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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > algextdeglem1 | Structured version Visualization version GIF version |
Description: Lemma for algextdeg 33316. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
Ref | Expression |
---|---|
algextdeg.k | β’ πΎ = (πΈ βΎs πΉ) |
algextdeg.l | β’ πΏ = (πΈ βΎs (πΈ fldGen (πΉ βͺ {π΄}))) |
algextdeg.d | β’ π· = ( deg1 βπΈ) |
algextdeg.m | β’ π = (πΈ minPoly πΉ) |
algextdeg.f | β’ (π β πΈ β Field) |
algextdeg.e | β’ (π β πΉ β (SubDRingβπΈ)) |
algextdeg.a | β’ (π β π΄ β (πΈ IntgRing πΉ)) |
Ref | Expression |
---|---|
algextdeglem1 | β’ (π β (πΏ βΎs πΉ) = πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algextdeg.l | . . . 4 β’ πΏ = (πΈ βΎs (πΈ fldGen (πΉ βͺ {π΄}))) | |
2 | 1 | oveq1i 7424 | . . 3 β’ (πΏ βΎs πΉ) = ((πΈ βΎs (πΈ fldGen (πΉ βͺ {π΄}))) βΎs πΉ) |
3 | ovex 7447 | . . . 4 β’ (πΈ fldGen (πΉ βͺ {π΄})) β V | |
4 | eqid 2727 | . . . . . 6 β’ (BaseβπΈ) = (BaseβπΈ) | |
5 | algextdeg.e | . . . . . . . 8 β’ (π β πΉ β (SubDRingβπΈ)) | |
6 | issdrg 20658 | . . . . . . . 8 β’ (πΉ β (SubDRingβπΈ) β (πΈ β DivRing β§ πΉ β (SubRingβπΈ) β§ (πΈ βΎs πΉ) β DivRing)) | |
7 | 5, 6 | sylib 217 | . . . . . . 7 β’ (π β (πΈ β DivRing β§ πΉ β (SubRingβπΈ) β§ (πΈ βΎs πΉ) β DivRing)) |
8 | 7 | simp1d 1140 | . . . . . 6 β’ (π β πΈ β DivRing) |
9 | 7 | simp2d 1141 | . . . . . . . 8 β’ (π β πΉ β (SubRingβπΈ)) |
10 | subrgsubg 20498 | . . . . . . . 8 β’ (πΉ β (SubRingβπΈ) β πΉ β (SubGrpβπΈ)) | |
11 | 4 | subgss 19066 | . . . . . . . 8 β’ (πΉ β (SubGrpβπΈ) β πΉ β (BaseβπΈ)) |
12 | 9, 10, 11 | 3syl 18 | . . . . . . 7 β’ (π β πΉ β (BaseβπΈ)) |
13 | eqid 2727 | . . . . . . . . . 10 β’ (πΈ evalSub1 πΉ) = (πΈ evalSub1 πΉ) | |
14 | algextdeg.k | . . . . . . . . . 10 β’ πΎ = (πΈ βΎs πΉ) | |
15 | eqid 2727 | . . . . . . . . . 10 β’ (0gβπΈ) = (0gβπΈ) | |
16 | algextdeg.f | . . . . . . . . . . 11 β’ (π β πΈ β Field) | |
17 | 16 | fldcrngd 20619 | . . . . . . . . . 10 β’ (π β πΈ β CRing) |
18 | 13, 14, 4, 15, 17, 9 | irngssv 33285 | . . . . . . . . 9 β’ (π β (πΈ IntgRing πΉ) β (BaseβπΈ)) |
19 | algextdeg.a | . . . . . . . . 9 β’ (π β π΄ β (πΈ IntgRing πΉ)) | |
20 | 18, 19 | sseldd 3979 | . . . . . . . 8 β’ (π β π΄ β (BaseβπΈ)) |
21 | 20 | snssd 4808 | . . . . . . 7 β’ (π β {π΄} β (BaseβπΈ)) |
22 | 12, 21 | unssd 4182 | . . . . . 6 β’ (π β (πΉ βͺ {π΄}) β (BaseβπΈ)) |
23 | 4, 8, 22 | fldgenssid 32927 | . . . . 5 β’ (π β (πΉ βͺ {π΄}) β (πΈ fldGen (πΉ βͺ {π΄}))) |
24 | 23 | unssad 4183 | . . . 4 β’ (π β πΉ β (πΈ fldGen (πΉ βͺ {π΄}))) |
25 | ressabs 17215 | . . . 4 β’ (((πΈ fldGen (πΉ βͺ {π΄})) β V β§ πΉ β (πΈ fldGen (πΉ βͺ {π΄}))) β ((πΈ βΎs (πΈ fldGen (πΉ βͺ {π΄}))) βΎs πΉ) = (πΈ βΎs πΉ)) | |
26 | 3, 24, 25 | sylancr 586 | . . 3 β’ (π β ((πΈ βΎs (πΈ fldGen (πΉ βͺ {π΄}))) βΎs πΉ) = (πΈ βΎs πΉ)) |
27 | 2, 26 | eqtrid 2779 | . 2 β’ (π β (πΏ βΎs πΉ) = (πΈ βΎs πΉ)) |
28 | 27, 14 | eqtr4di 2785 | 1 β’ (π β (πΏ βΎs πΉ) = πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 Vcvv 3469 βͺ cun 3942 β wss 3944 {csn 4624 βcfv 6542 (class class class)co 7414 Basecbs 17165 βΎs cress 17194 0gc0g 17406 SubGrpcsubg 19059 SubRingcsubrg 20488 DivRingcdr 20606 Fieldcfield 20607 SubDRingcsdrg 20656 evalSub1 ces1 22206 deg1 cdg1 25961 fldGen cfldgen 32924 IntgRing cirng 33280 minPoly cminply 33289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-ghm 19152 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-srg 20111 df-ring 20159 df-cring 20160 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-drng 20608 df-field 20609 df-sdrg 20657 df-lmod 20727 df-lss 20798 df-lsp 20838 df-assa 21767 df-asp 21768 df-ascl 21769 df-psr 21822 df-mvr 21823 df-mpl 21824 df-opsr 21826 df-evls 21996 df-psr1 22073 df-ply1 22075 df-evls1 22208 df-mon1 26040 df-fldgen 32925 df-irng 33281 |
This theorem is referenced by: algextdeglem4 33311 |
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