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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 19962 | . . . 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 2818 | . . . . . 6 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
16 | eqid 2818 | . . . . . 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 24683 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ (( deg1 ‘𝑅)‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) |
19 | 18 | simp1d 1134 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅)) |
20 | eqid 2818 | . . . . 5 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅) | |
21 | 20, 15 | mon1puc1p 24671 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic1p‘𝑅)) → 𝐺 ∈ (Unic1p‘𝑅)) |
22 | 3, 19, 21 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Unic1p‘𝑅)) |
23 | facth1.d | . . . 4 ⊢ ∥ = (∥r‘𝑃) | |
24 | eqid 2818 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
25 | eqid 2818 | . . . 4 ⊢ (rem1p‘𝑅) = (rem1p‘𝑅) | |
26 | 5, 23, 6, 20, 24, 25 | dvdsr1p 24682 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐺 ∥ 𝐹 ↔ (𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃))) |
27 | 3, 4, 22, 26 | syl3anc 1363 | . 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 24684 | . . 3 ⊢ (𝜑 → (𝐹(rem1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁))) |
29 | 5, 10, 17, 24 | ply1scl0 20386 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0g‘𝑃)) |
30 | 3, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0g‘𝑃)) |
31 | 30 | eqcomd 2824 | . . 3 ⊢ (𝜑 → (0g‘𝑃) = (𝐴‘ 0 )) |
32 | 28, 31 | eqeq12d 2834 | . 2 ⊢ (𝜑 → ((𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ))) |
33 | 5, 10, 7, 6 | ply1sclf1 20385 | . . . 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 20424 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾) |
36 | 7, 17 | ring0cl 19248 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
37 | 3, 36 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐾) |
38 | f1fveq 7011 | . . 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 832 | . 2 ⊢ (𝜑 → ((𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ) ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
40 | 27, 32, 39 | 3bitrd 306 | 1 ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {csn 4557 class class class wbr 5057 ◡ccnv 5547 “ cima 5551 –1-1→wf1 6345 ‘cfv 6348 (class class class)co 7145 1c1 10526 Basecbs 16471 0gc0g 16701 -gcsg 18043 Ringcrg 19226 CRingccrg 19227 ∥rcdsr 19317 NzRingcnzr 19958 algSccascl 20012 var1cv1 20272 Poly1cpl1 20273 eval1ce1 20405 deg1 cdg1 24575 Monic1pcmn1 24646 Unic1pcuc1p 24647 rem1pcr1p 24649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-srg 19185 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-nzr 19959 df-rlreg 19984 df-assa 20013 df-asp 20014 df-ascl 20015 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-evls 20214 df-evl 20215 df-psr1 20276 df-vr1 20277 df-ply1 20278 df-coe1 20279 df-evl1 20407 df-cnfld 20474 df-mdeg 24576 df-deg1 24577 df-mon1 24651 df-uc1p 24652 df-q1p 24653 df-r1p 24654 |
This theorem is referenced by: fta1glem1 24686 fta1glem2 24687 |
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