 ∈ Ring ∧ (𝐿s 𝐹) ∈ Ring) ∧ (𝐹 ⊆ (Base‘𝐿) ∧ (1r𝐿) ∈ 𝐹)))
5533, 37, 51, 54syl21anbrc 1343 . . 3 (𝜑𝐹 ∈ (SubRing‘𝐿))
5628, 55eqeltrrd 2840 . 2 (𝜑 → (Base‘𝐾) ∈ (SubRing‘𝐿))
57 brfldext 33675 . . 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 839 1 (𝜑𝐿/FldExt𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  wss 3963   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  1rcur 20199  Ringcrg 20251  SubRingcsubrg 20586  Fieldcfield 20747  SubDRingcsdrg 20804   fldGen cfldgen 33292  /FldExtcfldext 33666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-subrng 20563  df-subrg 20587  df-drng 20748  df-field 20749  df-sdrg 20805  df-fldgen 33293  df-fldext 33670
This theorem is referenced by:  rtelextdg2  33733
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