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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 20532 | . . . 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 2738 | . . . . . 6 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
| 16 | eqid 2738 | . . . . . 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 25327 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ (( deg1 ‘𝑅)‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) |
| 19 | 18 | simp1d 1141 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅)) |
| 20 | eqid 2738 | . . . . 5 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅) | |
| 21 | 20, 15 | mon1puc1p 25315 | . . . 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 2738 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 25 | eqid 2738 | . . . 4 ⊢ (rem1p‘𝑅) = (rem1p‘𝑅) | |
| 26 | 5, 23, 6, 20, 24, 25 | dvdsr1p 25326 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐺 ∥ 𝐹 ↔ (𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃))) |
| 27 | 3, 4, 22, 26 | syl3anc 1370 | . 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 25328 | . . 3 ⊢ (𝜑 → (𝐹(rem1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁))) |
| 29 | 5, 10, 17, 24 | ply1scl0 21461 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0g‘𝑃)) |
| 30 | 3, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0g‘𝑃)) |
| 31 | 30 | eqcomd 2744 | . . 3 ⊢ (𝜑 → (0g‘𝑃) = (𝐴‘ 0 )) |
| 32 | 28, 31 | eqeq12d 2754 | . 2 ⊢ (𝜑 → ((𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ))) |
| 33 | 5, 10, 7, 6 | ply1sclf1 21460 | . . . 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 21499 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾) |
| 36 | 7, 17 | ring0cl 19808 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
| 37 | 3, 36 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐾) |
| 38 | f1fveq 7135 | . . 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 834 | . 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 205 = wceq 1539 ∈ wcel 2106 {csn 4561 class class class wbr 5074 ◡ccnv 5588 “ cima 5592 –1-1→wf1 6430 ‘cfv 6433 (class class class)co 7275 1c1 10872 Basecbs 16912 0gc0g 17150 -gcsg 18579 Ringcrg 19783 CRingccrg 19784 ∥rcdsr 19880 NzRingcnzr 20528 algSccascl 21059 var1cv1 21347 Poly1cpl1 21348 eval1ce1 21480 deg1 cdg1 25216 Monic1pcmn1 25290 Unic1pcuc1p 25291 rem1pcr1p 25293 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-srg 19742 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-rnghom 19959 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-nzr 20529 df-rlreg 20554 df-cnfld 20598 df-assa 21060 df-asp 21061 df-ascl 21062 df-psr 21112 df-mvr 21113 df-mpl 21114 df-opsr 21116 df-evls 21282 df-evl 21283 df-psr1 21351 df-vr1 21352 df-ply1 21353 df-coe1 21354 df-evl1 21482 df-mdeg 25217 df-deg1 25218 df-mon1 25295 df-uc1p 25296 df-q1p 25297 df-r1p 25298 |
| This theorem is referenced by: fta1glem1 25330 fta1glem2 25331 |
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