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Theorem extdgmul 33689
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 2735 . . 3 ((subringAlg ‘𝐸)‘(Base‘𝐾)) = ((subringAlg ‘𝐸)‘(Base‘𝐾))
2 eqid 2735 . . 3 ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
3 eqid 2735 . . 3 ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)) = ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾))
4 eqid 2735 . . 3 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
5 eqid 2735 . . 3 (𝐸s (Base‘𝐾)) = (𝐸s (Base‘𝐾))
6 simpl 482 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
7 fldextfld1 33677 . . . . 5 (𝐸/FldExt𝐹𝐸 ∈ Field)
86, 7syl 17 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸 ∈ Field)
9 isfld 20757 . . . . 5 (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
109simplbi 497 . . . 4 (𝐸 ∈ Field → 𝐸 ∈ DivRing)
118, 10syl 17 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸 ∈ DivRing)
12 fldextfld1 33677 . . . . . . . 8 (𝐹/FldExt𝐾𝐹 ∈ Field)
1312adantl 481 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 ∈ Field)
14 brfldext 33675 . . . . . . 7 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
158, 13, 14syl2anc 584 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
166, 15mpbid 232 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹 = (𝐸s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
1716simpld 494 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 = (𝐸s (Base‘𝐹)))
18 isfld 20757 . . . . . 6 (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
1918simplbi 497 . . . . 5 (𝐹 ∈ Field → 𝐹 ∈ DivRing)
2013, 19syl 17 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 ∈ DivRing)
2117, 20eqeltrrd 2840 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s (Base‘𝐹)) ∈ DivRing)
22 fldexttr 33686 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
23 fldextfld2 33678 . . . . . . . 8 (𝐹/FldExt𝐾𝐾 ∈ Field)
2423adantl 481 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 ∈ Field)
25 brfldext 33675 . . . . . . 7 ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
268, 24, 25syl2anc 584 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
2722, 26mpbid 232 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐾 = (𝐸s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸)))
2827simpld 494 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 = (𝐸s (Base‘𝐾)))
29 isfld 20757 . . . . . 6 (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
3029simplbi 497 . . . . 5 (𝐾 ∈ Field → 𝐾 ∈ DivRing)
3124, 30syl 17 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 ∈ DivRing)
3228, 31eqeltrrd 2840 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s (Base‘𝐾)) ∈ DivRing)
3316simprd 495 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34 eqid 2735 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3534fldextsubrg 33679 . . . . 5 (𝐹/FldExt𝐾 → (Base‘𝐾) ∈ (SubRing‘𝐹))
3635adantl 481 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
3717fveq2d 6911 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸s (Base‘𝐹))))
3836, 37eleqtrd 2841 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))))
391, 2, 3, 4, 5, 11, 21, 32, 33, 38fedgmul 33659 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))) = ((dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ·e (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)))))
40 extdgval 33682 . . 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 33682 . . . 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 33682 . . . . 5 (𝐹/FldExt𝐾 → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
4544adantl 481 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
4617fveq2d 6911 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (subringAlg ‘𝐹) = (subringAlg ‘(𝐸s (Base‘𝐹))))
4746fveq1d 6909 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((subringAlg ‘𝐹)‘(Base‘𝐾)) = ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)))
4847fveq2d 6911 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))) = (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾))))
4945, 48eqtrd 2775 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾))))
5043, 49oveq12d 7449 . 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 2785 1 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431   ·e cxmu 13151  Basecbs 17245  s cress 17274  CRingccrg 20252  SubRingcsubrg 20586  DivRingcdr 20746  Fieldcfield 20747  subringAlg csra 21188  dimcldim 33626  /FldExtcfldext 33666  [:]cextdg 33669
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-reg 9630  ax-inf2 9679  ax-ac2 10501  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-tp 4636  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-se 5642  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-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-r1 9802  df-rank 9803  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  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-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-xmul 13154  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ocomp 17319  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-mri 17633  df-acs 17634  df-proset 18352  df-drs 18353  df-poset 18371  df-ipo 18586  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-drng 20748  df-field 20749  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-lindf 21844  df-linds 21845  df-dim 33627  df-fldext 33670  df-extdg 33671
This theorem is referenced by:  finexttrb  33690  fldext2chn  33734
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