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| Mirrors > Home > MPE Home > Th. List > Mathboxes > algextdeglem5 | Structured version Visualization version GIF version | ||
| Description: Lemma for algextdeg 33766. The subspace 𝑍 of annihilators of 𝐴 is a principal ideal generated by the minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| Ref | Expression |
|---|---|
| algextdeg.k | ⊢ 𝐾 = (𝐸 ↾s 𝐹) |
| algextdeg.l | ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
| algextdeg.d | ⊢ 𝐷 = (deg1‘𝐸) |
| algextdeg.m | ⊢ 𝑀 = (𝐸 minPoly 𝐹) |
| algextdeg.f | ⊢ (𝜑 → 𝐸 ∈ Field) |
| algextdeg.e | ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) |
| algextdeg.a | ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹)) |
| algextdeglem.o | ⊢ 𝑂 = (𝐸 evalSub1 𝐹) |
| algextdeglem.y | ⊢ 𝑃 = (Poly1‘𝐾) |
| algextdeglem.u | ⊢ 𝑈 = (Base‘𝑃) |
| algextdeglem.g | ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) |
| algextdeglem.n | ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍)) |
| algextdeglem.z | ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)}) |
| algextdeglem.q | ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍)) |
| algextdeglem.j | ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝)) |
| Ref | Expression |
|---|---|
| algextdeglem5 | ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{(𝑀‘𝐴)})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | algextdeglem.o | . . 3 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
| 2 | algextdeglem.y | . . . 4 ⊢ 𝑃 = (Poly1‘𝐾) | |
| 3 | algextdeg.k | . . . . 5 ⊢ 𝐾 = (𝐸 ↾s 𝐹) | |
| 4 | 3 | fveq2i 6909 | . . . 4 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹)) |
| 5 | 2, 4 | eqtri 2765 | . . 3 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) |
| 6 | eqid 2737 | . . 3 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
| 7 | algextdeg.f | . . 3 ⊢ (𝜑 → 𝐸 ∈ Field) | |
| 8 | algextdeg.e | . . 3 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (0g‘𝐸) = (0g‘𝐸) | |
| 10 | 7 | fldcrngd 20742 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ CRing) |
| 11 | issdrg 20789 | . . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) | |
| 12 | 8, 11 | sylib 218 | . . . . . 6 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) |
| 13 | 12 | simp2d 1144 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
| 14 | 1, 3, 6, 9, 10, 13 | irngssv 33738 | . . . 4 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸)) |
| 15 | algextdeg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹)) | |
| 16 | 14, 15 | sseldd 3984 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸)) |
| 17 | eqid 2737 | . . 3 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} | |
| 18 | eqid 2737 | . . 3 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃) | |
| 19 | eqid 2737 | . . 3 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹)) | |
| 20 | 1, 5, 6, 7, 8, 16, 9, 17, 18, 19 | ply1annig1p 33747 | . 2 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = ((RSpan‘𝑃)‘{((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})})) |
| 21 | algextdeglem.z | . . . 4 ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)}) | |
| 22 | 10 | crnggrpd 20244 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Grp) |
| 23 | 22 | grpmndd 18964 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Mnd) |
| 24 | 7 | flddrngd 20741 | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ DivRing) |
| 25 | subrgsubg 20577 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸)) | |
| 26 | 6 | subgss 19145 | . . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸)) |
| 27 | 13, 25, 26 | 3syl 18 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸)) |
| 28 | 16 | snssd 4809 | . . . . . . . . . 10 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸)) |
| 29 | 27, 28 | unssd 4192 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸)) |
| 30 | 6, 24, 29 | fldgensdrg 33316 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸)) |
| 31 | sdrgsubrg 20792 | . . . . . . . 8 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubDRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸)) | |
| 32 | subrgsubg 20577 | . . . . . . . 8 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubRing‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸)) | |
| 33 | 9 | subg0cl 19152 | . . . . . . . 8 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
| 34 | 30, 31, 32, 33 | 4syl 19 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
| 35 | 6, 24, 29 | fldgenssv 33317 | . . . . . . 7 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸)) |
| 36 | algextdeg.l | . . . . . . . 8 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) | |
| 37 | 36, 6, 9 | ress0g 18775 | . . . . . . 7 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})) ∧ (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸)) → (0g‘𝐸) = (0g‘𝐿)) |
| 38 | 23, 34, 35, 37 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐿)) |
| 39 | 38 | sneqd 4638 | . . . . 5 ⊢ (𝜑 → {(0g‘𝐸)} = {(0g‘𝐿)}) |
| 40 | 39 | imaeq2d 6078 | . . . 4 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐸)}) = (◡𝐺 “ {(0g‘𝐿)})) |
| 41 | 21, 40 | eqtr4id 2796 | . . 3 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘𝐸)})) |
| 42 | algextdeglem.g | . . . . 5 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) | |
| 43 | algextdeglem.u | . . . . . 6 ⊢ 𝑈 = (Base‘𝑃) | |
| 44 | 43 | mpteq1i 5238 | . . . . 5 ⊢ (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴)) |
| 45 | 42, 44 | eqtri 2765 | . . . 4 ⊢ 𝐺 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴)) |
| 46 | 1, 5, 6, 10, 13, 16, 9, 17, 45 | ply1annidllem 33744 | . . 3 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)} = (◡𝐺 “ {(0g‘𝐸)})) |
| 47 | 41, 46 | eqtr4d 2780 | . 2 ⊢ (𝜑 → 𝑍 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)}) |
| 48 | algextdeg.m | . . . . 5 ⊢ 𝑀 = (𝐸 minPoly 𝐹) | |
| 49 | 1, 5, 6, 7, 8, 16, 9, 17, 18, 19, 48 | minplyval 33748 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})) |
| 50 | 49 | sneqd 4638 | . . 3 ⊢ (𝜑 → {(𝑀‘𝐴)} = {((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})}) |
| 51 | 50 | fveq2d 6910 | . 2 ⊢ (𝜑 → ((RSpan‘𝑃)‘{(𝑀‘𝐴)}) = ((RSpan‘𝑃)‘{((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = (0g‘𝐸)})})) |
| 52 | 20, 47, 51 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{(𝑀‘𝐴)})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 ∪ cun 3949 ⊆ wss 3951 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 ◡ccnv 5684 dom cdm 5685 “ cima 5688 ‘cfv 6561 (class class class)co 7431 [cec 8743 Basecbs 17247 ↾s cress 17274 0gc0g 17484 /s cqus 17550 Mndcmnd 18747 SubGrpcsubg 19138 ~QG cqg 19140 SubRingcsubrg 20569 DivRingcdr 20729 Fieldcfield 20730 SubDRingcsdrg 20787 RSpancrsp 21217 Poly1cpl1 22178 evalSub1 ces1 22317 deg1cdg1 26093 idlGen1pcig1p 26169 fldGen cfldgen 33312 IntgRing cirng 33733 minPoly cminply 33742 |
| 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-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 ax-pre-sup 11233 ax-addf 11234 |
| 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-ofr 7698 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-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 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-z 12614 df-dec 12734 df-uz 12879 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-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 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-acs 17632 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-srg 20184 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-drng 20731 df-field 20732 df-sdrg 20788 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-cnfld 21365 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-evls1 22319 df-evl1 22320 df-mdeg 26094 df-deg1 26095 df-mon1 26170 df-uc1p 26171 df-q1p 26172 df-r1p 26173 df-ig1p 26174 df-fldgen 33313 df-irng 33734 df-minply 33743 |
| This theorem is referenced by: algextdeglem6 33763 |
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