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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 6906 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐻‘𝑥) = (𝐻‘𝑋)) | |
3 | 2 | breq2d 5159 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝐹) < (𝐻‘𝑥) ↔ (𝐷‘𝐹) < (𝐻‘𝑋))) |
4 | fveqeq2 6915 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 0 ↔ (𝐹‘𝑋) = 0 )) | |
5 | 3, 4 | imbi12d 344 | . . 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 26116 | . . . . . . 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 6090 | . . . . . . . 8 ⊢ (𝐻 “ (𝐹 supp 0 )) ⊆ ran 𝐻 | |
16 | 11, 12 | tdeglem1 26111 | . . . . . . . . . 10 ⊢ 𝐻:𝐴⟶ℕ0 |
17 | frn 6743 | . . . . . . . . . 10 ⊢ (𝐻:𝐴⟶ℕ0 → ran 𝐻 ⊆ ℕ0) | |
18 | 16, 17 | mp1i 13 | . . . . . . . . 9 ⊢ (𝜑 → ran 𝐻 ⊆ ℕ0) |
19 | nn0ssre 12527 | . . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ | |
20 | ressxr 11302 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
21 | 19, 20 | sstri 4004 | . . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ* |
22 | 18, 21 | sstrdi 4007 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐻 ⊆ ℝ*) |
23 | 15, 22 | sstrid 4006 | . . . . . . 7 ⊢ (𝜑 → (𝐻 “ (𝐹 supp 0 )) ⊆ ℝ*) |
24 | supxrcl 13353 | . . . . . . 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 2838 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
27 | 26 | xrleidd 13190 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
28 | 7, 8, 9, 10, 11, 12 | mdegleb 26117 | . . . . 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 232 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ((𝐷‘𝐹) < (𝐻‘𝑥) → (𝐹‘𝑥) = 0 )) |
31 | medglt.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
32 | 5, 30, 31 | rspcdva 3622 | . 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 206 = wceq 1536 ∈ wcel 2105 ∀wral 3058 {crab 3432 ⊆ wss 3962 class class class wbr 5147 ↦ cmpt 5230 ◡ccnv 5687 ran crn 5689 “ cima 5691 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supp csupp 8183 ↑m cmap 8864 Fincfn 8983 supcsup 9477 ℝcr 11151 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 ℕcn 12263 ℕ0cn0 12523 Basecbs 17244 0gc0g 17485 Σg cgsu 17486 ℂfldccnfld 21381 mPoly cmpl 21943 mDeg cmdg 26106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-gsum 17488 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-ur 20199 df-ring 20252 df-cring 20253 df-cnfld 21382 df-psr 21946 df-mpl 21948 df-mdeg 26108 |
This theorem is referenced by: mdegaddle 26127 mdegvscale 26128 mdegmullem 26131 |
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