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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > minplycl | Structured version Visualization version GIF version |
Description: The minimal polynomial is a polynomial. (Contributed by Thierry Arnoux, 22-Mar-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) |
ply1annig1p.0 | ⊢ 0 = (0g‘𝐸) |
ply1annig1p.q | ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } |
ply1annig1p.k | ⊢ 𝐾 = (RSpan‘𝑃) |
ply1annig1p.g | ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹)) |
minplyval.1 | ⊢ 𝑀 = (𝐸 minPoly 𝐹) |
Ref | Expression |
---|---|
minplycl | ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1annig1p.o | . . 3 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
2 | ply1annig1p.p | . . 3 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) | |
3 | ply1annig1p.b | . . 3 ⊢ 𝐵 = (Base‘𝐸) | |
4 | ply1annig1p.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ Field) | |
5 | ply1annig1p.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
6 | ply1annig1p.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
7 | ply1annig1p.0 | . . 3 ⊢ 0 = (0g‘𝐸) | |
8 | ply1annig1p.q | . . 3 ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } | |
9 | ply1annig1p.k | . . 3 ⊢ 𝐾 = (RSpan‘𝑃) | |
10 | ply1annig1p.g | . . 3 ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹)) | |
11 | minplyval.1 | . . 3 ⊢ 𝑀 = (𝐸 minPoly 𝐹) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | minplyval 33070 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄)) |
13 | 4 | fldcrngd 20517 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing) |
14 | issdrg 20551 | . . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) | |
15 | 5, 14 | sylib 217 | . . . . . 6 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) |
16 | 15 | simp2d 1142 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
17 | 1, 2, 3, 13, 16, 6, 7, 8 | ply1annidl 33067 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (LIdeal‘𝑃)) |
18 | eqid 2731 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
19 | eqid 2731 | . . . . 5 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃) | |
20 | 18, 19 | lidlss 20982 | . . . 4 ⊢ (𝑄 ∈ (LIdeal‘𝑃) → 𝑄 ⊆ (Base‘𝑃)) |
21 | 17, 20 | syl 17 | . . 3 ⊢ (𝜑 → 𝑄 ⊆ (Base‘𝑃)) |
22 | 15 | simp3d 1143 | . . . 4 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing) |
23 | 2, 10, 19 | ig1pcl 25942 | . . . 4 ⊢ (((𝐸 ↾s 𝐹) ∈ DivRing ∧ 𝑄 ∈ (LIdeal‘𝑃)) → (𝐺‘𝑄) ∈ 𝑄) |
24 | 22, 17, 23 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝐺‘𝑄) ∈ 𝑄) |
25 | 21, 24 | sseldd 3983 | . 2 ⊢ (𝜑 → (𝐺‘𝑄) ∈ (Base‘𝑃)) |
26 | 12, 25 | eqeltrd 2832 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 {crab 3431 ⊆ wss 3948 dom cdm 5676 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ↾s cress 17180 0gc0g 17392 SubRingcsubrg 20461 DivRingcdr 20504 Fieldcfield 20505 SubDRingcsdrg 20549 LIdealclidl 20932 RSpancrsp 20933 Poly1cpl1 21933 evalSub1 ces1 22065 idlGen1pcig1p 25896 minPoly cminply 33060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-subg 19043 df-ghm 19132 df-cntz 19226 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-srg 20085 df-ring 20133 df-cring 20134 df-oppr 20229 df-dvdsr 20252 df-unit 20253 df-invr 20283 df-rhm 20367 df-subrng 20438 df-subrg 20463 df-drng 20506 df-field 20507 df-sdrg 20550 df-lmod 20620 df-lss 20691 df-lsp 20731 df-sra 20934 df-rgmod 20935 df-lidl 20936 df-rlreg 21103 df-cnfld 21149 df-assa 21631 df-asp 21632 df-ascl 21633 df-psr 21685 df-mvr 21686 df-mpl 21687 df-opsr 21689 df-evls 21859 df-evl 21860 df-psr1 21936 df-vr1 21937 df-ply1 21938 df-coe1 21939 df-evls1 22067 df-evl1 22068 df-mdeg 25819 df-deg1 25820 df-mon1 25897 df-uc1p 25898 df-ig1p 25901 df-minply 33061 |
This theorem is referenced by: minplyirred 33074 minplym1p 33076 algextdeglem4 33080 algextdeglem6 33082 algextdeglem7 33083 algextdeglem8 33084 |
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