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Theorem extdgmul 33714
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 2737 . . 3 ((subringAlg ‘𝐸)‘(Base‘𝐾)) = ((subringAlg ‘𝐸)‘(Base‘𝐾))
2 eqid 2737 . . 3 ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
3 eqid 2737 . . 3 ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)) = ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾))
4 eqid 2737 . . 3 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
5 eqid 2737 . . 3 (𝐸s (Base‘𝐾)) = (𝐸s (Base‘𝐾))
6 simpl 482 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
7 fldextfld1 33700 . . . . 5 (𝐸/FldExt𝐹𝐸 ∈ Field)
86, 7syl 17 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸 ∈ Field)
9 isfld 20740 . . . . 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 33700 . . . . . . . 8 (𝐹/FldExt𝐾𝐹 ∈ Field)
1312adantl 481 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐹 ∈ Field)
14 brfldext 33698 . . . . . . 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 20740 . . . . . 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 2842 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s (Base‘𝐹)) ∈ DivRing)
22 fldexttr 33709 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
23 fldextfld2 33701 . . . . . . . 8 (𝐹/FldExt𝐾𝐾 ∈ Field)
2423adantl 481 . . . . . . 7 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → 𝐾 ∈ Field)
25 brfldext 33698 . . . . . . 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 20740 . . . . . 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 2842 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐸s (Base‘𝐾)) ∈ DivRing)
3316simprd 495 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34 eqid 2737 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3534fldextsubrg 33702 . . . . 5 (𝐹/FldExt𝐾 → (Base‘𝐾) ∈ (SubRing‘𝐹))
3635adantl 481 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
3717fveq2d 6910 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸s (Base‘𝐹))))
3836, 37eleqtrd 2843 . . 3 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸s (Base‘𝐹))))
391, 2, 3, 4, 5, 11, 21, 32, 33, 38fedgmul 33682 . 2 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))) = ((dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ·e (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)))))
40 extdgval 33705 . . 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 33705 . . . 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 33705 . . . . 5 (𝐹/FldExt𝐾 → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
4544adantl 481 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
4617fveq2d 6910 . . . . . 6 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (subringAlg ‘𝐹) = (subringAlg ‘(𝐸s (Base‘𝐹))))
4746fveq1d 6908 . . . . 5 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → ((subringAlg ‘𝐹)‘(Base‘𝐾)) = ((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾)))
4847fveq2d 6910 . . . 4 ((𝐸/FldExt𝐹𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))) = (dim‘((subringAlg ‘(𝐸s (Base‘𝐹)))‘(Base‘𝐾))))
4945, 48eqtrd 2777 . . 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 2787 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 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431   ·e cxmu 13153  Basecbs 17247  s cress 17274  CRingccrg 20231  SubRingcsubrg 20569  DivRingcdr 20729  Fieldcfield 20730  subringAlg csra 21170  dimcldim 33649  /FldExtcfldext 33689  [:]cextdg 33692
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-rpss 7743  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-r1 9804  df-rank 9805  df-dju 9941  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-xmul 13156  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ocomp 17318  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-mri 17631  df-acs 17632  df-proset 18340  df-drs 18341  df-poset 18359  df-ipo 18573  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-drng 20731  df-field 20732  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lmhm 21021  df-lbs 21074  df-lvec 21102  df-sra 21172  df-rgmod 21173  df-dsmm 21752  df-frlm 21767  df-uvc 21803  df-lindf 21826  df-linds 21827  df-dim 33650  df-fldext 33693  df-extdg 33694
This theorem is referenced by:  finexttrb  33715  fldextrspundglemul  33729  fldextrspundgdvdslem  33730  fldextrspundgdvds  33731  fldext2rspun  33732  fldext2chn  33769
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