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Mirrors > Home > MPE Home > Th. List > Mathboxes > minplyann | Structured version Visualization version GIF version |
Description: The minimal polynomial for 𝐴 annihilates 𝐴 (Contributed by Thierry Arnoux, 25-Apr-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) |
minplyann.1 | ⊢ 0 = (0g‘𝐸) |
minplyann.m | ⊢ 𝑀 = (𝐸 minPoly 𝐹) |
Ref | Expression |
---|---|
minplyann | ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1annig1p.o | . . . . 5 ⊢ 𝑂 = (𝐸 evalSub1 𝐹) | |
2 | ply1annig1p.p | . . . . 5 ⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹)) | |
3 | ply1annig1p.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐸) | |
4 | ply1annig1p.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ Field) | |
5 | ply1annig1p.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
6 | ply1annig1p.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
7 | minplyann.1 | . . . . 5 ⊢ 0 = (0g‘𝐸) | |
8 | eqid 2735 | . . . . 5 ⊢ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } | |
9 | eqid 2735 | . . . . 5 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃) | |
10 | eqid 2735 | . . . . 5 ⊢ (idlGen1p‘(𝐸 ↾s 𝐹)) = (idlGen1p‘(𝐸 ↾s 𝐹)) | |
11 | minplyann.m | . . . . 5 ⊢ 𝑀 = (𝐸 minPoly 𝐹) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | minplyval 33713 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })) |
13 | eqid 2735 | . . . . . . 7 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹) | |
14 | 13 | sdrgdrng 20808 | . . . . . 6 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing) |
15 | 5, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing) |
16 | 4 | fldcrngd 20759 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ CRing) |
17 | sdrgsubrg 20809 | . . . . . . 7 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸)) | |
18 | 5, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸)) |
19 | 1, 2, 3, 16, 18, 6, 7, 8 | ply1annidl 33710 | . . . . 5 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃)) |
20 | eqid 2735 | . . . . . 6 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃) | |
21 | 2, 10, 20 | ig1pcl 26233 | . . . . 5 ⊢ (((𝐸 ↾s 𝐹) ∈ DivRing ∧ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ∈ (LIdeal‘𝑃)) → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) |
22 | 15, 19, 21 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((idlGen1p‘(𝐸 ↾s 𝐹))‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) |
23 | 12, 22 | eqeltrd 2839 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) |
24 | fveq2 6907 | . . . . . 6 ⊢ (𝑞 = (𝑀‘𝐴) → (𝑂‘𝑞) = (𝑂‘(𝑀‘𝐴))) | |
25 | 24 | fveq1d 6909 | . . . . 5 ⊢ (𝑞 = (𝑀‘𝐴) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘(𝑀‘𝐴))‘𝐴)) |
26 | 25 | eqeq1d 2737 | . . . 4 ⊢ (𝑞 = (𝑀‘𝐴) → (((𝑂‘𝑞)‘𝐴) = 0 ↔ ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )) |
27 | 26 | elrab 3695 | . . 3 ⊢ ((𝑀‘𝐴) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ↔ ((𝑀‘𝐴) ∈ dom 𝑂 ∧ ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )) |
28 | 23, 27 | sylib 218 | . 2 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ dom 𝑂 ∧ ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )) |
29 | 28 | simprd 495 | 1 ⊢ (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 0gc0g 17486 SubRingcsubrg 20586 DivRingcdr 20746 Fieldcfield 20747 SubDRingcsdrg 20804 LIdealclidl 21234 RSpancrsp 21235 Poly1cpl1 22194 evalSub1 ces1 22333 idlGen1pcig1p 26184 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-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-subrng 20563 df-subrg 20587 df-rlreg 20711 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-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-ig1p 26189 df-minply 33708 |
This theorem is referenced by: irredminply 33722 |
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