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Mirrors > Home > MPE Home > Th. List > facth1 | Structured version Visualization version GIF version |
Description: The factor theorem and its converse. A polynomial πΉ has a root at π΄ iff πΊ = π₯ β π΄ is a factor of πΉ. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
ply1rem.p | β’ π = (Poly1βπ ) |
ply1rem.b | β’ π΅ = (Baseβπ) |
ply1rem.k | β’ πΎ = (Baseβπ ) |
ply1rem.x | β’ π = (var1βπ ) |
ply1rem.m | β’ β = (-gβπ) |
ply1rem.a | β’ π΄ = (algScβπ) |
ply1rem.g | β’ πΊ = (π β (π΄βπ)) |
ply1rem.o | β’ π = (eval1βπ ) |
ply1rem.1 | β’ (π β π β NzRing) |
ply1rem.2 | β’ (π β π β CRing) |
ply1rem.3 | β’ (π β π β πΎ) |
ply1rem.4 | β’ (π β πΉ β π΅) |
facth1.z | β’ 0 = (0gβπ ) |
facth1.d | β’ β₯ = (β₯rβπ) |
Ref | Expression |
---|---|
facth1 | β’ (π β (πΊ β₯ πΉ β ((πβπΉ)βπ) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1rem.1 | . . . 4 β’ (π β π β NzRing) | |
2 | nzrring 20455 | . . . 4 β’ (π β NzRing β π β Ring) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β π β Ring) |
4 | ply1rem.4 | . . 3 β’ (π β πΉ β π΅) | |
5 | ply1rem.p | . . . . . 6 β’ π = (Poly1βπ ) | |
6 | ply1rem.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
7 | ply1rem.k | . . . . . 6 β’ πΎ = (Baseβπ ) | |
8 | ply1rem.x | . . . . . 6 β’ π = (var1βπ ) | |
9 | ply1rem.m | . . . . . 6 β’ β = (-gβπ) | |
10 | ply1rem.a | . . . . . 6 β’ π΄ = (algScβπ) | |
11 | ply1rem.g | . . . . . 6 β’ πΊ = (π β (π΄βπ)) | |
12 | ply1rem.o | . . . . . 6 β’ π = (eval1βπ ) | |
13 | ply1rem.2 | . . . . . 6 β’ (π β π β CRing) | |
14 | ply1rem.3 | . . . . . 6 β’ (π β π β πΎ) | |
15 | eqid 2728 | . . . . . 6 β’ (Monic1pβπ ) = (Monic1pβπ ) | |
16 | eqid 2728 | . . . . . 6 β’ ( deg1 βπ ) = ( deg1 βπ ) | |
17 | facth1.z | . . . . . 6 β’ 0 = (0gβπ ) | |
18 | 5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17 | ply1remlem 26112 | . . . . 5 β’ (π β (πΊ β (Monic1pβπ ) β§ (( deg1 βπ )βπΊ) = 1 β§ (β‘(πβπΊ) β { 0 }) = {π})) |
19 | 18 | simp1d 1140 | . . . 4 β’ (π β πΊ β (Monic1pβπ )) |
20 | eqid 2728 | . . . . 5 β’ (Unic1pβπ ) = (Unic1pβπ ) | |
21 | 20, 15 | mon1puc1p 26099 | . . . 4 β’ ((π β Ring β§ πΊ β (Monic1pβπ )) β πΊ β (Unic1pβπ )) |
22 | 3, 19, 21 | syl2anc 583 | . . 3 β’ (π β πΊ β (Unic1pβπ )) |
23 | facth1.d | . . . 4 β’ β₯ = (β₯rβπ) | |
24 | eqid 2728 | . . . 4 β’ (0gβπ) = (0gβπ) | |
25 | eqid 2728 | . . . 4 β’ (rem1pβπ ) = (rem1pβπ ) | |
