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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 | ⊢ 𝐷 = (𝐼 mDeg 𝑅) |
mdegval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mdegval.b | ⊢ 𝐵 = (Base‘𝑃) |
mdegval.z | ⊢ 0 = (0g‘𝑅) |
mdegval.a | ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} |
mdegval.h | ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) |
mdeglt.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
medglt.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
mdeglt.lt | ⊢ (𝜑 → (𝐷‘𝐹) < (𝐻‘𝑋)) |
Ref | Expression |
---|---|
mdeglt | ⊢ (𝜑 → (𝐹‘𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdeglt.lt | . 2 ⊢ (𝜑 → (𝐷‘𝐹) < (𝐻‘𝑋)) | |
2 | fveq2 6774 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐻‘𝑥) = (𝐻‘𝑋)) | |
3 | 2 | breq2d 5086 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝐹) < (𝐻‘𝑥) ↔ (𝐷‘𝐹) < (𝐻‘𝑋))) |
4 | fveqeq2 6783 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 0 ↔ (𝐹‘𝑋) = 0 )) | |
5 | 3, 4 | imbi12d 345 | . . 3 ⊢ (𝑥 = 𝑋 → (((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ) ↔ ((𝐷‘𝐹) < (𝐻‘𝑋) → (𝐹‘𝑋) = 0 ))) |
6 | mdeglt.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
7 | mdegval.d | . . . . . . . 8 ⊢ 𝐷 = (𝐼 mDeg 𝑅) | |
8 | mdegval.p | . . . . . . . 8 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
9 | mdegval.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑃) | |
10 | mdegval.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
11 | mdegval.a | . . . . . . . 8 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
12 | mdegval.h | . . . . . . . 8 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) | |
13 | 7, 8, 9, 10, 11, 12 | mdegval 25228 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < )) |
14 | 6, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < )) |
15 | imassrn 5980 | . . . . . . . 8 ⊢ (𝐻 “ (𝐹 supp 0 )) ⊆ ran 𝐻 | |
16 | 11, 12 | tdeglem1 25220 | . . . . . . . . . 10 ⊢ 𝐻:𝐴⟶ℕ0 |
17 | frn 6607 | . . . . . . . . . 10 ⊢ (𝐻:𝐴⟶ℕ0 → ran 𝐻 ⊆ ℕ0) | |
18 | 16, 17 | mp1i 13 | . . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ ℕ0) |
19 | nn0ssre 12237 | . . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ | |
20 | ressxr 11019 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
21 | 19, 20 | sstri 3930 | . . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ* |
22 | 18, 21 | sstrdi 3933 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐻 ⊆ ℝ*) |
23 | 15, 22 | sstrid 3932 | . . . . . . 7 ⊢ (𝜑 → (𝐻 “ (𝐹 supp 0 )) ⊆ ℝ*) |
24 | supxrcl 13049 | . . . . . . 7 ⊢ ((𝐻 “ (𝐹 supp 0 )) ⊆ ℝ* → sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < ) ∈ ℝ*) | |
25 | 23, 24 | syl 17 | . . . . . 6 ⊢ (𝜑 → sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < ) ∈ ℝ*) |
26 | 14, 25 | eqeltrd 2839 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
27 | 26 | xrleidd 12886 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
28 | 7, 8, 9, 10, 11, 12 | mdegleb 25229 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ (𝐷‘𝐹) ∈ ℝ*) → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))) |
29 | 6, 26, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))) |
30 | 27, 29 | mpbid 231 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 )) |
31 | medglt.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
32 | 5, 30, 31 | rspcdva 3562 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) < (𝐻‘𝑋) → (𝐹‘𝑋) = 0 )) |
33 | 1, 32 | mpd 15 | 1 ⊢ (𝜑 → (𝐹‘𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 ◡ccnv 5588 ran crn 5590 “ cima 5592 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 supp csupp 7977 ↑m cmap 8615 Fincfn 8733 supcsup 9199 ℝcr 10870 ℝ*cxr 11008 < clt 11009 ≤ cle 11010 ℕcn 11973 ℕ0cn0 12233 Basecbs 16912 0gc0g 17150 Σg cgsu 17151 ℂfldccnfld 20597 mPoly cmpl 21109 mDeg cmdg 25215 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-gsum 17153 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-cnfld 20598 df-psr 21112 df-mpl 21114 df-mdeg 25217 |
This theorem is referenced by: mdegaddle 25239 mdegvscale 25240 mdegmullem 25243 |
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