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Mirrors > Home > MPE Home > Th. List > Mathboxes > algextdeglem7 | Structured version Visualization version GIF version |
Description: Lemma for algextdeg 33731. 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 6910 | . . . . . . . 8 ⊢ (Poly1‘𝐾) = (Poly1‘(𝐸 ↾s 𝐹)) |
7 | 3, 6 | eqtri 2763 | . . . . . . 7 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) |
8 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
9 | algextdeg.f | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field) | |
10 | algextdeg.e | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
11 | eqid 2735 | . . . . . . . . 9 ⊢ (0g‘𝐸) = (0g‘𝐸) | |
12 | 9 | fldcrngd 20759 | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ CRing) |
13 | sdrgsubrg 20809 | . . . . . . . . . 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 33703 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 IntgRing 𝐹) ⊆ (Base‘𝐸)) |
16 | algextdeg.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹)) | |
17 | 15, 16 | sseldd 3996 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐸)) |
18 | eqid 2735 | . . . . . . 7 ⊢ {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} = {𝑝 ∈ dom 𝑂 ∣ ((𝑂‘𝑝)‘𝐴) = (0g‘𝐸)} | |
19 | eqid 2735 | . . . . . . 7 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃) | |
20 | eqid 2735 | . . . . . . 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 33714 | . . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃)) |
23 | 22, 4 | eleqtrrdi 2850 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑈) |
24 | 1, 2, 3, 4, 23, 14 | ressdeg1 33571 | . . . 4 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) = ((deg1‘𝐾)‘(𝑀‘𝐴))) |
25 | 24 | breq2d 5160 | . . 3 ⊢ (𝜑 → (((deg1‘𝐾)‘𝑋) < (𝐷‘(𝑀‘𝐴)) ↔ ((deg1‘𝐾)‘𝑋) < ((deg1‘𝐾)‘(𝑀‘𝐴)))) |
26 | algextdeglem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
27 | eqid 2735 | . . . . 5 ⊢ (deg1‘𝐾) = (deg1‘𝐾) | |
28 | algextdeglem.t | . . . . 5 ⊢ 𝑇 = (◡(deg1‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴)))) | |
29 | 9 | flddrngd 20758 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ DivRing) |
30 | 29 | drngringd 20754 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Ring) |
31 | eqid 2735 | . . . . . . . . 9 ⊢ (Poly1‘𝐸) = (Poly1‘𝐸) | |
32 | eqid 2735 | . . . . . . . . 9 ⊢ (PwSer1‘𝐾) = (PwSer1‘𝐾) | |
33 | eqid 2735 | . . . . . . . . 9 ⊢ (Base‘(PwSer1‘𝐾)) = (Base‘(PwSer1‘𝐾)) | |
34 | eqid 2735 | . . . . . . . . 9 ⊢ (Base‘(Poly1‘𝐸)) = (Base‘(Poly1‘𝐸)) | |
35 | 31, 1, 3, 4, 14, 32, 33, 34 | ressply1bas2 22245 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 = ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸)))) |
36 | inss2 4246 | . . . . . . . 8 ⊢ ((Base‘(PwSer1‘𝐾)) ∩ (Base‘(Poly1‘𝐸))) ⊆ (Base‘(Poly1‘𝐸)) | |
37 | 35, 36 | eqsstrdi 4050 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (Base‘(Poly1‘𝐸))) |
38 | 37, 23 | sseldd 3996 | . . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸))) |
39 | eqid 2735 | . . . . . . 7 ⊢ (0g‘(Poly1‘𝐸)) = (0g‘(Poly1‘𝐸)) | |
40 | 39, 9, 10, 21, 16 | irngnminplynz 33720 | . . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) |
41 | 2, 31, 39, 34 | deg1nn0cl 26142 | . . . . . 6 ⊢ ((𝐸 ∈ Ring ∧ (𝑀‘𝐴) ∈ (Base‘(Poly1‘𝐸)) ∧ (𝑀‘𝐴) ≠ (0g‘(Poly1‘𝐸))) → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0) |
42 | 30, 38, 40, 41 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝐷‘(𝑀‘𝐴)) ∈ ℕ0) |
43 | fldsdrgfld 20816 | . . . . . . . . 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 2843 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Field) |
46 | fldidom 20788 | . . . . . . 7 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn) | |
47 | 45, 46 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ IDomn) |
48 | 47 | idomringd 20745 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Ring) |
49 | 3, 27, 28, 42, 48, 4 | ply1degleel 33596 | . . . 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 2735 | . . . 4 ⊢ (Unic1p‘𝐾) = (Unic1p‘𝐾) | |
52 | algextdeglem.r | . . . 4 ⊢ 𝑅 = (rem1p‘𝐾) | |
53 | 47 | idomdomd 20743 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Domn) |
54 | 1 | fveq2i 6910 | . . . . . 6 ⊢ (Monic1p‘𝐾) = (Monic1p‘(𝐸 ↾s 𝐹)) |
55 | 39, 9, 10, 21, 16, 54 | minplym1p 33721 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ (Monic1p‘𝐾)) |
56 | eqid 2735 | . . . . . 6 ⊢ (Monic1p‘𝐾) = (Monic1p‘𝐾) | |
57 | 51, 56 | mon1puc1p 26205 | . . . . 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 26216 | . . 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 2737 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 {crab 3433 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 {csn 4631 ∪ cuni 4912 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5688 dom cdm 5689 “ cima 5692 ‘cfv 6563 (class class class)co 7431 [cec 8742 -∞cmnf 11291 < clt 11293 ℕ0cn0 12524 [,)cico 13386 Basecbs 17245 ↾s cress 17274 0gc0g 17486 /s cqus 17552 ~QG cqg 19153 Ringcrg 20251 SubRingcsubrg 20586 IDomncidom 20710 Fieldcfield 20747 SubDRingcsdrg 20804 RSpancrsp 21235 PwSer1cps1 22192 Poly1cpl1 22194 evalSub1 ces1 22333 deg1cdg1 26108 Monic1pcmn1 26180 Unic1pcuc1p 26181 rem1pcr1p 26183 idlGen1pcig1p 26184 fldGen cfldgen 33292 IntgRing cirng 33698 minPoly cminply 33707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-ico 13390 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-rhm 20489 df-nzr 20530 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-domn 20712 df-idom 20713 df-drng 20748 df-field 20749 df-sdrg 20805 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-cnfld 21383 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-evl1 22336 df-mdeg 26109 df-deg1 26110 df-mon1 26185 df-uc1p 26186 df-q1p 26187 df-r1p 26188 df-ig1p 26189 df-irng 33699 df-minply 33708 |
This theorem is referenced by: algextdeglem8 33730 |
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