Theorem extdgmul 33705

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 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 ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))
4		eqid 2735	. . 3 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
5		eqid 2735	. . 3 ⊢ (𝐸 ↾s (Base‘𝐾)) = (𝐸 ↾s (Base‘𝐾))
6		simpl 482	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
7		fldextfld1 33689	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
8	6, 7	syl 17	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
9		isfld 20700	. . . . 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 33689	. . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐹 ∈ Field)
13	12	adantl 481	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
14		brfldext 33687	. . . . . . 7 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
15	8, 13, 14	syl2anc 584	. . . . . 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 20700	. . . . . 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 2835	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s (Base‘𝐹)) ∈ DivRing)
22		fldexttr 33700	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
23		fldextfld2 33690	. . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
24	23	adantl 481	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
25		brfldext 33687	. . . . . . 7 ⊢ ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
26	8, 24, 25	syl2anc 584	. . . . . 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 20700	. . . . . 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 2835	. . 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 2735	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
35	34	fldextsubrg 33691	. . . . 5 ⊢ (𝐹/FldExt𝐾 → (Base‘𝐾) ∈ (SubRing‘𝐹))
36	35	adantl 481	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
37	17	fveq2d 6880	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
38	36, 37	eleqtrd 2836	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
39	1, 2, 3, 4, 5, 11, 21, 32, 33, 38	fedgmul 33671	. 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 33695	. . 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 33695	. . . 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 33695	. . . . 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 6880	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (subringAlg ‘𝐹) = (subringAlg ‘(𝐸 ↾s (Base‘𝐹))))
47	46	fveq1d 6878	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((subringAlg ‘𝐹)‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)))
48	47	fveq2d 6880	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))))
49	45, 48	eqtrd 2770	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))))
50	43, 49	oveq12d 7423	. 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 2780	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405   ·_e cxmu 13127  Basecbs 17228   ↾_s cress 17251  CRingccrg 20194  SubRingcsubrg 20529  DivRingcdr 20689  Fieldcfield 20690  subringAlg csra 21129  dimcldim 33638  /_FldExtcfldext 33678  [:]cextdg 33681

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-reg 9606  ax-inf2 9655  ax-ac2 10477  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-rpss 7717  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8719  df-map 8842  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-sup 9454  df-oi 9524  df-r1 9778  df-rank 9779  df-dju 9915  df-card 9953  df-acn 9956  df-ac 10130  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-xnn0 12575  df-z 12589  df-dec 12709  df-uz 12853  df-xmul 13130  df-fz 13525  df-fzo 13672  df-seq 14020  df-hash 14349  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ocomp 17292  df-ds 17293  df-hom 17295  df-cco 17296  df-0g 17455  df-gsum 17456  df-prds 17461  df-pws 17463  df-mre 17598  df-mrc 17599  df-mri 17600  df-acs 17601  df-proset 18306  df-drs 18307  df-poset 18325  df-ipo 18538  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-mulg 19051  df-subg 19106  df-ghm 19196  df-cntz 19300  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-oppr 20297  df-dvdsr 20317  df-unit 20318  df-invr 20348  df-nzr 20473  df-subrng 20506  df-subrg 20530  df-drng 20691  df-field 20692  df-lmod 20819  df-lss 20889  df-lsp 20929  df-lmhm 20980  df-lbs 21033  df-lvec 21061  df-sra 21131  df-rgmod 21132  df-dsmm 21692  df-frlm 21707  df-uvc 21743  df-lindf 21766  df-linds 21767  df-dim 33639  df-fldext 33682  df-extdg 33683

This theorem is referenced by:  finexttrb  33706  fldextrspundglemul  33720  fldextrspundgdvdslem  33721  fldextrspundgdvds  33722  fldext2rspun  33723  fldext2chn  33762  constrext2chnlem  33784