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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > minplymindeg | Structured version Visualization version GIF version |
Description: The minimal polynomial of 𝐴 is minimal among the nonzero annihilators of 𝐴 with regard to degree. (Contributed by Thierry Arnoux, 22-Jun-2025.) |
Ref | Expression |
---|---|
ply1annig1p.o | ⊢ 𝑂 = (𝐸 evalSub1 𝐹) |
ply1annig1p.p | ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) |
ply1annig1p.b | ⊢ 𝐵 = (Base‘𝐸) |
ply1annig1p.e | ⊢ (𝜑 → 𝐸 ∈ Field) |
ply1annig1p.f | ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) |
ply1annig1p.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
minplymindeg.0 | ⊢ 0 = (0g‘𝐸) |
minplymindeg.m | ⊢ 𝑀 = (𝐸 minPoly 𝐹) |
minplymindeg.d | ⊢ 𝐷 = (deg1‘(𝐸 ↾s 𝐹)) |
minplymindeg.z | ⊢ 𝑍 = (0g‘𝑃) |
minplymindeg.u | ⊢ 𝑈 = (Base‘𝑃) |
minplymindeg.1 | ⊢ (𝜑 → ((𝑂‘𝐻)‘𝐴) = 0 ) |
minplymindeg.h | ⊢ (𝜑 → 𝐻 ∈ 𝑈) |
minplymindeg.2 | ⊢ (𝜑 → 𝐻 ≠ 𝑍) |
Ref | Expression |
---|---|
minplymindeg | ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ≤ (𝐷‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1annig1p.o | . . . 4 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
2 | ply1annig1p.p | . . . 4 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) | |
3 | ply1annig1p.b | . . . 4 ⊢ 𝐵 = (Base‘𝐸) | |
4 | ply1annig1p.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Field) | |
5 | ply1annig1p.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
6 | ply1annig1p.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
7 | minplymindeg.0 | . . . 4 ⊢ 0 = (0g‘𝐸) | |
8 | eqid 2736 | . . . 4 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } | |
9 | eqid 2736 | . . . 4 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃) | |
10 | eqid 2736 | . . . 4 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹)) | |
11 | minplymindeg.m | . . . 4 ⊢ 𝑀 = (𝐸 minPoly 𝐹) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | minplyval 33729 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })) |
13 | 12 | fveq2d 6908 | . 2 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) = (𝐷‘((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }))) |
14 | minplymindeg.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
15 | eqid 2736 | . . . . 5 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹) | |
16 | 15 | sdrgdrng 20783 | . . . 4 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing) |
17 | 5, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing) |
18 | 4 | fldcrngd 20734 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing) |
19 | sdrgsubrg 20784 | . . . . 5 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸)) | |
20 | 5, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
21 | 1, 2, 3, 18, 20, 6, 7, 8 | ply1annidl 33726 | . . 3 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃)) |
22 | minplymindeg.d | . . 3 ⊢ 𝐷 = (deg1‘(𝐸 ↾s 𝐹)) | |
23 | minplymindeg.z | . . 3 ⊢ 𝑍 = (0g‘𝑃) | |
24 | fveq2 6904 | . . . . . 6 ⊢ (𝑞 = 𝐻 → (𝑂‘𝑞) = (𝑂‘𝐻)) | |
25 | 24 | fveq1d 6906 | . . . . 5 ⊢ (𝑞 = 𝐻 → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘𝐻)‘𝐴)) |
26 | 25 | eqeq1d 2738 | . . . 4 ⊢ (𝑞 = 𝐻 → (((𝑂‘𝑞)‘𝐴) = 0 ↔ ((𝑂‘𝐻)‘𝐴) = 0 )) |
27 | minplymindeg.h | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ 𝑈) | |
28 | 1, 2, 14, 18, 20 | evls1dm 33574 | . . . . 5 ⊢ (𝜑 → dom 𝑂 = 𝑈) |
29 | 27, 28 | eleqtrrd 2843 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ dom 𝑂) |
30 | minplymindeg.1 | . . . 4 ⊢ (𝜑 → ((𝑂‘𝐻)‘𝐴) = 0 ) | |
31 | 26, 29, 30 | elrabd 3693 | . . 3 ⊢ (𝜑 → 𝐻 ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) |
32 | minplymindeg.2 | . . 3 ⊢ (𝜑 → 𝐻 ≠ 𝑍) | |
33 | 2, 10, 14, 17, 21, 22, 23, 31, 32 | ig1pmindeg 33609 | . 2 ⊢ (𝜑 → (𝐷‘((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })) ≤ (𝐷‘𝐻)) |
34 | 13, 33 | eqbrtrd 5163 | 1 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ≤ (𝐷‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 {crab 3435 class class class wbr 5141 dom cdm 5683 ‘cfv 6559 (class class class)co 7429 ≤ cle 11292 Basecbs 17243 ↾s cress 17270 0gc0g 17480 SubRingcsubrg 20561 DivRingcdr 20721 Fieldcfield 20722 SubDRingcsdrg 20779 RSpancrsp 21209 Poly1cpl1 22168 evalSub1 ces1 22307 deg1cdg1 26083 idlGen1pcig1p 26159 minPoly cminply 33723 |
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 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 ax-addf 11230 |
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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-ofr 7695 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-tpos 8247 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-er 8741 df-map 8864 df-pm 8865 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9398 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-starv 17308 df-sca 17309 df-vsca 17310 df-ip 17311 df-tset 17312 df-ple 17313 df-ds 17315 df-unif 17316 df-hom 17317 df-cco 17318 df-0g 17482 df-gsum 17483 df-prds 17488 df-pws 17490 df-mre 17625 df-mrc 17626 df-acs 17628 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-mhm 18792 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-mulg 19082 df-subg 19137 df-ghm 19227 df-cntz 19331 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-srg 20180 df-ring 20228 df-cring 20229 df-oppr 20326 df-dvdsr 20349 df-unit 20350 df-invr 20380 df-rhm 20464 df-subrng 20538 df-subrg 20562 df-rlreg 20686 df-drng 20723 df-field 20724 df-sdrg 20780 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21164 df-rgmod 21165 df-lidl 21210 df-cnfld 21357 df-assa 21865 df-asp 21866 df-ascl 21867 df-psr 21921 df-mvr 21922 df-mpl 21923 df-opsr 21925 df-evls 22090 df-evl 22091 df-psr1 22171 df-vr1 22172 df-ply1 22173 df-coe1 22174 df-evls1 22309 df-evl1 22310 df-mdeg 26084 df-deg1 26085 df-mon1 26160 df-uc1p 26161 df-ig1p 26164 df-minply 33724 |
This theorem is referenced by: rtelextdg2lem 33748 |
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