Theorem extdgmul 34062

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 2762	. . 3 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐾)) = ((subringAlg ‘𝐸)‘(Base‘𝐾))
2		eqid 2762	. . 3 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
3		eqid 2762	. . 3 ⊢ ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))
4		eqid 2762	. . 3 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
5		eqid 2762	. . 3 ⊢ (𝐸 ↾s (Base‘𝐾)) = (𝐸 ↾s (Base‘𝐾))
6		simpl 487	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
7		fldextfld1 34046	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
8	6, 7	syl 18	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
9		isfld 20851	. . . . 5 ⊢ (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
10	9	simplbi 501	. . . 4 ⊢ (𝐸 ∈ Field → 𝐸 ∈ DivRing)
11	8, 10	syl 18	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ DivRing)
12		fldextfld1 34046	. . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐹 ∈ Field)
13	12	adantl 486	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
14		brfldext 34044	. . . . . . 7 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
15	8, 13, 14	syl2anc 595	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
16	6, 15	mpbid 235	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))
17	16	simpld 499	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 = (𝐸 ↾s (Base‘𝐹)))
18		isfld 20851	. . . . . 6 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
19	18	simplbi 501	. . . . 5 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
20	13, 19	syl 18	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ DivRing)
21	17, 20	eqeltrrd 2863	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s (Base‘𝐹)) ∈ DivRing)
22		fldexttr 34057	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
23		fldextfld2 34047	. . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
24	23	adantl 486	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
25		brfldext 34044	. . . . . . 7 ⊢ ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
26	8, 24, 25	syl2anc 595	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
27	22, 26	mpbid 235	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸)))
28	27	simpld 499	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 = (𝐸 ↾s (Base‘𝐾)))
29		isfld 20851	. . . . . 6 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
30	29	simplbi 501	. . . . 5 ⊢ (𝐾 ∈ Field → 𝐾 ∈ DivRing)
31	24, 30	syl 18	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ DivRing)
32	28, 31	eqeltrrd 2863	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s (Base‘𝐾)) ∈ DivRing)
33	16	simprd 500	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐹) ∈ (SubRing‘𝐸))
34		eqid 2762	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
35	34	fldextsubrg 34048	. . . . 5 ⊢ (𝐹/FldExt𝐾 → (Base‘𝐾) ∈ (SubRing‘𝐹))
36	35	adantl 486	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
37	17	fveq2d 6885	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
38	36, 37	eleqtrd 2864	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
39	1, 2, 3, 4, 5, 11, 21, 32, 33, 38	fedgmul 34030	. 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 34052	. . 3 ⊢ (𝐸/FldExt𝐾 → (𝐸[:]𝐾) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))))
41	22, 40	syl 18	. 2 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐾))))
42		extdgval 34052	. . . 4 ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
43	6, 42	syl 18	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
44		extdgval 34052	. . . . 5 ⊢ (𝐹/FldExt𝐾 → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
45	44	adantl 486	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))))
46	17	fveq2d 6885	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (subringAlg ‘𝐹) = (subringAlg ‘(𝐸 ↾s (Base‘𝐹))))
47	46	fveq1d 6883	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((subringAlg ‘𝐹)‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)))
48	47	fveq2d 6885	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))))
49	45, 48	eqtrd 2797	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))))
50	43, 49	oveq12d 7430	. 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 2807	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412   ·_e cxmu 13142  Basecbs 17275   ↾_s cress 17296  CRingccrg 20322  SubRingcsubrg 20679  DivRingcdr 20838  Fieldcfield 20839  subringAlg csra 21303  dimcldim 33998  /_FldExtcfldext 34037  [:]cextdg 34039

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-reg 9552  ax-inf2 9608  ax-ac2 10453  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-rpss 7722  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8155  df-tpos 8220  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-er 8692  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9320  df-sup 9400  df-oi 9470  df-r1 9734  df-rank 9735  df-scott 9856  df-dju 9894  df-card 9932  df-acn 9935  df-ac 10107  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-xnn0 12584  df-z 12598  df-dec 12718  df-uz 12869  df-xmul 13145  df-fz 13542  df-fzo 13690  df-seq 14045  df-hash 14374  df-struct 17213  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-plusg 17329  df-mulr 17330  df-sca 17332  df-vsca 17333  df-ip 17334  df-tset 17335  df-ple 17336  df-ocomp 17337  df-ds 17338  df-hom 17340  df-cco 17341  df-0g 17500  df-gsum 17501  df-prds 17506  df-pws 17508  df-mre 17644  df-mrc 17645  df-mri 17646  df-acs 17647  df-proset 18356  df-drs 18357  df-poset 18375  df-ipo 18590  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-mhm 18847  df-submnd 18848  df-grp 19009  df-minusg 19010  df-sbg 19011  df-mulg 19140  df-subg 19195  df-ghm 19290  df-cntz 19393  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-oppr 20426  df-dvdsr 20446  df-unit 20447  df-invr 20477  df-nzr 20621  df-subrng 20656  df-subrg 20680  df-drng 20840  df-field 20841  df-lmod 20994  df-lss 21064  df-lsp 21104  df-lmhm 21154  df-lbs 21207  df-lvec 21235  df-sra 21305  df-rgmod 21306  df-dsmm 21893  df-frlm 21908  df-uvc 21944  df-lindf 21967  df-linds 21968  df-dim 33999  df-fldext 34040  df-extdg 34041

This theorem is used by:  finexttrb  34064  fldextrspundglemul  34078  fldextrspundgdvdslem  34079  fldextrspundgdvds  34080  fldext2rspun  34081  fldext2chn  34127  constrext2chnlem  34149