Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  extdgmul Structured version   Visualization version   GIF version

Theorem extdgmul 31114
 Description: The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, the degree of the extension 𝐸/FldExt𝐾 is the product of the degrees of the extensions 𝐸/FldExt𝐹 and 𝐹/FldExt𝐾. Proposition 1.2 of [Lang], p. 224. (Contributed by Thierry Arnoux, 30-Jul-2023.)
Assertion
Ref Expression
extdgmul ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))

Proof of Theorem extdgmul
StepHypRef Expression
1 eqid 2824 . . 3 ((subringAlg ‘𝐸)‘(Base‘𝐾)) = ((subringAlg ‘𝐸)‘(Base‘𝐾))
2 eqid 2824 . . 3 ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
3 eqid 2824 . . 3 ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)) = ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾))
4 eqid 2824 . . 3 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
5 eqid 2824 . . 3 (𝐸s (Base‘𝐾)) = (𝐸s (Base‘𝐾))
6 simpl 486 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
7 fldextfld1 31102 . . . . 5 (𝐸/FldExt𝐹𝐸 ∈ Field)
86, 7syl 17 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸 ∈ Field)
9 isfld 19514 . . . . 5 (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
109simplbi 501 . . . 4 (𝐸 ∈ Field → 𝐸 ∈ DivRing)
118, 10syl 17 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸 ∈ DivRing)
12 fldextfld1 31102 . . . . . . . 8 (𝐹/FldExt𝐾𝐹 ∈ Field)
1312adantl 485 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 ∈ Field)
14 brfldext 31100 . . . . . . 7 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
158, 13, 14syl2anc 587 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
166, 15mpbid 235 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
1716simpld 498 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 = (𝐸s (Base‘𝐹)))
18 isfld 19514 . . . . . 6 (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
1918simplbi 501 . . . . 5 (𝐹 ∈ Field → 𝐹 ∈ DivRing)
2013, 19syl 17 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 ∈ DivRing)
2117, 20eqeltrrd 2917 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s (Base‘𝐹)) ∈ DivRing)
22 fldexttr 31111 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
23 fldextfld2 31103 . . . . . . . 8 (𝐹/FldExt𝐾𝐾 ∈ Field)
2423adantl 485 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 ∈ Field)
25 brfldext 31100 . . . . . . 7 ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
268, 24, 25syl2anc 587 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
2722, 26mpbid 235 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸)))
2827simpld 498 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 = (𝐸s (Base‘𝐾)))
29 isfld 19514 . . . . . 6 (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
3029simplbi 501 . . . . 5 (𝐾 ∈ Field → 𝐾 ∈ DivRing)
3124, 30syl 17 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 ∈ DivRing)
3228, 31eqeltrrd 2917 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s (Base‘𝐾)) ∈ DivRing)
3316simprd 499 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34 eqid 2824 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3534fldextsubrg 31104 . . . . 5 (𝐹/FldExt𝐾 → (Base‘𝐾) ∈ (SubRing‘𝐹))
3635adantl 485 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
3717fveq2d 6666 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸s (Base‘𝐹))))
3836, 37eleqtrd 2918 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))))
391, 2, 3, 4, 5, 11, 21, 32, 33, 38fedgmul 31090 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))) = ((dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ·e (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)))))
40 extdgval 31107 . . 3 (𝐸/FldExt𝐾 → (𝐸[:]𝐾) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))))
4122, 40syl 17 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸[:]𝐾) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))))
42 extdgval 31107 . . . 4 (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
436, 42syl 17 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
44 extdgval 31107 . . . . 5 (𝐹/FldExt𝐾 → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
4544adantl 485 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
4617fveq2d 6666 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (subringAlg ‘𝐹) = (subringAlg ‘(𝐸s (Base‘𝐹))))
4746fveq1d 6664 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((subringAlg ‘𝐹)‘(Base‘𝐾)) = ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)))
4847fveq2d 6666 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))) = (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾))))
4945, 48eqtrd 2859 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾))))
5043, 49oveq12d 7168 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)) = ((dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ·e (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)))))
5139, 41, 503eqtr4d 2869 1 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   class class class wbr 5053  ‘cfv 6344  (class class class)co 7150   ·e cxmu 12506  Basecbs 16486   ↾s cress 16487  CRingccrg 19301  DivRingcdr 19505  Fieldcfield 19506  SubRingcsubrg 19534  subringAlg csra 19943  dimcldim 31062  /FldExtcfldext 31091  [:]cextdg 31094 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-reg 9054  ax-inf2 9102  ax-ac2 9884  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-int 4864  df-iun 4908  df-iin 4909  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-isom 6353  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7404  df-rpss 7444  df-om 7576  df-1st 7685  df-2nd 7686  df-supp 7828  df-tpos 7889  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-1o 8099  df-oadd 8103  df-er 8286  df-map 8405  df-ixp 8459  df-en 8507  df-dom 8508  df-sdom 8509  df-fin 8510  df-fsupp 8832  df-sup 8904  df-oi 8972  df-r1 9191  df-rank 9192  df-dju 9328  df-card 9366  df-acn 9369  df-ac 9541  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11700  df-3 11701  df-4 11702  df-5 11703  df-6 11704  df-7 11705  df-8 11706  df-9 11707  df-n0 11898  df-xnn0 11968  df-z 11982  df-dec 12099  df-uz 12244  df-xmul 12509  df-fz 12898  df-fzo 13041  df-seq 13377  df-hash 13699  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-sca 16584  df-vsca 16585  df-ip 16586  df-tset 16587  df-ple 16588  df-ocomp 16589  df-ds 16590  df-hom 16592  df-cco 16593  df-0g 16718  df-gsum 16719  df-prds 16724  df-pws 16726  df-mre 16860  df-mrc 16861  df-mri 16862  df-acs 16863  df-proset 17541  df-drs 17542  df-poset 17559  df-ipo 17765  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959  df-submnd 17960  df-grp 18109  df-minusg 18110  df-sbg 18111  df-mulg 18228  df-subg 18279  df-ghm 18359  df-cntz 18450  df-cmn 18911  df-abl 18912  df-mgp 19243  df-ur 19255  df-ring 19302  df-oppr 19379  df-dvdsr 19397  df-unit 19398  df-invr 19428  df-drng 19507  df-field 19508  df-subrg 19536  df-lmod 19639  df-lss 19707  df-lsp 19747  df-lmhm 19797  df-lbs 19850  df-lvec 19878  df-sra 19947  df-rgmod 19948  df-nzr 20034  df-dsmm 20879  df-frlm 20894  df-uvc 20930  df-lindf 20953  df-linds 20954  df-dim 31063  df-fldext 31095  df-extdg 31096 This theorem is referenced by:  finexttrb  31115
 Copyright terms: Public domain W3C validator