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| Mirrors > Home > MPE Home > Th. List > Mathboxes > algextdeglem7 | Structured version Visualization version GIF version | ||
| Description: Lemma for algextdeg 33766. The polynomials 𝑋 of lower degree than the minimal polynomial are left unchanged when taking the remainder of the division by that 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)) |
| algextdeglem.r | ⊢ 𝑅 = (rem1p‘𝐾) |
| algextdeglem.h | ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴))) |
| algextdeglem.t | ⊢ 𝑇 = (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴)))) |
| algextdeglem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
| Ref | Expression |
|---|---|
| algextdeglem7 | ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝐻‘𝑋) = 𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | algextdeg.k | . . . . 5 ⊢ 𝐾 = (𝐸 ↾s 𝐹) | |
| 2 | algextdeg.d | . . . . 5 ⊢ 𝐷 = (deg1‘𝐸) | |
| 3 | algextdeglem.y | . . . . 5 ⊢ 𝑃 = (Poly1‘𝐾) | |
| 4 | algextdeglem.u | . . . . 5 ⊢ 𝑈 = (Base‘𝑃) | |
| 5 | algextdeglem.o | . . . . . . 7 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
| 6 | 1 | fveq2i 6909 | . . . . . . . 8 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹)) |
| 7 | 3, 6 | eqtri 2765 | . . . . . . 7 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) |
| 8 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
| 9 | algextdeg.f | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field) | |
| 10 | algextdeg.e | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
| 11 | eqid 2737 | . . . . . . . . 9 ⊢ (0g‘𝐸) = (0g‘𝐸) | |
| 12 | 9 | fldcrngd 20742 | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing) |
| 13 | sdrgsubrg 20792 | . . . . . . . . . 10 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸)) | |
| 14 | 10, 13 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
| 15 | 5, 1, 8, 11, 12, 14 | irngssv 33738 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸)) |
| 16 | algextdeg.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹)) | |
| 17 | 15, 16 | sseldd 3984 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸)) |
| 18 | eqid 2737 | . . . . . . 7 ⊢ {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} = {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} | |
| 19 | eqid 2737 | . . . . . . 7 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃) | |
| 20 | eqid 2737 | . . . . . . 7 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹)) | |
| 21 | algextdeg.m | . . . . . . 7 ⊢ 𝑀 = (𝐸 minPoly 𝐹) | |
| 22 | 5, 7, 8, 9, 10, 17, 11, 18, 19, 20, 21 | minplycl 33749 | . . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃)) |
| 23 | 22, 4 | eleqtrrdi 2852 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈) |
| 24 | 1, 2, 3, 4, 23, 14 | ressdeg1 33591 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) = ((deg1‘𝐾)‘(𝑀‘𝐴))) |
| 25 | 24 | breq2d 5155 | . . 3 ⊢ (𝜑 → (((deg1‘𝐾)‘𝑋) < (𝐷‘(𝑀‘𝐴)) ↔ ((deg1‘𝐾)‘𝑋) < ((deg1‘𝐾)‘(𝑀‘𝐴)))) |
| 26 | algextdeglem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
| 27 | eqid 2737 | . . . . 5 ⊢ (deg1‘𝐾) = (deg1‘𝐾) | |
| 28 | algextdeglem.t | . . . . 5 ⊢ 𝑇 = (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴)))) | |
| 29 | 9 | flddrngd 20741 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing) |
| 30 | 29 | drngringd 20737 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Ring) |
| 31 | eqid 2737 | . . . . . . . . 9 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸) | |
| 32 | eqid 2737 | . . . . . . . . 9 ⊢ (PwSer1‘𝐾) = (PwSer1‘𝐾) | |
| 33 | eqid 2737 | . . . . . . . . 9 ⊢ (Base‘(PwSer1‘𝐾)) = (Base‘(PwSer1‘𝐾)) | |
| 34 | eqid 2737 | . . . . . . . . 9 ⊢ (Base‘(Poly1‘𝐸)) = (Base‘(Poly1‘𝐸)) | |
| 35 | 31, 1, 3, 4, 14, 32, 33, 34 | ressply1bas2 22229 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 = ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸)))) |
