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Mirrors > Home > MPE Home > Th. List > mdeglt | Structured version Visualization version GIF version |
Description: If there is an upper limit on the degree of a polynomial that is lower than the degree of some exponent bag, then that exponent bag is unrepresented in the polynomial. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
Ref | Expression |
---|---|
mdegval.d | β’ π· = (πΌ mDeg π ) |
mdegval.p | β’ π = (πΌ mPoly π ) |
mdegval.b | β’ π΅ = (Baseβπ) |
mdegval.z | β’ 0 = (0gβπ ) |
mdegval.a | β’ π΄ = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} |
mdegval.h | β’ π» = (β β π΄ β¦ (βfld Ξ£g β)) |
mdeglt.f | β’ (π β πΉ β π΅) |
medglt.x | β’ (π β π β π΄) |
mdeglt.lt | β’ (π β (π·βπΉ) < (π»βπ)) |
Ref | Expression |
---|---|
mdeglt | β’ (π β (πΉβπ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdeglt.lt | . 2 β’ (π β (π·βπΉ) < (π»βπ)) | |
2 | fveq2 6846 | . . . . 5 β’ (π₯ = π β (π»βπ₯) = (π»βπ)) | |
3 | 2 | breq2d 5121 | . . . 4 β’ (π₯ = π β ((π·βπΉ) < (π»βπ₯) β (π·βπΉ) < (π»βπ))) |
4 | fveqeq2 6855 | . . . 4 β’ (π₯ = π β ((πΉβπ₯) = 0 β (πΉβπ) = 0 )) | |
5 | 3, 4 | imbi12d 345 | . . 3 β’ (π₯ = π β (((π·βπΉ) < (π»βπ₯) β (πΉβπ₯) = 0 ) β ((π·βπΉ) < (π»βπ) β (πΉβπ) = 0 ))) |
6 | mdeglt.f | . . . . . . 7 β’ (π β πΉ β π΅) | |
7 | mdegval.d | . . . . . . . 8 β’ π· = (πΌ mDeg π ) | |
8 | mdegval.p | . . . . . . . 8 β’ π = (πΌ mPoly π ) | |
9 | mdegval.b | . . . . . . . 8 β’ π΅ = (Baseβπ) | |
10 | mdegval.z | . . . . . . . 8 β’ 0 = (0gβπ ) | |
11 | mdegval.a | . . . . . . . 8 β’ π΄ = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} | |
12 | mdegval.h | . . . . . . . 8 β’ π» = (β β π΄ β¦ (βfld Ξ£g β)) | |
13 | 7, 8, 9, 10, 11, 12 | mdegval 25451 | . . . . . . 7 β’ (πΉ β π΅ β (π·βπΉ) = sup((π» β (πΉ supp 0 )), β*, < )) |
14 | 6, 13 | syl 17 | . . . . . 6 β’ (π β (π·βπΉ) = sup((π» β (πΉ supp 0 )), β*, < )) |
15 | imassrn 6028 | . . . . . . . 8 β’ (π» β (πΉ supp 0 )) β ran π» | |
16 | 11, 12 | tdeglem1 25443 | . . . . . . . . . 10 β’ π»:π΄βΆβ0 |
17 | frn 6679 | . . . . . . . . . 10 β’ (π»:π΄βΆβ0 β ran π» β β0) | |
18 | 16, 17 | mp1i 13 | . . . . . . . . 9 β’ (π β ran π» β β0) |
19 | nn0ssre 12425 | . . . . . . . . . 10 β’ β0 β β | |
20 | ressxr 11207 | . . . . . . . . . 10 β’ β β β* | |
21 | 19, 20 | sstri 3957 | . . . . . . . . 9 β’ β0 β β* |
22 | 18, 21 | sstrdi 3960 | . . . . . . . 8 β’ (π β ran π» β β*) |
23 | 15, 22 | sstrid 3959 | . . . . . . 7 β’ (π β (π» β (πΉ supp 0 )) β β*) |
24 | supxrcl 13243 | . . . . . . 7 β’ ((π» β (πΉ supp 0 )) β β* β sup((π» β (πΉ supp 0 )), β*, < ) β β*) | |
25 | 23, 24 | syl 17 | . . . . . 6 β’ (π β sup((π» β (πΉ supp 0 )), β*, < ) β β*) |
26 | 14, 25 | eqeltrd 2834 | . . . . 5 β’ (π β (π·βπΉ) β β*) |
27 | 26 | xrleidd 13080 | . . . 4 β’ (π β (π·βπΉ) β€ (π·βπΉ)) |
28 | 7, 8, 9, 10, 11, 12 | mdegleb 25452 | . . . . 5 β’ ((πΉ β π΅ β§ (π·βπΉ) β β*) β ((π·βπΉ) β€ (π·βπΉ) β βπ₯ β π΄ ((π·βπΉ) < (π»βπ₯) β (πΉβπ₯) = 0 ))) |
29 | 6, 26, 28 | syl2anc 585 | . . . 4 β’ (π β ((π·βπΉ) β€ (π·βπΉ) β βπ₯ β π΄ ((π·βπΉ) < (π»βπ₯) β (πΉβπ₯) = 0 ))) |
30 | 27, 29 | mpbid 231 | . . 3 β’ (π β βπ₯ β π΄ ((π·βπΉ) < (π»βπ₯) β (πΉβπ₯) = 0 )) |
31 | medglt.x | . . 3 β’ (π β π β π΄) | |
32 | 5, 30, 31 | rspcdva 3584 | . 2 β’ (π β ((π·βπΉ) < (π»βπ) β (πΉβπ) = 0 )) |
33 | 1, 32 | mpd 15 | 1 β’ (π β (πΉβπ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 βwral 3061 {crab 3406 β wss 3914 class class class wbr 5109 β¦ cmpt 5192 β‘ccnv 5636 ran crn 5638 β cima 5640 βΆwf 6496 βcfv 6500 (class class class)co 7361 supp csupp 8096 βm cmap 8771 Fincfn 8889 supcsup 9384 βcr 11058 β*cxr 11196 < clt 11197 β€ cle 11198 βcn 12161 β0cn0 12421 Basecbs 17091 0gc0g 17329 Ξ£g cgsu 17330 βfldccnfld 20819 mPoly cmpl 21331 mDeg cmdg 25438 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-gsum 17332 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-cnfld 20820 df-psr 21334 df-mpl 21336 df-mdeg 25440 |
This theorem is referenced by: mdegaddle 25462 mdegvscale 25463 mdegmullem 25466 |
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