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| Mirrors > Home > MPE Home > Th. List > Mathboxes > algextdeglem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for algextdeg 33766. The quotient 𝑃 / 𝑍 of the vector space 𝑃 of polynomials by the subspace 𝑍 of polynomials annihilating 𝐴 is itself a vector space. (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 |
|---|---|
| algextdeglem3 | ⊢ (𝜑 → 𝑄 ∈ LVec) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | algextdeglem.q | . 2 ⊢ 𝑄 = (𝑃 /s (𝑃 ~QG 𝑍)) | |
| 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 | algextdeg.e | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
| 7 | issdrg 20789 | . . . . 5 ⊢ (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) | |
| 8 | 6, 7 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸 ↾s 𝐹) ∈ DivRing)) |
| 9 | 8 | simp3d 1145 | . . 3 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing) |
| 10 | 5, 9 | ply1lvec 33585 | . 2 ⊢ (𝜑 → 𝑃 ∈ LVec) |
| 11 | algextdeglem.z | . . . 4 ⊢ 𝑍 = (◡𝐺 “ {(0g‘𝐿)}) | |
| 12 | eqidd 2738 | . . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐹) = ((subringAlg ‘𝐿)‘𝐹)) | |
| 13 | eqidd 2738 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐿)) | |
| 14 | eqid 2737 | . . . . . . . . . 10 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
| 15 | algextdeg.f | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Field) | |
| 16 | 15 | flddrngd 20741 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ DivRing) |
| 17 | 8 | simp2d 1144 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
| 18 | subrgsubg 20577 | . . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸)) | |
| 19 | 14 | subgss 19145 | . . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → 𝐹 ⊆ (Base‘𝐸)) |
| 20 | 17, 18, 19 | 3syl 18 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸)) |
| 21 | algextdeglem.o | . . . . . . . . . . . . . 14 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
| 22 | eqid 2737 | . . . . . . . . . . . . . 14 ⊢ (0g‘𝐸) = (0g‘𝐸) | |
| 23 | 15 | fldcrngd 20742 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ CRing) |
| 24 | 21, 3, 14, 22, 23, 17 | irngssv 33738 | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸)) |
| 25 | algextdeg.a | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹)) | |
| 26 | 24, 25 | sseldd 3984 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸)) |
| 27 | 26 | snssd 4809 | . . . . . . . . . . 11 ⊢ (𝜑 → {𝐴} ⊆ (Base‘𝐸)) |
| 28 | 20, 27 | unssd 4192 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (Base‘𝐸)) |
| 29 | 14, 16, 28 | fldgenssid 33315 | . . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∪ {𝐴}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
| 30 | 29 | unssad 4193 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ⊆ (𝐸 fldGen (𝐹 ∪ {𝐴}))) |
| 31 | 14, 16, 28 | fldgenssv 33317 | . . . . . . . . 9 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸)) |
| 32 | algextdeg.l | . . . . . . . . . 10 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴}))) | |
| 33 | 32, 14 | ressbas2 17283 | . . . . . . . . 9 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝐴})) ⊆ (Base‘𝐸) → (𝐸 fldGen (𝐹 ∪ {𝐴})) = (Base‘𝐿)) |
| 34 | 31, 33 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝐴})) = (Base‘𝐿)) |
| 35 | 30, 34 | sseqtrd 4020 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿)) |
| 36 | 12, 13, 35 | sralmod0 21195 | . . . . . 6 ⊢ (𝜑 → (0g‘𝐿) = (0g‘((subringAlg ‘𝐿)‘𝐹))) |
| 37 | 36 | sneqd 4638 | . . . . 5 ⊢ (𝜑 → {(0g‘𝐿)} = {(0g‘((subringAlg ‘𝐿)‘𝐹))}) |
| 38 | 37 | imaeq2d 6078 | . . . 4 ⊢ (𝜑 → (◡𝐺 “ {(0g‘𝐿)}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})) |
| 39 | 11, 38 | eqtrid 2789 | . . 3 ⊢ (𝜑 → 𝑍 = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))})) |
| 40 | algextdeg.d | . . . . 5 ⊢ 𝐷 = (deg1‘𝐸) | |
| 41 | algextdeg.m | . . . . 5 ⊢ 𝑀 = (𝐸 minPoly 𝐹) | |
| 42 | algextdeglem.u | . . . . 5 ⊢ 𝑈 = (Base‘𝑃) | |
| 43 | algextdeglem.g | . . . . 5 ⊢ 𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴)) | |
| 44 | algextdeglem.n | . . . . 5 ⊢ 𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍)) | |
| 45 | algextdeglem.j | . . . . 5 ⊢ 𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝)) | |
| 46 | 3, 32, 40, 41, 15, 6, 25, 21, 2, 42, 43, 44, 11, 1, 45 | algextdeglem2 33759 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹))) |
| 47 | eqid 2737 | . . . . 5 ⊢ (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) = (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) | |
| 48 | eqid 2737 | . . . . 5 ⊢ (0g‘((subringAlg ‘𝐿)‘𝐹)) = (0g‘((subringAlg ‘𝐿)‘𝐹)) | |
| 49 | eqid 2737 | . . . . 5 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃) | |
| 50 | 47, 48, 49 | lmhmkerlss 21050 | . . . 4 ⊢ (𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)) → (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) ∈ (LSubSp‘𝑃)) |
| 51 | 46, 50 | syl 17 | . . 3 ⊢ (𝜑 → (◡𝐺 “ {(0g‘((subringAlg ‘𝐿)‘𝐹))}) ∈ (LSubSp‘𝑃)) |
| 52 | 39, 51 | eqeltrd 2841 | . 2 ⊢ (𝜑 → 𝑍 ∈ (LSubSp‘𝑃)) |
| 53 | 1, 10, 52 | quslvec 33388 | 1 ⊢ (𝜑 → 𝑄 ∈ LVec) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 [cec 8743 Basecbs 17247 ↾s cress 17274 0gc0g 17484 /s cqus 17550 SubGrpcsubg 19138 ~QG cqg 19140 SubRingcsubrg 20569 DivRingcdr 20729 Fieldcfield 20730 SubDRingcsdrg 20787 LSubSpclss 20929 LMHom clmhm 21018 LVecclvec 21101 subringAlg csra 21170 Poly1cpl1 22178 evalSub1 ces1 22317 deg1cdg1 26093 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 |
| 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-ec 8747 df-qs 8751 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-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-imas 17553 df-qus 17554 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-nsg 19142 df-eqg 19143 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-drng 20731 df-field 20732 df-sdrg 20788 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lmhm 21021 df-lvec 21102 df-sra 21172 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-mon1 26170 df-fldgen 33313 df-irng 33734 |
| This theorem is referenced by: algextdeglem4 33761 algextdeglem6 33763 |
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