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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 𝐹 has a root at 𝐴 iff 𝐺 = 𝑥 − 𝐴 is a factor of 𝐹. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
ply1rem.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1rem.b | ⊢ 𝐵 = (Base‘𝑃) |
ply1rem.k | ⊢ 𝐾 = (Base‘𝑅) |
ply1rem.x | ⊢ 𝑋 = (var1‘𝑅) |
ply1rem.m | ⊢ − = (-g‘𝑃) |
ply1rem.a | ⊢ 𝐴 = (algSc‘𝑃) |
ply1rem.g | ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁)) |
ply1rem.o | ⊢ 𝑂 = (eval1‘𝑅) |
ply1rem.1 | ⊢ (𝜑 → 𝑅 ∈ NzRing) |
ply1rem.2 | ⊢ (𝜑 → 𝑅 ∈ CRing) |
ply1rem.3 | ⊢ (𝜑 → 𝑁 ∈ 𝐾) |
ply1rem.4 | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
facth1.z | ⊢ 0 = (0g‘𝑅) |
facth1.d | ⊢ ∥ = (∥r‘𝑃) |
Ref | Expression |
---|---|
facth1 | ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1rem.1 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ NzRing) | |
2 | nzrring 19261 | . . . 4 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) |
4 | ply1rem.4 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
5 | ply1rem.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | ply1rem.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
7 | ply1rem.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑅) | |
8 | ply1rem.x | . . . . . 6 ⊢ 𝑋 = (var1‘𝑅) | |
9 | ply1rem.m | . . . . . 6 ⊢ − = (-g‘𝑃) | |
10 | ply1rem.a | . . . . . 6 ⊢ 𝐴 = (algSc‘𝑃) | |
11 | ply1rem.g | . . . . . 6 ⊢ 𝐺 = (𝑋 − (𝐴‘𝑁)) | |
12 | ply1rem.o | . . . . . 6 ⊢ 𝑂 = (eval1‘𝑅) | |
13 | ply1rem.2 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
14 | ply1rem.3 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ 𝐾) | |
15 | eqid 2622 | . . . . . 6 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
16 | eqid 2622 | . . . . . 6 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
17 | facth1.z | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
18 | 5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17 | ply1remlem 23922 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ (( deg1 ‘𝑅)‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) |
19 | 18 | simp1d 1073 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅)) |
20 | eqid 2622 | . . . . 5 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅) | |
21 | 20, 15 | mon1puc1p 23910 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic1p‘𝑅)) → 𝐺 ∈ (Unic1p‘𝑅)) |
22 | 3, 19, 21 | syl2anc 693 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Unic1p‘𝑅)) |
23 | facth1.d | . . . 4 ⊢ ∥ = (∥r‘𝑃) | |
24 | eqid 2622 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
25 | eqid 2622 | . . . 4 ⊢ (rem1p‘𝑅) = (rem1p‘𝑅) | |
26 | 5, 23, 6, 20, 24, 25 | dvdsr1p 23921 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐺 ∥ 𝐹 ↔ (𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃))) |
27 | 3, 4, 22, 26 | syl3anc 1326 | . 2 ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ (𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃))) |
28 | 5, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 4, 25 | ply1rem 23923 | . . 3 ⊢ (𝜑 → (𝐹(rem1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁))) |
29 | 5, 10, 17, 24 | ply1scl0 19660 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0g‘𝑃)) |
30 | 3, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0g‘𝑃)) |
31 | 30 | eqcomd 2628 | . . 3 ⊢ (𝜑 → (0g‘𝑃) = (𝐴‘ 0 )) |
32 | 28, 31 | eqeq12d 2637 | . 2 ⊢ (𝜑 → ((𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ))) |
33 | 5, 10, 7, 6 | ply1sclf1 19659 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝐴:𝐾–1-1→𝐵) |
34 | 3, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴:𝐾–1-1→𝐵) |
35 | 12, 5, 7, 6, 13, 14, 4 | fveval1fvcl 19697 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾) |
36 | 7, 17 | ring0cl 18569 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
37 | 3, 36 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐾) |
38 | f1fveq 6519 | . . 3 ⊢ ((𝐴:𝐾–1-1→𝐵 ∧ (((𝑂‘𝐹)‘𝑁) ∈ 𝐾 ∧ 0 ∈ 𝐾)) → ((𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ) ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) | |
39 | 34, 35, 37, 38 | syl12anc 1324 | . 2 ⊢ (𝜑 → ((𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ) ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
40 | 27, 32, 39 | 3bitrd 294 | 1 ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 {csn 4177 class class class wbr 4653 ◡ccnv 5113 “ cima 5117 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 1c1 9937 Basecbs 15857 0gc0g 16100 -gcsg 17424 Ringcrg 18547 CRingccrg 18548 ∥rcdsr 18638 NzRingcnzr 19257 algSccascl 19311 var1cv1 19546 Poly1cpl1 19547 eval1ce1 19679 deg1 cdg1 23814 Monic1pcmn1 23885 Unic1pcuc1p 23886 rem1pcr1p 23888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-nzr 19258 df-rlreg 19283 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-evls 19506 df-evl 19507 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-evl1 19681 df-cnfld 19747 df-mdeg 23815 df-deg1 23816 df-mon1 23890 df-uc1p 23891 df-q1p 23892 df-r1p 23893 |
This theorem is referenced by: fta1glem1 23925 fta1glem2 23926 |
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