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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 6664 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐻‘𝑥) = (𝐻‘𝑋)) | |
3 | 2 | breq2d 5070 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝐹) < (𝐻‘𝑥) ↔ (𝐷‘𝐹) < (𝐻‘𝑋))) |
4 | fveqeq2 6673 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 0 ↔ (𝐹‘𝑋) = 0 )) | |
5 | 3, 4 | imbi12d 347 | . . 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 24651 | . . . . . . 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 5934 | . . . . . . . 8 ⊢ (𝐻 “ (𝐹 supp 0 )) ⊆ ran 𝐻 | |
16 | 8, 9 | mplrcl 20264 | . . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V) |
17 | 11, 12 | tdeglem1 24646 | . . . . . . . . . 10 ⊢ (𝐼 ∈ V → 𝐻:𝐴⟶ℕ0) |
18 | frn 6514 | . . . . . . . . . 10 ⊢ (𝐻:𝐴⟶ℕ0 → ran 𝐻 ⊆ ℕ0) | |
19 | 6, 16, 17, 18 | 4syl 19 | . . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ ℕ0) |
20 | nn0ssre 11895 | . . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ | |
21 | ressxr 10679 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
22 | 20, 21 | sstri 3975 | . . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ* |
23 | 19, 22 | sstrdi 3978 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐻 ⊆ ℝ*) |
24 | 15, 23 | sstrid 3977 | . . . . . . 7 ⊢ (𝜑 → (𝐻 “ (𝐹 supp 0 )) ⊆ ℝ*) |
25 | supxrcl 12702 | . . . . . . 7 ⊢ ((𝐻 “ (𝐹 supp 0 )) ⊆ ℝ* → sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < ) ∈ ℝ*) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝜑 → sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < ) ∈ ℝ*) |
27 | 14, 26 | eqeltrd 2913 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
28 | 27 | xrleidd 12539 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
29 | 7, 8, 9, 10, 11, 12 | mdegleb 24652 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ (𝐷‘𝐹) ∈ ℝ*) → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))) |
30 | 6, 27, 29 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))) |
31 | 28, 30 | mpbid 234 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 )) |
32 | medglt.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
33 | 5, 31, 32 | rspcdva 3624 | . 2 ⊢ (𝜑 → ((𝐷‘𝐹) < (𝐻‘𝑋) → (𝐹‘𝑋) = 0 )) |
34 | 1, 33 | mpd 15 | 1 ⊢ (𝜑 → (𝐹‘𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 Vcvv 3494 ⊆ wss 3935 class class class wbr 5058 ↦ cmpt 5138 ◡ccnv 5548 ran crn 5550 “ cima 5552 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 supp csupp 7824 ↑m cmap 8400 Fincfn 8503 supcsup 8898 ℝcr 10530 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 ℕcn 11632 ℕ0cn0 11891 Basecbs 16477 0gc0g 16707 Σg cgsu 16708 mPoly cmpl 20127 ℂfldccnfld 20539 mDeg cmdg 24641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-gsum 16710 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-psr 20130 df-mpl 20132 df-cnfld 20540 df-mdeg 24643 |
This theorem is referenced by: mdegaddle 24662 mdegvscale 24663 mdegmullem 24666 |
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