26 | 5, 23, 6, 20, 24, 25 | dvdsr1p 26111 | . . 3 β’ ((π β Ring β§ πΉ β π΅ β§ πΊ β (Unic1pβπ )) β (πΊ β₯ πΉ β (πΉ(rem1pβπ )πΊ) = (0gβπ))) |
27 | 3, 4, 22, 26 | syl3anc 1369 | . 2 β’ (π β (πΊ β₯ πΉ β (πΉ(rem1pβπ )πΊ) = (0gβπ))) |
28 | 5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 4, 25 | ply1rem 26113 | . . 3 β’ (π β (πΉ(rem1pβπ )πΊ) = (π΄β((πβπΉ)βπ))) |
29 | 5, 10, 17, 24 | ply1scl0 22209 | . . . . 5 β’ (π β Ring β (π΄β 0 ) = (0gβπ)) |
30 | 3, 29 | syl 17 | . . . 4 β’ (π β (π΄β 0 ) = (0gβπ)) |
31 | 30 | eqcomd 2734 | . . 3 β’ (π β (0gβπ) = (π΄β 0 )) |
32 | 28, 31 | eqeq12d 2744 | . 2 β’ (π β ((πΉ(rem1pβπ )πΊ) = (0gβπ) β (π΄β((πβπΉ)βπ)) = (π΄β 0 ))) |
33 | 5, 10, 7, 6 | ply1sclf1 22208 | . . . 4 β’ (π β Ring β π΄:πΎβ1-1βπ΅) |
34 | 3, 33 | syl 17 | . . 3 β’ (π β π΄:πΎβ1-1βπ΅) |
35 | 12, 5, 7, 6, 13, 14, 4 | fveval1fvcl 22252 | . . 3 β’ (π β ((πβπΉ)βπ) β πΎ) |
36 | 7, 17 | ring0cl 20203 | . . . 4 β’ (π β Ring β 0 β πΎ) |
37 | 3, 36 | syl 17 | . . 3 β’ (π β 0 β πΎ) |
38 | f1fveq 7272 | . . 3 β’ ((π΄:πΎβ1-1βπ΅ β§ (((πβπΉ)βπ) β πΎ β§ 0 β πΎ)) β ((π΄β((πβπΉ)βπ)) = (π΄β 0 ) β ((πβπΉ)βπ) = 0 )) | |
39 | 34, 35, 37, 38 | syl12anc 836 | . 2 β’ (π β ((π΄β((πβπΉ)βπ)) = (π΄β 0 ) β ((πβπΉ)βπ) = 0 )) |
40 | 27, 32, 39 | 3bitrd 305 | 1 β’ (π β (πΊ β₯ πΉ β ((πβπΉ)βπ) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1534 β wcel 2099 {csn 4629 class class class wbr 5148 β‘ccnv 5677 β cima 5681 β1-1βwf1 6545 βcfv 6548 (class class class)co 7420 1c1 11140 Basecbs 17180 0gc0g 17421 -gcsg 18892 Ringcrg 20173 CRingccrg 20174 β₯rcdsr 20293 NzRingcnzr 20451 algSccascl 21786 var1cv1 22095 Poly1cpl1 22096 eval1ce1 22233 deg1 cdg1 26000 Monic1pcmn1 26074 Unic1pcuc1p 26075 rem1pcr1p 26077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-srg 20127 df-ring 20175 df-cring 20176 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-rhm 20411 df-nzr 20452 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-lsp 20856 df-rlreg 21230 df-cnfld 21280 df-assa 21787 df-asp 21788 df-ascl 21789 df-psr 21842 df-mvr 21843 df-mpl 21844 df-opsr 21846 df-evls 22018 df-evl 22019 df-psr1 22099 df-vr1 22100 df-ply1 22101 df-coe1 22102 df-evl1 22235 df-mdeg 26001 df-deg1 26002 df-mon1 26079 df-uc1p 26080 df-q1p 26081 df-r1p 26082 |
This theorem is referenced by: fta1glem1 26115 fta1glem2 26116 |
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