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Theorem fldexttr 33025
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 483 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹/FldExt𝐾)
2 simpl 481 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
3 fldextfld2 33017 . . . . . . 7 (𝐸/FldExt𝐹𝐹 ∈ Field)
42, 3syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 ∈ Field)
5 fldextfld2 33017 . . . . . . 7 (𝐹/FldExt𝐾𝐾 ∈ Field)
61, 5syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 ∈ Field)
7 brfldext 33014 . . . . . 6 ((𝐹 ∈ Field ∧ 𝐾 ∈ Field) → (𝐹/FldExt𝐾 ↔ (𝐾 = (𝐹s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐹))))
84, 6, 7syl2anc 582 . . . . 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 493 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 = (𝐹s (Base‘𝐾)))
11 fldextfld1 33016 . . . . . . . . 9 (𝐸/FldExt𝐹𝐸 ∈ Field)
122, 11syl 17 . . . . . . . 8 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸 ∈ Field)
13 brfldext 33014 . . . . . . . 8 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
1412, 4, 13syl2anc 582 . . . . . . 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 493 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 = (𝐸s (Base‘𝐹)))
1716oveq1d 7426 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹s (Base‘𝐾)) = ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)))
18 fvex 6903 . . . . 5 (Base‘𝐹) ∈ V
19 fvex 6903 . . . . 5 (Base‘𝐾) ∈ V
20 ressress 17197 . . . . 5 (((Base‘𝐹) ∈ V ∧ (Base‘𝐾) ∈ V) → ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))))
2118, 19, 20mp2an 688 . . . 4 ((𝐸s (Base‘𝐹)) ↾s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾)))
2217, 21eqtrdi 2786 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹s (Base‘𝐾)) = (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))))
23 incom 4200 . . . . 5 ((Base‘𝐾) ∩ (Base‘𝐹)) = ((Base‘𝐹) ∩ (Base‘𝐾))
249simprd 494 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
25 eqid 2730 . . . . . . . 8 (Base‘𝐹) = (Base‘𝐹)
2625subrgss 20462 . . . . . . 7 ((Base‘𝐾) ∈ (SubRing‘𝐹) → (Base‘𝐾) ⊆ (Base‘𝐹))
2724, 26syl 17 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ⊆ (Base‘𝐹))
28 df-ss 3964 . . . . . 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 2784 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((Base‘𝐹) ∩ (Base‘𝐾)) = (Base‘𝐾))
3130oveq2d 7427 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s ((Base‘𝐹) ∩ (Base‘𝐾))) = (𝐸s (Base‘𝐾)))
3210, 22, 313eqtrd 2774 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 = (𝐸s (Base‘𝐾)))
3315simprd 494 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
3416fveq2d 6894 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸s (Base‘𝐹))))
3524, 34eleqtrd 2833 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))))
36 eqid 2730 . . . . 5 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
3736subsubrg 20488 . . . 4 ((Base‘𝐹) ∈ (SubRing‘𝐸) → ((Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))) ↔ ((Base‘𝐾) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ⊆ (Base‘𝐹))))
3837simprbda 497 . . 3 (((Base‘𝐹) ∈ (SubRing‘𝐸) ∧ (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹)))) → (Base‘𝐾) ∈ (SubRing‘𝐸))
3933, 35, 38syl2anc 582 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐸))
40 brfldext 33014 . . 3 ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
4112, 6, 40syl2anc 582 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
4232, 39, 41mpbir2and 709 1 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  Vcvv 3472  cin 3946  wss 3947   class class class wbr 5147  cfv 6542  (class class class)co 7411  Basecbs 17148  s cress 17177  SubRingcsubrg 20457  Fieldcfield 20501  /FldExtcfldext 33005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-subg 19039  df-mgp 20029  df-ur 20076  df-ring 20129  df-subrg 20459  df-fldext 33009
This theorem is referenced by:  extdgmul  33028  finexttrb  33029
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