| 36 | inss2 4238 | . . . . . . . 8 ⊢ ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))) ⊆ (Base‘(Poly1‘𝐸)) | |
| 37 | 35, 36 | eqsstrdi 4028 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(Poly1‘𝐸))) |
| 38 | 37, 23 | sseldd 3984 | . . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸))) |
| 39 | eqid 2737 | . . . . . . 7 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸)) | |
| 40 | 39, 9, 10, 21, 16 | irngnminplynz 33755 | . . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) |
| 41 | 2, 31, 39, 34 | deg1nn0cl 26127 | . . . . . 6 ⊢ ((𝐸 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)) ∧ (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0) |
| 42 | 30, 38, 40, 41 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0) |
| 43 | fldsdrgfld 20799 | . . . . . . . . 9 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸 ↾s 𝐹) ∈ Field) | |
| 44 | 9, 10, 43 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ Field) |
| 45 | 1, 44 | eqeltrid 2845 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Field) |
| 46 | fldidom 20771 | . . . . . . 7 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn) | |
| 47 | 45, 46 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ IDomn) |
| 48 | 47 | idomringd 20728 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring) |
| 49 | 3, 27, 28, 42, 48, 4 | ply1degleel 33616 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝑋 ∈ 𝑈 ∧ ((deg1‘𝐾)‘𝑋) < (𝐷‘(𝑀‘𝐴))))) |
| 50 | 26, 49 | mpbirand 707 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ ((deg1‘𝐾)‘𝑋) < (𝐷‘(𝑀‘𝐴)))) |
| 51 | eqid 2737 | . . . 4 ⊢ (Unic1p‘𝐾) = (Unic1p‘𝐾) | |
| 52 | algextdeglem.r | . . . 4 ⊢ 𝑅 = (rem1p‘𝐾) | |
| 53 | 47 | idomdomd 20726 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Domn) |
| 54 | 1 | fveq2i 6909 | . . . . . 6 ⊢ (Monic1p‘𝐾) = (Monic1p‘(𝐸 ↾s 𝐹)) |
| 55 | 39, 9, 10, 21, 16, 54 | minplym1p 33756 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘𝐾)) |
| 56 | eqid 2737 | . . . . . 6 ⊢ (Monic1p‘𝐾) = (Monic1p‘𝐾) | |
| 57 | 51, 56 | mon1puc1p 26190 | . . . . 5 ⊢ ((𝐾 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Monic1p‘𝐾)) → (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) |
| 58 | 48, 55, 57 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Unic1p‘𝐾)) |
| 59 | 3, 4, 51, 52, 27, 53, 26, 58 | r1pid2 26201 | . . 3 ⊢ (𝜑 → ((𝑋𝑅(𝑀‘𝐴)) = 𝑋 ↔ ((deg1‘𝐾)‘𝑋) < ((deg1‘𝐾)‘(𝑀‘𝐴)))) |
| 60 | 25, 50, 59 | 3bitr4d 311 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝑋𝑅(𝑀‘𝐴)) = 𝑋)) |
| 61 | algextdeglem.h | . . . 4 ⊢ 𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴))) | |
| 62 | oveq1 7438 | . . . 4 ⊢ (𝑝 = 𝑋 → (𝑝𝑅(𝑀‘𝐴)) = (𝑋𝑅(𝑀‘𝐴))) | |
| 63 | ovexd 7466 | . . . 4 ⊢ (𝜑 → (𝑋𝑅(𝑀‘𝐴)) ∈ V) | |
| 64 | 61, 62, 26, 63 | fvmptd3 7039 | . . 3 ⊢ (𝜑 → (𝐻‘𝑋) = (𝑋𝑅(𝑀‘𝐴))) |
| 65 | 64 | eqeq1d 2739 | . 2 ⊢ (𝜑 → ((𝐻‘𝑋) = 𝑋 ↔ (𝑋𝑅(𝑀‘𝐴)) = 𝑋)) |
| 66 | 60, 65 | bitr4d 282 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝐻‘𝑋) = 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 Vcvv 3480 ∪ cun 3949 ∩ cin 3950 {csn 4626 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 dom cdm 5685 “ cima 5688 ‘cfv 6561 (class class class)co 7431 [cec 8743 -∞cmnf 11293 < clt 11295 ℕ0cn0 12526 [,)cico 13389 Basecbs 17247 ↾s cress 17274 0gc0g 17484 /s cqus 17550 ~QG cqg 19140 Ringcrg 20230 SubRingcsubrg 20569 IDomncidom 20693 Fieldcfield 20730 SubDRingcsdrg 20787 RSpancrsp 21217 PwSer1cps1 22176 Poly1cpl1 22178 evalSub1 ces1 22317 deg1cdg1 26093 Monic1pcmn1 26165 Unic1pcuc1p 26166 rem1pcr1p 26168 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-ico 13393 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-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-idom 20696 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-irng 33734 df-minply 33743 |
| This theorem is referenced by: algextdeglem8 33765 |
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