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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 20416 | . . . 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 2726 | . . . . . 6 β’ (Monic1pβπ ) = (Monic1pβπ ) | |
16 | eqid 2726 | . . . . . 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 26050 | . . . . 5 β’ (π β (πΊ β (Monic1pβπ ) β§ (( deg1 βπ )βπΊ) = 1 β§ (β‘(πβπΊ) β { 0 }) = {π})) |
19 | 18 | simp1d 1139 | . . . 4 β’ (π β πΊ β (Monic1pβπ )) |
20 | eqid 2726 | . . . . 5 β’ (Unic1pβπ ) = (Unic1pβπ ) | |
21 | 20, 15 | mon1puc1p 26037 | . . . 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 2726 | . . . 4 β’ (0gβπ) = (0gβπ) | |
25 | eqid 2726 | . . . 4 β’ (rem1pβπ ) = (rem1pβπ ) | |
26 | 5, 23, 6, 20, 24, 25 | dvdsr1p 26049 | . . 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 1368 | . 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 26051 | . . 3 β’ (π β (πΉ(rem1pβπ )πΊ) = (π΄β((πβπΉ)βπ))) |
29 | 5, 10, 17, 24 | ply1scl0 22160 | . . . . 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 2732 | . . 3 β’ (π β (0gβπ) = (π΄β 0 )) |
32 | 28, 31 | eqeq12d 2742 | . 2 β’ (π β ((πΉ(rem1pβπ )πΊ) = (0gβπ) β (π΄β((πβπΉ)βπ)) = (π΄β 0 ))) |
33 | 5, 10, 7, 6 | ply1sclf1 22159 | . . . 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 22203 | . . 3 β’ (π β ((πβπΉ)βπ) β πΎ) |
36 | 7, 17 | ring0cl 20164 | . . . 4 β’ (π β Ring β 0 β πΎ) |
37 | 3, 36 | syl 17 | . . 3 β’ (π β 0 β πΎ) |
38 | f1fveq 7256 | . . 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 834 | . 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 1533 β wcel 2098 {csn 4623 class class class wbr 5141 β‘ccnv 5668 β cima 5672 β1-1βwf1 6533 βcfv 6536 (class class class)co 7404 1c1 11110 Basecbs 17151 0gc0g 17392 -gcsg 18863 Ringcrg 20136 CRingccrg 20137 β₯rcdsr 20254 NzRingcnzr 20412 algSccascl 21743 var1cv1 22046 Poly1cpl1 22047 eval1ce1 22184 deg1 cdg1 25938 Monic1pcmn1 26012 Unic1pcuc1p 26013 rem1pcr1p 26015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14294 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19048 df-ghm 19137 df-cntz 19231 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-srg 20090 df-ring 20138 df-cring 20139 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-rhm 20372 df-nzr 20413 df-subrng 20444 df-subrg 20469 df-lmod 20706 df-lss 20777 df-lsp 20817 df-rlreg 21191 df-cnfld 21237 df-assa 21744 df-asp 21745 df-ascl 21746 df-psr 21799 df-mvr 21800 df-mpl 21801 df-opsr 21803 df-evls 21973 df-evl 21974 df-psr1 22050 df-vr1 22051 df-ply1 22052 df-coe1 22053 df-evl1 22186 df-mdeg 25939 df-deg1 25940 df-mon1 26017 df-uc1p 26018 df-q1p 26019 df-r1p 26020 |
This theorem is referenced by: fta1glem1 26053 fta1glem2 26054 |
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