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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 20445 | . . . 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 25232 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ (( deg1 ‘𝑅)‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) |
| 19 | 18 | simp1d 1140 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅)) |
| 20 | eqid 2738 | . . . . 5 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅) | |
| 21 | 20, 15 | mon1puc1p 25220 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ (Monic1p‘𝑅)) → 𝐺 ∈ (Unic1p‘𝑅)) |
| 22 | 3, 19, 21 | syl2anc 583 | . . 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 25231 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ (Unic1p‘𝑅)) → (𝐺 ∥ 𝐹 ↔ (𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃))) |
| 27 | 3, 4, 22, 26 | syl3anc 1369 | . 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 25233 | . . 3 ⊢ (𝜑 → (𝐹(rem1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁))) |
| 29 | 5, 10, 17, 24 | ply1scl0 21371 | . . . . 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 21370 | . . . 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 21409 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾) |
| 36 | 7, 17 | ring0cl 19723 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
| 37 | 3, 36 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐾) |
| 38 | f1fveq 7116 | . . 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 833 | . 2 ⊢ (𝜑 → ((𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ) ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
| 40 | 27, 32, 39 | 3bitrd 304 | 1 ⊢ (𝜑 → (𝐺 ∥ 𝐹 ↔ ((𝑂‘𝐹)‘𝑁) = 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2108 {csn 4558 class class class wbr 5070 ◡ccnv 5579 “ cima 5583 –1-1→wf1 6415 ‘cfv 6418 (class class class)co 7255 1c1 10803 Basecbs 16840 0gc0g 17067 -gcsg 18494 Ringcrg 19698 CRingccrg 19699 ∥rcdsr 19795 NzRingcnzr 20441 algSccascl 20969 var1cv1 21257 Poly1cpl1 21258 eval1ce1 21390 deg1 cdg1 25121 Monic1pcmn1 25195 Unic1pcuc1p 25196 rem1pcr1p 25198 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-srg 19657 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-rnghom 19874 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-nzr 20442 df-rlreg 20467 df-cnfld 20511 df-assa 20970 df-asp 20971 df-ascl 20972 df-psr 21022 df-mvr 21023 df-mpl 21024 df-opsr 21026 df-evls 21192 df-evl 21193 df-psr1 21261 df-vr1 21262 df-ply1 21263 df-coe1 21264 df-evl1 21392 df-mdeg 25122 df-deg1 25123 df-mon1 25200 df-uc1p 25201 df-q1p 25202 df-r1p 25203 |
| This theorem is referenced by: fta1glem1 25235 fta1glem2 25236 |
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