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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 20533 | . . . 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 2735 | . . . . . 6 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
| 16 | eqid 2735 | . . . . . 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 26219 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (Monic1p‘𝑅) ∧ ((deg1‘𝑅)‘𝐺) = 1 ∧ (◡(𝑂‘𝐺) “ { 0 }) = {𝑁})) |
| 19 | 18 | simp1d 1141 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Monic1p‘𝑅)) |
| 20 | eqid 2735 | . . . . 5 ⊢ (Unic1p‘𝑅) = (Unic1p‘𝑅) | |
| 21 | 20, 15 | mon1puc1p 26205 | . . . 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 2735 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 25 | eqid 2735 | . . . 4 ⊢ (rem1p‘𝑅) = (rem1p‘𝑅) | |
| 26 | 5, 23, 6, 20, 24, 25 | dvdsr1p 26218 | . . 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 26220 | . . 3 ⊢ (𝜑 → (𝐹(rem1p‘𝑅)𝐺) = (𝐴‘((𝑂‘𝐹)‘𝑁))) |
| 29 | 5, 10, 17, 24 | ply1scl0 22309 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝐴‘ 0 ) = (0g‘𝑃)) |
| 30 | 3, 29 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴‘ 0 ) = (0g‘𝑃)) |
| 31 | 30 | eqcomd 2741 | . . 3 ⊢ (𝜑 → (0g‘𝑃) = (𝐴‘ 0 )) |
| 32 | 28, 31 | eqeq12d 2751 | . 2 ⊢ (𝜑 → ((𝐹(rem1p‘𝑅)𝐺) = (0g‘𝑃) ↔ (𝐴‘((𝑂‘𝐹)‘𝑁)) = (𝐴‘ 0 ))) |
| 33 | 5, 10, 7, 6 | ply1sclf1 22308 | . . . 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 22353 | . . 3 ⊢ (𝜑 → ((𝑂‘𝐹)‘𝑁) ∈ 𝐾) |
| 36 | 7, 17 | ring0cl 20281 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾) |
| 37 | 3, 36 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐾) |
| 38 | f1fveq 7282 | . . 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 837 | . 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 1537 ∈ wcel 2106 {csn 4631 class class class wbr 5148 ◡ccnv 5688 “ cima 5692 –1-1→wf1 6560 ‘cfv 6563 (class class class)co 7431 1c1 11154 Basecbs 17245 0gc0g 17486 -gcsg 18966 Ringcrg 20251 CRingccrg 20252 ∥rcdsr 20371 NzRingcnzr 20529 algSccascl 21890 var1cv1 22193 Poly1cpl1 22194 eval1ce1 22334 deg1cdg1 26108 Monic1pcmn1 26180 Unic1pcuc1p 26181 rem1pcr1p 26183 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-rhm 20489 df-nzr 20530 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-lmod 20877 df-lss 20948 df-lsp 20988 df-cnfld 21383 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evl1 22336 df-mdeg 26109 df-deg1 26110 df-mon1 26185 df-uc1p 26186 df-q1p 26187 df-r1p 26188 |
| This theorem is referenced by: fta1glem1 26222 fta1glem2 26223 |
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