Theorem extdgmul 33807

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 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 ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))
4		eqid 2737	. . 3 ⊢ (𝐸 ↾s (Base‘𝐹)) = (𝐸 ↾s (Base‘𝐹))
5		eqid 2737	. . 3 ⊢ (𝐸 ↾s (Base‘𝐾)) = (𝐸 ↾s (Base‘𝐾))
6		simpl 482	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐹)
7		fldextfld1 33791	. . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field)
8	6, 7	syl 17	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field)
9		isfld 20717	. . . . 5 ⊢ (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
10	9	simplbi 496	. . . 4 ⊢ (𝐸 ∈ Field → 𝐸 ∈ DivRing)
11	8, 10	syl 17	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ DivRing)
12		fldextfld1 33791	. . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐹 ∈ Field)
13	12	adantl 481	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field)
14		brfldext 33789	. . . . . . 7 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸))))
15	8, 13, 14	syl2anc 585	. . . . . 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 20717	. . . . . 6 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
19	18	simplbi 496	. . . . 5 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
20	13, 19	syl 17	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ DivRing)
21	17, 20	eqeltrrd 2838	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s (Base‘𝐹)) ∈ DivRing)
22		fldexttr 33802	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾)
23		fldextfld2 33792	. . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field)
24	23	adantl 481	. . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field)
25		brfldext 33789	. . . . . . 7 ⊢ ((𝐸 ∈ Field ∧ 𝐾 ∈ Field) → (𝐸/FldExt𝐾 ↔ (𝐾 = (𝐸 ↾s (Base‘𝐾)) ∧ (Base‘𝐾) ∈ (SubRing‘𝐸))))
26	8, 24, 25	syl2anc 585	. . . . . 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 20717	. . . . . 6 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
30	29	simplbi 496	. . . . 5 ⊢ (𝐾 ∈ Field → 𝐾 ∈ DivRing)
31	24, 30	syl 17	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ DivRing)
32	28, 31	eqeltrrd 2838	. . 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 2737	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
35	34	fldextsubrg 33793	. . . . 5 ⊢ (𝐹/FldExt𝐾 → (Base‘𝐾) ∈ (SubRing‘𝐹))
36	35	adantl 481	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹))
37	17	fveq2d 6845	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹))))
38	36, 37	eleqtrd 2839	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹))))
39	1, 2, 3, 4, 5, 11, 21, 32, 33, 38	fedgmul 33775	. 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 33797	. . 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 33797	. . . 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 33797	. . . . 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 6845	. . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (subringAlg ‘𝐹) = (subringAlg ‘(𝐸 ↾s (Base‘𝐹))))
47	46	fveq1d 6843	. . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((subringAlg ‘𝐹)‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)))
48	47	fveq2d 6845	. . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))))
49	45, 48	eqtrd 2772	. . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))))
50	43, 49	oveq12d 7385	. 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 2782	1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6499  (class class class)co 7367   ·_e cxmu 13062  Basecbs 17179   ↾_s cress 17200  CRingccrg 20215  SubRingcsubrg 20546  DivRingcdr 20706  Fieldcfield 20707  subringAlg csra 21166  dimcldim 33743  /_FldExtcfldext 33782  [:]cextdg 33784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-reg 9507  ax-inf2 9562  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-rpss 7677  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-r1 9688  df-rank 9689  df-dju 9825  df-card 9863  df-acn 9866  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-xmul 13065  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ocomp 17241  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-mri 17550  df-acs 17551  df-proset 18260  df-drs 18261  df-poset 18279  df-ipo 18494  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-invr 20368  df-nzr 20490  df-subrng 20523  df-subrg 20547  df-drng 20708  df-field 20709  df-lmod 20857  df-lss 20927  df-lsp 20967  df-lmhm 21017  df-lbs 21070  df-lvec 21098  df-sra 21168  df-rgmod 21169  df-dsmm 21712  df-frlm 21727  df-uvc 21763  df-lindf 21786  df-linds 21787  df-dim 33744  df-fldext 33785  df-extdg 33786

This theorem is referenced by:  finexttrb  33809  fldextrspundglemul  33823  fldextrspundgdvdslem  33824  fldextrspundgdvds  33825  fldext2rspun  33826  fldext2chn  33872  constrext2chnlem  33894