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| Mirrors > Home > MPE Home > Th. List > r1pid2 | Structured version Visualization version GIF version | ||
| Description: Identity law for polynomial remainder operation: it leaves a polynomial 𝐴 unchanged iff the degree of 𝐴 is less than the degree of the divisor 𝐵. (Contributed by Thierry Arnoux, 2-Apr-2025.) Generalize to domains. (Revised by SN, 21-Jun-2025.) |
| Ref | Expression |
|---|---|
| r1pid2.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| r1pid2.u | ⊢ 𝑈 = (Base‘𝑃) |
| r1pid2.n | ⊢ 𝑁 = (Unic1p‘𝑅) |
| r1pid2.e | ⊢ 𝐸 = (rem1p‘𝑅) |
| r1pid2.d | ⊢ 𝐷 = (deg1‘𝑅) |
| r1pid2.r | ⊢ (𝜑 → 𝑅 ∈ Domn) |
| r1pid2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| r1pid2.b | ⊢ (𝜑 → 𝐵 ∈ 𝑁) |
| Ref | Expression |
|---|---|
| r1pid2 | ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐷‘𝐴) < (𝐷‘𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r1pid2.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
| 2 | eqid 2737 | . . 3 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 3 | eqid 2737 | . . 3 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
| 4 | r1pid2.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Domn) | |
| 5 | domnring 20707 | . . . . 5 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | |
| 6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
| 7 | r1pid2.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 8 | r1pid2.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑁) | |
| 9 | eqid 2737 | . . . . 5 ⊢ (quot1p‘𝑅) = (quot1p‘𝑅) | |
| 10 | r1pid2.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 11 | r1pid2.n | . . . . 5 ⊢ 𝑁 = (Unic1p‘𝑅) | |
| 12 | 9, 10, 1, 11 | q1pcl 26196 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑁) → (𝐴(quot1p‘𝑅)𝐵) ∈ 𝑈) |
| 13 | 6, 7, 8, 12 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝐴(quot1p‘𝑅)𝐵) ∈ 𝑈) |
| 14 | 10, 1, 11 | uc1pcl 26183 | . . . . 5 ⊢ (𝐵 ∈ 𝑁 → 𝐵 ∈ 𝑈) |
| 15 | 8, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
| 16 | 10, 2, 11 | uc1pn0 26185 | . . . . 5 ⊢ (𝐵 ∈ 𝑁 → 𝐵 ≠ (0g‘𝑃)) |
| 17 | 8, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ (0g‘𝑃)) |
| 18 | 15, 17 | eldifsnd 4787 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∖ {(0g‘𝑃)})) |
| 19 | 10 | ply1domn 26163 | . . . 4 ⊢ (𝑅 ∈ Domn → 𝑃 ∈ Domn) |
| 20 | 4, 19 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Domn) |
| 21 | 1, 2, 3, 13, 18, 20 | domneq0r 20724 | . 2 ⊢ (𝜑 → (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵) = (0g‘𝑃) ↔ (𝐴(quot1p‘𝑅)𝐵) = (0g‘𝑃))) |
| 22 | r1pid2.e | . . . . . . 7 ⊢ 𝐸 = (rem1p‘𝑅) | |
| 23 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
| 24 | 10, 1, 11, 9, 22, 3, 23 | r1pid 26200 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑁) → 𝐴 = (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵))) |
| 25 | 6, 7, 8, 24 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → 𝐴 = (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵))) |
| 26 | 25 | eqeq2d 2748 | . . . 4 ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐴𝐸𝐵) = (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵)))) |
| 27 | eqcom 2744 | . . . 4 ⊢ ((((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵)) = (𝐴𝐸𝐵) ↔ (𝐴𝐸𝐵) = (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵))) | |
| 28 | 26, 27 | bitr4di 289 | . . 3 ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵)) = (𝐴𝐸𝐵))) |
| 29 | domnring 20707 | . . . . . . 7 ⊢ (𝑃 ∈ Domn → 𝑃 ∈ Ring) | |
| 30 | 20, 29 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring) |
| 31 | 30 | ringgrpd 20239 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ Grp) |
| 32 | 22, 10, 1, 11 | r1pcl 26198 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑁) → (𝐴𝐸𝐵) ∈ 𝑈) |
