Nearby theorems
|
|
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > r1p0 | Structured version Visualization version GIF version | ||
| Description: Polynomial remainder operation of a division of the zero polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| Ref | Expression |
|---|---|
| r1padd1.p | β’ π = (Poly1βπ ) |
| r1padd1.u | β’ π = (Baseβπ) |
| r1padd1.n | β’ π = (Unic1pβπ ) |
| r1padd1.e | β’ πΈ = (rem1pβπ ) |
| r1p0.r | β’ (π β π β Ring) |
| r1p0.d | β’ (π β π· β π) |
| r1p0.0 | β’ 0 = (0gβπ) |
| Ref | Expression |
|---|---|
| r1p0 | β’ (π β ( 0 πΈπ·) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r1p0.r | . . . . . . 7 β’ (π β π β Ring) | |
| 2 | r1padd1.p | . . . . . . . 8 β’ π = (Poly1βπ ) | |
| 3 | 2 | ply1sca 21997 | . . . . . . 7 β’ (π β Ring β π = (Scalarβπ)) |
| 4 | 1, 3 | syl 17 | . . . . . 6 β’ (π β π = (Scalarβπ)) |
| 5 | 4 | fveq2d 6896 | . . . . 5 β’ (π β (0gβπ ) = (0gβ(Scalarβπ))) |
| 6 | 5 | oveq1d 7428 | . . . 4 β’ (π β ((0gβπ )( Β·π βπ) 0 ) = ((0gβ(Scalarβπ))( Β·π βπ) 0 )) |
| 7 | 2 | ply1lmod 21996 | . . . . . 6 β’ (π β Ring β π β LMod) |
| 8 | 1, 7 | syl 17 | . . . . 5 β’ (π β π β LMod) |
| 9 | 2 | ply1ring 21992 | . . . . . 6 β’ (π β Ring β π β Ring) |
| 10 | r1padd1.u | . . . . . . 7 β’ π = (Baseβπ) | |
| 11 | r1p0.0 | . . . . . . 7 β’ 0 = (0gβπ) | |
| 12 | 10, 11 | ring0cl 20157 | . . . . . 6 β’ (π β Ring β 0 β π) |
| 13 | 1, 9, 12 | 3syl 18 | . . . . 5 β’ (π β 0 β π) |
| 14 | eqid 2730 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 15 | eqid 2730 | . . . . . 6 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 16 | eqid 2730 | . . . . . 6 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
| 17 | 10, 14, 15, 16, 11 | lmod0vs 20651 | . . . . 5 β’ ((π β LMod β§ 0 β π) β ((0gβ(Scalarβπ))( Β·π βπ) 0 ) = 0 ) |
| 18 | 8, 13, 17 | syl2anc 582 | . . . 4 β’ (π β ((0gβ(Scalarβπ))( Β·π βπ) 0 ) = 0 ) |
| 19 | 6, 18 | eqtrd 2770 | . . 3 β’ (π β ((0gβπ )( Β·π βπ) 0 ) = 0 ) |
| 20 | 19 | oveq1d 7428 | . 2 β’ (π β (((0gβπ )( Β·π βπ) 0 )πΈπ·) = ( 0 πΈπ·)) |
| 21 | r1padd1.n | . . . 4 β’ π = (Unic1pβπ ) | |
| 22 | r1padd1.e | . . . 4 β’ πΈ = (rem1pβπ ) | |
| 23 | r1p0.d | . . . 4 β’ (π β π· β π) | |
| 24 | eqid 2730 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
| 25 | eqid 2730 | . . . . . 6 β’ (0gβπ ) = (0gβπ ) | |
| 26 | 24, 25 | ring0cl 20157 | . . . . 5 β’ (π β Ring β (0gβπ ) β (Baseβπ )) |
| 27 | 1, 26 | syl 17 | . . . 4 β’ (π β (0gβπ ) β (Baseβπ )) |
| 28 | 2, 10, 21, 22, 1, 13, 23, 15, 24, 27 | r1pvsca 32948 | . . 3 β’ (π β (((0gβπ )( Β·π βπ) 0 )πΈπ·) = ((0gβπ )( Β·π βπ)( 0 πΈπ·))) |
| 29 | 5 | oveq1d 7428 | . . 3 β’ (π β ((0gβπ )( Β·π βπ)( 0 πΈπ·)) = ((0gβ(Scalarβπ))( Β·π βπ)( 0 πΈπ·))) |
| 30 | 22, 2, 10, 21 | r1pcl 25909 | . . . . 5 β’ ((π β Ring β§ 0 β π β§ π· β π) β ( 0 πΈπ·) β π) |
| 31 | 1, 13, 23, 30 | syl3anc 1369 | . . . 4 β’ (π β ( 0 πΈπ·) β π) |
| 32 | 10, 14, 15, 16, 11 | lmod0vs 20651 | . . . 4 β’ ((π β LMod β§ ( 0 πΈπ·) β π) β ((0gβ(Scalarβπ))( Β·π βπ)( 0 πΈπ·)) = 0 ) |
| 33 | 8, 31, 32 | syl2anc 582 | . . 3 β’ (π β ((0gβ(Scalarβπ))( Β·π βπ)( 0 πΈπ·)) = 0 ) |
| 34 | 28, 29, 33 | 3eqtrd 2774 | . 2 β’ (π β (((0gβπ )( Β·π βπ) 0 )πΈπ·) = 0 ) |
| 35 | 20, 34 | eqtr3d 2772 | 1 β’ (π β ( 0 πΈπ·) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βcfv 6544 (class class class)co 7413 Basecbs 17150 Scalarcsca 17206 Β·π cvsca 17207 0gc0g 17391 Ringcrg 20129 LModclmod 20616 Poly1cpl1 21922 Unic1pcuc1p 25878 rem1pcr1p 25880 |
| 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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12479 df-z 12565 df-dec 12684 df-uz 12829 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14297 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18707 df-submnd 18708 df-grp 18860 df-minusg 18861 df-sbg 18862 df-mulg 18989 df-subg 19041 df-ghm 19130 df-cntz 19224 df-cmn 19693 df-abl 19694 df-mgp 20031 df-rng 20049 df-ur 20078 df-ring 20131 df-cring 20132 df-oppr 20227 df-dvdsr 20250 df-unit 20251 df-invr 20281 df-subrng 20436 df-subrg 20461 df-lmod 20618 df-lss 20689 df-rlreg 21101 df-cnfld 21147 df-psr 21683 df-mvr 21684 df-mpl 21685 df-opsr 21687 df-psr1 21925 df-vr1 21926 df-ply1 21927 df-coe1 21928 df-mdeg 25804 df-deg1 25805 df-uc1p 25883 df-q1p 25884 df-r1p 25885 |
| This theorem is referenced by: r1pquslmic 32954 |
| Copyright terms: Public domain | W3C validator |