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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 5155 | . . . 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 26102 | . . . . . . 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 6089 | . . . . . . . 8 ⊢ (𝐻 “ (𝐹 supp 0 )) ⊆ ran 𝐻 | |
| 16 | 11, 12 | tdeglem1 26097 | . . . . . . . . . 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 12530 | . . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ | |
| 20 | ressxr 11305 | . . . . . . . . . 10 ⊢ ℝ ⊆ ℝ* | |
| 21 | 19, 20 | sstri 3993 | . . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ* |
| 22 | 18, 21 | sstrdi 3996 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐻 ⊆ ℝ*) |
| 23 | 15, 22 | sstrid 3995 | . . . . . . 7 ⊢ (𝜑 → (𝐻 “ (𝐹 supp 0 )) ⊆ ℝ*) |
| 24 | supxrcl 13357 | . . . . . . 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 2841 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ*) |
| 27 | 26 | xrleidd 13194 | . . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
| 28 | 7, 8, 9, 10, 11, 12 | mdegleb 26103 | . . . . 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 3623 | . 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 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 ran crn 5686 “ cima 5688 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 ↑m cmap 8866 Fincfn 8985 supcsup 9480 ℝcr 11154 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 ℂfldccnfld 21364 mPoly cmpl 21926 mDeg cmdg 26092 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-ur 20179 df-ring 20232 df-cring 20233 df-cnfld 21365 df-psr 21929 df-mpl 21931 df-mdeg 26094 |
| This theorem is referenced by: mdegaddle 26113 mdegvscale 26114 mdegmullem 26117 |
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