![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2734 | . . 3 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
3 | eqid 2734 | . . 3 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
4 | r1pid2.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Domn) | |
5 | domnring 20723 | . . . . 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 2734 | . . . . 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 26210 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑁) → (𝐴(quot1p‘𝑅)𝐵) ∈ 𝑈) |
13 | 6, 7, 8, 12 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝐴(quot1p‘𝑅)𝐵) ∈ 𝑈) |
14 | 10, 1, 11 | uc1pcl 26197 | . . . . 5 ⊢ (𝐵 ∈ 𝑁 → 𝐵 ∈ 𝑈) |
15 | 8, 14 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
16 | 10, 2, 11 | uc1pn0 26199 | . . . . 5 ⊢ (𝐵 ∈ 𝑁 → 𝐵 ≠ (0g‘𝑃)) |
17 | 8, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ (0g‘𝑃)) |
18 | 15, 17 | eldifsnd 4791 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∖ {(0g‘𝑃)})) |
19 | 10 | ply1domn 26177 | . . . 4 ⊢ (𝑅 ∈ Domn → 𝑃 ∈ Domn) |
20 | 4, 19 | syl 17 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Domn) |
21 | 1, 2, 3, 13, 18, 20 | domneq0r 20740 | . 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 2734 | . . . . . . 7 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
24 | 10, 1, 11, 9, 22, 3, 23 | r1pid 26214 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑁) → 𝐴 = (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵))) |
25 | 6, 7, 8, 24 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → 𝐴 = (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵))) |
26 | 25 | eqeq2d 2745 | . . . 4 ⊢ (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐴𝐸𝐵) = (((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵)(+g‘𝑃)(𝐴𝐸𝐵)))) |
27 | eqcom 2741 | . . . 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 20723 | . . . . . . 7 ⊢ (𝑃 ∈ Domn → 𝑃 ∈ Ring) | |
30 | 20, 29 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring) |
31 | 30 | ringgrpd 20259 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ Grp) |
32 | 22, 10, 1, 11 | r1pcl 26212 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑁) → (𝐴𝐸𝐵) ∈ 𝑈) |
33 | 6, 7, 8, 32 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝐴𝐸𝐵) ∈ 𝑈) |
34 | 1, 23, 2, 31, 33 | grplidd 18999 | . . . 4 ⊢ (𝜑 → ((0g‘𝑃)(+g‘𝑃)(𝐴𝐸𝐵)) = (𝐴𝐸𝐵)) |
35 | 34 | eqeq2d 2745 | . . 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 20276 | . . . 4 ⊢ (𝜑 → ((𝐴(quot1p‘𝑅)𝐵)(.r‘𝑃)𝐵) ∈ 𝑈) |
37 | 1, 2 | ring0cl 20280 | . . . . 5 ⊢ (𝑃 ∈ Ring → (0g‘𝑃) ∈ 𝑈) |
38 | 30, 37 | syl 17 | . . . 4 ⊢ (𝜑 → (0g‘𝑃) ∈ 𝑈) |
39 | 1, 23 | grprcan 19003 | . . . 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 1371 | . . 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 20308 | . . . . . . 7 ⊢ (𝜑 → ((0g‘𝑃)(.r‘𝑃)𝐵) = (0g‘𝑃)) |
43 | 42 | oveq2d 7446 | . . . . . 6 ⊢ (𝜑 → (𝐴(-g‘𝑃)((0g‘𝑃)(.r‘𝑃)𝐵)) = (𝐴(-g‘𝑃)(0g‘𝑃))) |
44 | eqid 2734 | . . . . . . . 8 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
45 | 1, 2, 44 | grpsubid1 19055 | . . . . . . 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 2775 | . . . . 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 5157 | . . 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 26209 | . . . 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 1370 | . . 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 1536 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 < clt 11292 Basecbs 17244 +gcplusg 17297 .rcmulr 17298 0gc0g 17485 Grpcgrp 18963 -gcsg 18965 Ringcrg 20250 Domncdomn 20708 Poly1cpl1 22193 deg1cdg1 26107 Unic1pcuc1p 26180 quot1pcq1p 26181 rem1pcr1p 26182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-ofr 7697 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 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 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-nzr 20529 df-subrng 20562 df-subrg 20586 df-rlreg 20710 df-domn 20711 df-lmod 20876 df-lss 20947 df-cnfld 21382 df-ascl 21892 df-psr 21946 df-mvr 21947 df-mpl 21948 df-opsr 21950 df-psr1 22196 df-vr1 22197 df-ply1 22198 df-coe1 22199 df-mdeg 26108 df-deg1 26109 df-uc1p 26185 df-q1p 26186 df-r1p 26187 |
This theorem is referenced by: algextdeglem7 33728 algextdeglem8 33729 |
Copyright terms: Public domain | W3C validator |