Theorem extdgmul 33674

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/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))

Proof of Theorem extdgmul

Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐾)) = ((subringAlg ‘𝐸)‘(Base‘𝐾))
2		eqid 2740	. . 3 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
3		eqid 2740	. . 3 ⊢ ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))
4		eqid 2740	. . 3 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
5		eqid 2740	. . 3 ⊢ (𝐸 ↾s (Base‘𝐾)) = (𝐸 ↾s (Base‘𝐾))
6		simpl 482	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
7		fldextfld1 33662	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
8	6, 7	syl 17	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
9		isfld 20762	. . . . 5 ⊢ (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
10	9	simplbi 497	. . . 4 ⊢ (𝐸 ∈ Field → 𝐸 ∈ DivRing)
11	8, 10	syl 17	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ DivRing)
12		fldextfld1 33662	. . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐹 ∈ Field)
13	12	adantl 481	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
14		brfldext 33660	. . . . . . 7 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
15	8, 13, 14	syl2anc 583	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
16	6, 15	mpbid 232	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
17	16	simpld 494	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
18		isfld 20762	. . . . . 6 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
19	18	simplbi 497	. . . . 5 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
20	13, 19	syl 17	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ DivRing)
21	17, 20	eqeltrrd 2845	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s (Base‘𝐹)) ∈ DivRing)
22		fldexttr 33671	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
23		fldextfld2 33663	. . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
24	23	adantl 481	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
25		brfldext 33660	. . . . . . 7 ⊢ ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
26	8, 24, 25	syl2anc 583	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
27	22, 26	mpbid 232	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸)))
28	27	simpld 494	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐸 ↾s (Base‘𝐾)))
29		isfld 20762	. . . . . 6 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
30	29	simplbi 497	. . . . 5 ⊢ (𝐾 ∈ Field → 𝐾 ∈ DivRing)
31	24, 30	syl 17	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ DivRing)
32	28, 31	eqeltrrd 2845	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s (Base‘𝐾)) ∈ DivRing)
33	16	simprd 495	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34		eqid 2740	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
35	34	fldextsubrg 33664	. . . . 5 ⊢ (𝐹/FldExt𝐾 → (Base‘𝐾) ∈ (SubRing‘𝐹))
36	35	adantl 481	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
37	17	fveq2d 6924	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
38	36, 37	eleqtrd 2846	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
39	1, 2, 3, 4, 5, 11, 21, 32, 33, 38	fedgmul 33644	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))) = ((dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ·e (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)))))
40		extdgval 33667	. . 3 ⊢ (𝐸/FldExt𝐾 → (𝐸[:]𝐾) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))))
41	22, 40	syl 17	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))))
42		extdgval 33667	. . . 4 ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
43	6, 42	syl 17	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
44		extdgval 33667	. . . . 5 ⊢ (𝐹/FldExt𝐾 → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
45	44	adantl 481	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
46	17	fveq2d 6924	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (subringAlg ‘𝐹) = (subringAlg ‘(𝐸 ↾s (Base‘𝐹))))
47	46	fveq1d 6922	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((subringAlg ‘𝐹)‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)))
48	47	fveq2d 6924	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))))
49	45, 48	eqtrd 2780	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))))
50	43, 49	oveq12d 7466	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)) = ((dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ·e (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)))))
51	39, 41, 50	3eqtr4d 2790	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448   ·_e cxmu 13174  Basecbs 17258   ↾_s cress 17287  CRingccrg 20261  SubRingcsubrg 20595  DivRingcdr 20751  Fieldcfield 20752  subringAlg csra 21193  dimcldim 33611  /_FldExtcfldext 33651  [:]cextdg 33654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710  ax-ac2 10532  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-rpss 7758  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-tpos 8267  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-oi 9579  df-r1 9833  df-rank 9834  df-dju 9970  df-card 10008  df-acn 10011  df-ac 10185  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-xnn0 12626  df-z 12640  df-dec 12759  df-uz 12904  df-xmul 13177  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ocomp 17332  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-mri 17646  df-acs 17647  df-proset 18365  df-drs 18366  df-poset 18383  df-ipo 18598  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-oppr 20360  df-dvdsr 20383  df-unit 20384  df-invr 20414  df-nzr 20539  df-subrng 20572  df-subrg 20597  df-drng 20753  df-field 20754  df-lmod 20882  df-lss 20953  df-lsp 20993  df-lmhm 21044  df-lbs 21097  df-lvec 21125  df-sra 21195  df-rgmod 21196  df-dsmm 21775  df-frlm 21790  df-uvc 21826  df-lindf 21849  df-linds 21850  df-dim 33612  df-fldext 33655  df-extdg 33656

This theorem is referenced by:  finexttrb  33675  fldext2chn  33719