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Theorem fldexttr 32575
Description: Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Assertion
Ref Expression
fldexttr ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)

Proof of Theorem fldexttr
StepHypRef Expression
1 simpr 485 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹/FldExt𝐾)
2 simpl 483 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
3 fldextfld2 32567 . . . . . . 7 (𝐸/FldExt𝐹𝐹 ∈ Field)
42, 3syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 ∈ Field)
5 fldextfld2 32567 . . . . . . 7 (𝐹/FldExt𝐾𝐾 ∈ Field)
61, 5syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 ∈ Field)
7 brfldext 32564 . . . . . 6 ((𝐹 ∈ Field ∧ 𝐾 ∈ Field) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
84, 6, 7syl2anc 584 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
91, 8mpbid 231 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐾 = (𝐹s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹)))
109simpld 495 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 = (𝐹s (Base‘𝐾)))
11 fldextfld1 32566 . . . . . . . . 9 (𝐸/FldExt𝐹𝐸 ∈ Field)
122, 11syl 17 . . . . . . . 8 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸 ∈ Field)
13 brfldext 32564 . . . . . . . 8 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
1412, 4, 13syl2anc 584 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
152, 14mpbid 231 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
1615simpld 495 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 = (𝐸s (Base‘𝐹)))
1716oveq1d 7408 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹s (Base‘𝐾)) = ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)))
18 fvex 6891 . . . . 5 (Base‘𝐹) ∈ V
19 fvex 6891 . . . . 5 (Base‘𝐾) ∈ V
20 ressress 17175 . . . . 5 (((Base‘𝐹) ∈ V ∧ (Base‘𝐾) ∈ V) → ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))))
2118, 19, 20mp2an 690 . . . 4 ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾)))
2217, 21eqtrdi 2787 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))))
23 incom 4197 . . . . 5 ((Base‘𝐾) ∩ (Base‘𝐹)) = ((Base‘𝐹) ∩ (Base‘𝐾))
249simprd 496 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
25 eqid 2731 . . . . . . . 8 (Base‘𝐹) = (Base‘𝐹)
2625subrgss 20313 . . . . . . 7 ((Base‘𝐾) ∈ (SubRing‘𝐹) → (Base‘𝐾) ⊆ (Base‘𝐹))
2724, 26syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ⊆ (Base‘𝐹))
28 df-ss 3961 . . . . . 6 ((Base‘𝐾) ⊆ (Base‘𝐹) ↔ ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
2927, 28sylib 217 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((Base‘𝐾) ∩ (Base‘𝐹)) = (Base‘𝐾))
3023, 29eqtr3id 2785 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((Base‘𝐹) ∩ (Base‘𝐾)) = (Base‘𝐾))
3130oveq2d 7409 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))) = (𝐸s (Base‘𝐾)))
3210, 22, 313eqtrd 2775 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 = (𝐸s (Base‘𝐾)))
3315simprd 496 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
3416fveq2d 6882 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸s (Base‘𝐹))))
3524, 34eleqtrd 2834 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))))
36 eqid 2731 . . . . 5 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
3736subsubrg 20339 . . . 4 ((Base‘𝐹) ∈ (SubRing‘𝐸) → ((Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))) ↔ ((Base‘𝐾) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ⊆ (Base‘𝐹))))
3837simprbda 499 . . 3 (((Base‘𝐹) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹)))) → (Base‘𝐾) ∈ (SubRing‘𝐸))
3933, 35, 38syl2anc 584 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐸))
40 brfldext 32564 . . 3 ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
4112, 6, 40syl2anc 584 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
4232, 39, 41mpbir2and 711 1 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3473  cin 3943  wss 3944   class class class wbr 5141  cfv 6532  (class class class)co 7393  Basecbs 17126  s cress 17155  Fieldcfield 20266  SubRingcsubrg 20308  /FldExtcfldext 32555
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-mulr 17193  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-subg 18975  df-mgp 19947  df-ur 19964  df-ring 20016  df-subrg 20310  df-fldext 32559
This theorem is referenced by:  extdgmul  32578  finexttrb  32579
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