Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 20431 | . . . 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 2731 | . . . . . 6 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
| 16 | eqid 2731 | . . . . . 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 26097 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ ((deg1‘𝑅)‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) |
| 19 | 18 | simp1d 1142 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅)) |
| 20 | eqid 2731 | . . . . 5 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅) | |
| 21 | 20, 15 | mon1puc1p 26083 | . . . 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 2731 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 25 | eqid 2731 | . . . 4 ⊢ (rem1p‘𝑅) = (rem1p‘𝑅) | |
| 26 | 5, 23, 6, 20, 24, 25 | dvdsr1p 26096 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐺 ∥ 𝐹 ↔ (𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃))) |
| 27 | 3, 4, 22, 26 | syl3anc 1373 | . 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 26098 | . . 3 ⊢ (𝜑 → (𝐹(rem1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁))) |
| 29 | 5, 10, 17, 24 | ply1scl0 22204 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0g‘𝑃)) |
| 30 | 3, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0g‘𝑃)) |
| 31 | 30 | eqcomd 2737 | . . 3 ⊢ (𝜑 → (0g‘𝑃) = (𝐴‘ 0 )) |
| 32 | 28, 31 | eqeq12d 2747 | . 2 ⊢ (𝜑 → ((𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ))) |
| 33 | 5, 10, 7, 6 | ply1sclf1 22203 | . . . 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 22248 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾) |
| 36 | 7, 17 | ring0cl 20185 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
| 37 | 3, 36 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐾) |
| 38 | f1fveq 7196 | . . 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 836 | . 2 ⊢ (𝜑 → ((𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ) ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
| 40 | 27, 32, 39 | 3bitrd 305 | 1 ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1541 ∈ wcel 2111 {csn 4573 class class class wbr 5089 ◡ccnv 5613 “ cima 5617 –1-1→wf1 6478 ‘cfv 6481 (class class class)co 7346 1c1 11007 Basecbs 17120 0gc0g 17343 -gcsg 18848 Ringcrg 20151 CRingccrg 20152 ∥rcdsr 20272 NzRingcnzr 20427 algSccascl 21789 var1cv1 22088 Poly1cpl1 22089 eval1ce1 22229 deg1cdg1 25986 Monic1pcmn1 26058 Unic1pcuc1p 26059 rem1pcr1p 26061 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-fzo 13555 df-seq 13909 df-hash 14238 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-srg 20105 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-rhm 20390 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rlreg 20609 df-lmod 20795 df-lss 20865 df-lsp 20905 df-cnfld 21292 df-assa 21790 df-asp 21791 df-ascl 21792 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-evls 22009 df-evl 22010 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-evl1 22231 df-mdeg 25987 df-deg1 25988 df-mon1 26063 df-uc1p 26064 df-q1p 26065 df-r1p 26066 |
| This theorem is referenced by: fta1glem1 26100 fta1glem2 26101 |
| Copyright terms: Public domain | W3C validator |