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Mirrors > Home > MPE Home > Th. List > mdegnn0cl | Structured version Visualization version GIF version |
Description: Degree of a nonzero polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
Ref | Expression |
---|---|
mdeg0.d | ⊢ 𝐷 = (𝐼 mDeg 𝑅) |
mdeg0.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mdeg0.z | ⊢ 0 = (0g‘𝑃) |
mdegnn0cl.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
mdegnn0cl | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdeg0.d | . . 3 ⊢ 𝐷 = (𝐼 mDeg 𝑅) | |
2 | mdeg0.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | mdegnn0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
4 | eqid 2736 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2736 | . . 3 ⊢ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
6 | eqid 2736 | . . 3 ⊢ (ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ)) = (ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ)) | |
7 | mdeg0.z | . . 3 ⊢ 0 = (0g‘𝑃) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mdegldg 25429 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ∃𝑥 ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ((𝐹‘𝑥) ≠ (0g‘𝑅) ∧ ((ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ))‘𝑥) = (𝐷‘𝐹))) |
9 | 5, 6 | tdeglem1 25418 | . . . . . . 7 ⊢ (ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ)):{𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}⟶ℕ0 |
10 | 9 | a1i 11 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ)):{𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}⟶ℕ0) |
11 | 10 | ffvelcdmda 7034 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) ∧ 𝑥 ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}) → ((ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ))‘𝑥) ∈ ℕ0) |
12 | eleq1 2825 | . . . . 5 ⊢ (((ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ))‘𝑥) = (𝐷‘𝐹) → (((ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ))‘𝑥) ∈ ℕ0 ↔ (𝐷‘𝐹) ∈ ℕ0)) | |
13 | 11, 12 | syl5ibcom 244 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) ∧ 𝑥 ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}) → (((ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ))‘𝑥) = (𝐷‘𝐹) → (𝐷‘𝐹) ∈ ℕ0)) |
14 | 13 | adantld 491 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) ∧ 𝑥 ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}) → (((𝐹‘𝑥) ≠ (0g‘𝑅) ∧ ((ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ))‘𝑥) = (𝐷‘𝐹)) → (𝐷‘𝐹) ∈ ℕ0)) |
15 | 14 | rexlimdva 3152 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (∃𝑥 ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ((𝐹‘𝑥) ≠ (0g‘𝑅) ∧ ((ℎ ∈ {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} ↦ (ℂfld Σg ℎ))‘𝑥) = (𝐷‘𝐹)) → (𝐷‘𝐹) ∈ ℕ0)) |
16 | 8, 15 | mpd 15 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 {crab 3407 ↦ cmpt 5188 ◡ccnv 5632 “ cima 5636 ⟶wf 6492 ‘cfv 6496 (class class class)co 7356 ↑m cmap 8764 Fincfn 8882 ℕcn 12152 ℕ0cn0 12412 Basecbs 17082 0gc0g 17320 Σg cgsu 17321 Ringcrg 19962 ℂfldccnfld 20794 mPoly cmpl 21306 mDeg cmdg 25413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-addf 11129 ax-mulf 11130 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-sup 9377 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-fzo 13567 df-seq 13906 df-hash 14230 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-starv 17147 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-hom 17156 df-cco 17157 df-0g 17322 df-gsum 17323 df-prds 17328 df-pws 17330 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-submnd 18601 df-grp 18750 df-minusg 18751 df-subg 18923 df-cntz 19095 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-ring 19964 df-cring 19965 df-cnfld 20795 df-psr 21309 df-mpl 21311 df-mdeg 25415 |
This theorem is referenced by: deg1nn0cl 25451 |
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