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| Mirrors > Home > MPE Home > Th. List > Mathboxes > extdgmul | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| extdgmul | ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾))) |
| 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 33700 | . . . . 5 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
| 8 | 6, 7 | syl 17 | . . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸 ∈ Field) |
| 9 | isfld 20740 | . . . . 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 33700 | . . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐹 ∈ Field) | |
| 13 | 12 | adantl 481 | . . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐹 ∈ Field) |
| 14 | brfldext 33698 | . . . . . . 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 20740 | . . . . . 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 2842 | . . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸 ↾s (Base‘𝐹)) ∈ DivRing) |
| 22 | fldexttr 33709 | . . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾) | |
| 23 | fldextfld2 33701 | . . . . . . . 8 ⊢ (𝐹/FldExt𝐾 → 𝐾 ∈ Field) | |
| 24 | 23 | adantl 481 | . . . . . . 7 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐾 ∈ Field) |
| 25 | brfldext 33698 | . . . . . . 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 20740 | . . . . . 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 2842 | . . 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 33702 | . . . . 5 ⊢ (𝐹/FldExt𝐾 → (Base‘𝐾) ∈ (SubRing‘𝐹)) |
| 36 | 35 | adantl 481 | . . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘𝐹)) |
| 37 | 17 | fveq2d 6910 | . . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (SubRing‘𝐹) = (SubRing‘(𝐸 ↾s (Base‘𝐹)))) |
| 38 | 36, 37 | eleqtrd 2843 | . . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (Base‘𝐾) ∈ (SubRing‘(𝐸 ↾s (Base‘𝐹)))) |
| 39 | 1, 2, 3, 4, 5, 11, 21, 32, 33, 38 | fedgmul 33682 | . 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 33705 | . . 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 33705 | . . . 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 33705 | . . . . 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 6910 | . . . . . 6 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (subringAlg ‘𝐹) = (subringAlg ‘(𝐸 ↾s (Base‘𝐹)))) |
| 47 | 46 | fveq1d 6908 | . . . . 5 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → ((subringAlg ‘𝐹)‘(Base‘𝐾)) = ((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾))) |
| 48 | 47 | fveq2d 6910 | . . . 4 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (dim‘((subringAlg ‘𝐹)‘(Base‘𝐾))) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)))) |
| 49 | 45, 48 | eqtrd 2777 | . . 3 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐹[:]𝐾) = (dim‘((subringAlg ‘(𝐸 ↾s (Base‘𝐹)))‘(Base‘𝐾)))) |
| 50 | 43, 49 | oveq12d 7449 | . 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 2787 | 1 ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/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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