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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 6434 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐻‘𝑥) = (𝐻‘𝑋)) | |
3 | 2 | breq2d 4886 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝐹) < (𝐻‘𝑥) ↔ (𝐷‘𝐹) < (𝐻‘𝑋))) |
4 | fveqeq2 6443 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 0 ↔ (𝐹‘𝑋) = 0 )) | |
5 | 3, 4 | imbi12d 336 | . . 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 24223 | . . . . . . 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 5719 | . . . . . . . 8 ⊢ (𝐻 “ (𝐹 supp 0 )) ⊆ ran 𝐻 | |
16 | 8, 9 | mplrcl 19851 | . . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V) |
17 | 11, 12 | tdeglem1 24218 | . . . . . . . . . 10 ⊢ (𝐼 ∈ V → 𝐻:𝐴⟶ℕ0) |
18 | frn 6285 | . . . . . . . . . 10 ⊢ (𝐻:𝐴⟶ℕ0 → ran 𝐻 ⊆ ℕ0) | |
19 | 6, 16, 17, 18 | 4syl 19 | . . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ ℕ0) |
20 | nn0ssre 11623 | . . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ | |
21 | ressxr 10401 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
22 | 20, 21 | sstri 3837 | . . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ* |
23 | 19, 22 | syl6ss 3840 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐻 ⊆ ℝ*) |
24 | 15, 23 | syl5ss 3839 | . . . . . . 7 ⊢ (𝜑 → (𝐻 “ (𝐹 supp 0 )) ⊆ ℝ*) |
25 | supxrcl 12434 | . . . . . . 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 2907 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
28 | 27 | xrleidd 12272 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
29 | 7, 8, 9, 10, 11, 12 | mdegleb 24224 | . . . . 5 ⊢ ((𝐹 ∈ 𝐵 ∧ (𝐷‘𝐹) ∈ ℝ*) → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))) |
30 | 6, 27, 29 | syl2anc 581 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐹) ≤ (𝐷‘𝐹) ↔ ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 ))) |
31 | 28, 30 | mpbid 224 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 )) |
32 | medglt.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
33 | 5, 31, 32 | rspcdva 3533 | . 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 198 = wceq 1658 ∈ wcel 2166 ∀wral 3118 {crab 3122 Vcvv 3415 ⊆ wss 3799 class class class wbr 4874 ↦ cmpt 4953 ◡ccnv 5342 ran crn 5344 “ cima 5346 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 supp csupp 7560 ↑𝑚 cmap 8123 Fincfn 8223 supcsup 8616 ℝcr 10252 ℝ*cxr 10391 < clt 10392 ≤ cle 10393 ℕcn 11351 ℕ0cn0 11619 Basecbs 16223 0gc0g 16454 Σg cgsu 16455 mPoly cmpl 19715 ℂfldccnfld 20107 mDeg cmdg 24213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-1st 7429 df-2nd 7430 df-supp 7561 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-fsupp 8546 df-sup 8618 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-fzo 12762 df-seq 13097 df-hash 13412 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-starv 16321 df-sca 16322 df-vsca 16323 df-tset 16325 df-ple 16326 df-ds 16328 df-unif 16329 df-0g 16456 df-gsum 16457 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-grp 17780 df-minusg 17781 df-cntz 18101 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-cring 18905 df-psr 19718 df-mpl 19720 df-cnfld 20108 df-mdeg 24215 |
This theorem is referenced by: mdegaddle 24234 mdegvscale 24235 mdegmullem 24238 |
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