| 33 | 6, 7, 8, 32 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝐴𝐸𝐵) ∈ 𝑈) |
| 34 | 1, 23, 2, 31, 33 | grplidd 18987 | . . . 4 ⊢ (𝜑 → ((0g‘𝑃)(+g‘𝑃)(𝐴𝐸𝐵)) = (𝐴𝐸𝐵)) |
| 35 | 34 | eqeq2d 2748 | . . 3 ⊢ (𝜑 → ((((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵)) = ((0g‘𝑃)(+g‘𝑃)(𝐴𝐸𝐵)) ↔ (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵)) = (𝐴𝐸𝐵))) |
| 36 | 1, 3, 30, 13, 15 | ringcld 20257 | . . . 4 ⊢ (𝜑 → ((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵) ∈ 𝑈) |
| 37 | 1, 2 | ring0cl 20264 | . . . . 5 ⊢ (𝑃 ∈ Ring → (0g‘𝑃) ∈ 𝑈) |
| 38 | 30, 37 | syl 17 | . . . 4 ⊢ (𝜑 → (0g‘𝑃) ∈ 𝑈) |
| 39 | 1, 23 | grprcan 18991 | . . . 4 ⊢ ((𝑃 ∈ Grp ∧ (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵) ∈ 𝑈 ∧ (0g‘𝑃) ∈ 𝑈 ∧ (𝐴𝐸𝐵) ∈ 𝑈)) → ((((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵)) = ((0g‘𝑃)(+g‘𝑃)(𝐴𝐸𝐵)) ↔ ((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵) = (0g‘𝑃))) |
| 40 | 31, 36, 38, 33, 39 | syl13anc 1374 | . . 3 ⊢ (𝜑 → ((((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵)) = ((0g‘𝑃)(+g‘𝑃)(𝐴𝐸𝐵)) ↔ ((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵) = (0g‘𝑃))) |
| 41 | 28, 35, 40 | 3bitr2d 307 | . 2 ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ ((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵) = (0g‘𝑃))) |
| 42 | 1, 3, 2, 30, 15 | ringlzd 20292 | . . . . . . 7 ⊢ (𝜑 → ((0g‘𝑃)(.r‘𝑃)𝐵) = (0g‘𝑃)) |
| 43 | 42 | oveq2d 7447 | . . . . . 6 ⊢ (𝜑 → (𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵)) = (𝐴(-g‘𝑃)(0g‘𝑃))) |
| 44 | eqid 2737 | . . . . . . . 8 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
| 45 | 1, 2, 44 | grpsubid1 19043 | . . . . . . 7 ⊢ ((𝑃 ∈ Grp ∧ 𝐴 ∈ 𝑈) → (𝐴(-g‘𝑃)(0g‘𝑃)) = 𝐴) |
| 46 | 31, 7, 45 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴(-g‘𝑃)(0g‘𝑃)) = 𝐴) |
| 47 | 43, 46 | eqtr2d 2778 | . . . . 5 ⊢ (𝜑 → 𝐴 = (𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵))) |
| 48 | 47 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐴) = (𝐷‘(𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵)))) |
| 49 | 48 | breq1d 5153 | . . 3 ⊢ (𝜑 → ((𝐷‘𝐴) < (𝐷‘𝐵) ↔ (𝐷‘(𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵))) < (𝐷‘𝐵))) |
| 50 | 38 | biantrurd 532 | . . 3 ⊢ (𝜑 → ((𝐷‘(𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵))) < (𝐷‘𝐵) ↔ ((0g‘𝑃) ∈ 𝑈 ∧ (𝐷‘(𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵))) < (𝐷‘𝐵)))) |
| 51 | r1pid2.d | . . . . 5 ⊢ 𝐷 = (deg1‘𝑅) | |
| 52 | 9, 10, 1, 51, 44, 3, 11 | q1peqb 26195 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑁) → (((0g‘𝑃) ∈ 𝑈 ∧ (𝐷‘(𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵))) < (𝐷‘𝐵)) ↔ (𝐴(quot1p‘𝑅)𝐵) = (0g‘𝑃))) |
| 53 | 6, 7, 8, 52 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (((0g‘𝑃) ∈ 𝑈 ∧ (𝐷‘(𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵))) < (𝐷‘𝐵)) ↔ (𝐴(quot1p‘𝑅)𝐵) = (0g‘𝑃))) |
| 54 | 49, 50, 53 | 3bitrd 305 | . 2 ⊢ (𝜑 → ((𝐷‘𝐴) < (𝐷‘𝐵) ↔ (𝐴(quot1p‘𝑅)𝐵) = (0g‘𝑃))) |
| 55 | 21, 41, 54 | 3bitr4d 311 | 1 ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐷‘𝐴) < (𝐷‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 < clt 11295 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 0gc0g 17484 Grpcgrp 18951 -gcsg 18953 Ringcrg 20230 Domncdomn 20692 Poly1cpl1 22178 deg1cdg1 26093 Unic1pcuc1p 26166 quot1pcq1p 26167 rem1pcr1p 26168 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-lmod 20860 df-lss 20930 df-cnfld 21365 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-mdeg 26094 df-deg1 26095 df-uc1p 26171 df-q1p 26172 df-r1p 26173 |
| This theorem is referenced by: algextdeglem7 33764 algextdeglem8 33765 |
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