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Mirrors > Home > MPE Home > Th. List > deg1mulle2 | Structured version Visualization version GIF version |
Description: Produce a bound on the product of two univariate polynomials given bounds on the factors. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
Ref | Expression |
---|---|
deg1addle.y | โข ๐ = (Poly1โ๐ ) |
deg1addle.d | โข ๐ท = ( deg1 โ๐ ) |
deg1addle.r | โข (๐ โ ๐ โ Ring) |
deg1mulle2.b | โข ๐ต = (Baseโ๐) |
deg1mulle2.t | โข ยท = (.rโ๐) |
deg1mulle2.f | โข (๐ โ ๐น โ ๐ต) |
deg1mulle2.g | โข (๐ โ ๐บ โ ๐ต) |
deg1mulle2.j1 | โข (๐ โ ๐ฝ โ โ0) |
deg1mulle2.k1 | โข (๐ โ ๐พ โ โ0) |
deg1mulle2.j2 | โข (๐ โ (๐ทโ๐น) โค ๐ฝ) |
deg1mulle2.k2 | โข (๐ โ (๐ทโ๐บ) โค ๐พ) |
Ref | Expression |
---|---|
deg1mulle2 | โข (๐ โ (๐ทโ(๐น ยท ๐บ)) โค (๐ฝ + ๐พ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . 2 โข (1o mPoly ๐ ) = (1o mPoly ๐ ) | |
2 | deg1addle.d | . . 3 โข ๐ท = ( deg1 โ๐ ) | |
3 | 2 | deg1fval 25598 | . 2 โข ๐ท = (1o mDeg ๐ ) |
4 | 1on 8478 | . . 3 โข 1o โ On | |
5 | 4 | a1i 11 | . 2 โข (๐ โ 1o โ On) |
6 | deg1addle.r | . 2 โข (๐ โ ๐ โ Ring) | |
7 | deg1addle.y | . . 3 โข ๐ = (Poly1โ๐ ) | |
8 | eqid 2733 | . . 3 โข (PwSer1โ๐ ) = (PwSer1โ๐ ) | |
9 | deg1mulle2.b | . . 3 โข ๐ต = (Baseโ๐) | |
10 | 7, 8, 9 | ply1bas 21719 | . 2 โข ๐ต = (Baseโ(1o mPoly ๐ )) |
11 | deg1mulle2.t | . . 3 โข ยท = (.rโ๐) | |
12 | 7, 1, 11 | ply1mulr 21749 | . 2 โข ยท = (.rโ(1o mPoly ๐ )) |
13 | deg1mulle2.f | . 2 โข (๐ โ ๐น โ ๐ต) | |
14 | deg1mulle2.g | . 2 โข (๐ โ ๐บ โ ๐ต) | |
15 | deg1mulle2.j1 | . 2 โข (๐ โ ๐ฝ โ โ0) | |
16 | deg1mulle2.k1 | . 2 โข (๐ โ ๐พ โ โ0) | |
17 | deg1mulle2.j2 | . 2 โข (๐ โ (๐ทโ๐น) โค ๐ฝ) | |
18 | deg1mulle2.k2 | . 2 โข (๐ โ (๐ทโ๐บ) โค ๐พ) | |
19 | 1, 3, 5, 6, 10, 12, 13, 14, 15, 16, 17, 18 | mdegmulle2 25597 | 1 โข (๐ โ (๐ทโ(๐น ยท ๐บ)) โค (๐ฝ + ๐พ)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 class class class wbr 5149 Oncon0 6365 โcfv 6544 (class class class)co 7409 1oc1o 8459 + caddc 11113 โค cle 11249 โ0cn0 12472 Basecbs 17144 .rcmulr 17198 Ringcrg 20056 mPoly cmpl 21459 PwSer1cps1 21699 Poly1cpl1 21701 deg1 cdg1 25569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-mulg 18951 df-subg 19003 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-subrg 20317 df-cnfld 20945 df-psr 21462 df-mpl 21464 df-opsr 21466 df-psr1 21704 df-ply1 21706 df-mdeg 25570 df-deg1 25571 |
This theorem is referenced by: deg1mul2 25632 ply1divex 25654 hbtlem4 41868 |
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