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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1annig1p | Structured version Visualization version GIF version |
Description: The ideal 𝑄 of polynomials annihilating an element 𝐴 is generated by the ideal's canonical generator. (Contributed by Thierry Arnoux, 9-Feb-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)) |
Ref | Expression |
---|---|
ply1annig1p | ⊢ (𝜑 → 𝑄 = (𝐾‘{(𝐺‘𝑄)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1annig1p.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
2 | issdrg 20658 | . . . 4 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) |
4 | 3 | simp3d 1142 | . 2 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing) |
5 | ply1annig1p.o | . . 3 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
6 | ply1annig1p.p | . . 3 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) | |
7 | ply1annig1p.b | . . 3 ⊢ 𝐵 = (Base‘𝐸) | |
8 | ply1annig1p.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Field) | |
9 | 8 | fldcrngd 20619 | . . 3 ⊢ (𝜑 → 𝐸 ∈ CRing) |
10 | 3 | simp2d 1141 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
11 | ply1annig1p.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
12 | ply1annig1p.0 | . . 3 ⊢ 0 = (0g‘𝐸) | |
13 | ply1annig1p.q | . . 3 ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } | |
14 | 5, 6, 7, 9, 10, 11, 12, 13 | ply1annidl 33296 | . 2 ⊢ (𝜑 → 𝑄 ∈ (LIdeal‘𝑃)) |
15 | ply1annig1p.g | . . 3 ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹)) | |
16 | eqid 2727 | . . 3 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃) | |
17 | ply1annig1p.k | . . 3 ⊢ 𝐾 = (RSpan‘𝑃) | |
18 | 6, 15, 16, 17 | ig1prsp 26089 | . 2 ⊢ (((𝐸 ↾s 𝐹) ∈ DivRing ∧ 𝑄 ∈ (LIdeal‘𝑃)) → 𝑄 = (𝐾‘{(𝐺‘𝑄)})) |
19 | 4, 14, 18 | syl2anc 583 | 1 ⊢ (𝜑 → 𝑄 = (𝐾‘{(𝐺‘𝑄)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {crab 3427 {csn 4624 dom cdm 5672 ‘cfv 6542 (class class class)co 7414 Basecbs 17165 ↾s cress 17194 0gc0g 17406 SubRingcsubrg 20488 DivRingcdr 20606 Fieldcfield 20607 SubDRingcsdrg 20656 LIdealclidl 21084 RSpancrsp 21085 Poly1cpl1 22070 evalSub1 ces1 22206 idlGen1pcig1p 26039 |
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 ax-pre-sup 11202 ax-addf 11203 |
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-tpos 8223 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-inf 9452 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-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 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-oppr 20255 df-dvdsr 20278 df-unit 20279 df-invr 20309 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-sra 21040 df-rgmod 21041 df-lidl 21086 df-rsp 21087 df-rlreg 21212 df-cnfld 21260 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-evl 21997 df-psr1 22073 df-vr1 22074 df-ply1 22075 df-coe1 22076 df-evls1 22208 df-evl1 22209 df-mdeg 25962 df-deg1 25963 df-mon1 26040 df-uc1p 26041 df-q1p 26042 df-r1p 26043 df-ig1p 26044 |
This theorem is referenced by: irngnminplynz 33305 minplym1p 33306 algextdeglem4 33311 algextdeglem5 33312 |
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