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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 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 2730 | . . . . . 6 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
| 16 | eqid 2730 | . . . . . 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 26076 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ ((deg1‘𝑅)‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) |
| 19 | 18 | simp1d 1142 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅)) |
| 20 | eqid 2730 | . . . . 5 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅) | |
| 21 | 20, 15 | mon1puc1p 26062 | . . . 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 2730 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 25 | eqid 2730 | . . . 4 ⊢ (rem1p‘𝑅) = (rem1p‘𝑅) | |
| 26 | 5, 23, 6, 20, 24, 25 | dvdsr1p 26075 | . . 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 26077 | . . 3 ⊢ (𝜑 → (𝐹(rem1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁))) |
| 29 | 5, 10, 17, 24 | ply1scl0 22182 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0g‘𝑃)) |
| 30 | 3, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0g‘𝑃)) |
| 31 | 30 | eqcomd 2736 | . . 3 ⊢ (𝜑 → (0g‘𝑃) = (𝐴‘ 0 )) |
| 32 | 28, 31 | eqeq12d 2746 | . 2 ⊢ (𝜑 → ((𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ))) |
| 33 | 5, 10, 7, 6 | ply1sclf1 22181 | . . . 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 22226 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾) |
| 36 | 7, 17 | ring0cl 20182 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
| 37 | 3, 36 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐾) |
| 38 | f1fveq 7239 | . . 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 1540 ∈ wcel 2109 {csn 4591 class class class wbr 5109 ◡ccnv 5639 “ cima 5643 –1-1→wf1 6510 ‘cfv 6513 (class class class)co 7389 1c1 11075 Basecbs 17185 0gc0g 17408 -gcsg 18873 Ringcrg 20148 CRingccrg 20149 ∥rcdsr 20269 NzRingcnzr 20427 algSccascl 21767 var1cv1 22066 Poly1cpl1 22067 eval1ce1 22207 deg1cdg1 25965 Monic1pcmn1 26037 Unic1pcuc1p 26038 rem1pcr1p 26040 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-ofr 7656 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-er 8673 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-sup 9399 df-oi 9469 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-fz 13475 df-fzo 13622 df-seq 13973 df-hash 14302 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17410 df-gsum 17411 df-prds 17416 df-pws 17418 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-ghm 19151 df-cntz 19255 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-srg 20102 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-rhm 20387 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rlreg 20609 df-lmod 20774 df-lss 20844 df-lsp 20884 df-cnfld 21271 df-assa 21768 df-asp 21769 df-ascl 21770 df-psr 21824 df-mvr 21825 df-mpl 21826 df-opsr 21828 df-evls 21987 df-evl 21988 df-psr1 22070 df-vr1 22071 df-ply1 22072 df-coe1 22073 df-evl1 22209 df-mdeg 25966 df-deg1 25967 df-mon1 26042 df-uc1p 26043 df-q1p 26044 df-r1p 26045 |
| This theorem is referenced by: fta1glem1 26079 fta1glem2 26080 |
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