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Mirrors > Home > MPE Home > Th. List > mdegvsca | Structured version Visualization version GIF version |
Description: The degree of a scalar multiple of a polynomial is exactly the degree of the original polynomial when the multiple is a nonzero-divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
Ref | Expression |
---|---|
mdegaddle.y | ⊢ 𝑌 = (𝐼 mPoly 𝑅) |
mdegaddle.d | ⊢ 𝐷 = (𝐼 mDeg 𝑅) |
mdegaddle.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mdegaddle.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mdegvsca.b | ⊢ 𝐵 = (Base‘𝑌) |
mdegvsca.e | ⊢ 𝐸 = (RLReg‘𝑅) |
mdegvsca.p | ⊢ · = ( ·𝑠 ‘𝑌) |
mdegvsca.f | ⊢ (𝜑 → 𝐹 ∈ 𝐸) |
mdegvsca.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
mdegvsca | ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdegaddle.y | . . . . . . 7 ⊢ 𝑌 = (𝐼 mPoly 𝑅) | |
2 | mdegvsca.p | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑌) | |
3 | eqid 2760 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | mdegvsca.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑌) | |
5 | eqid 2760 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | eqid 2760 | . . . . . . 7 ⊢ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} | |
7 | mdegvsca.e | . . . . . . . . 9 ⊢ 𝐸 = (RLReg‘𝑅) | |
8 | 7, 3 | rrgss 19494 | . . . . . . . 8 ⊢ 𝐸 ⊆ (Base‘𝑅) |
9 | mdegvsca.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝐸) | |
10 | 8, 9 | sseldi 3742 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑅)) |
11 | mdegvsca.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
12 | 1, 2, 3, 4, 5, 6, 10, 11 | mplvsca 19649 | . . . . . 6 ⊢ (𝜑 → (𝐹 · 𝐺) = (({𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {𝐹}) ∘𝑓 (.r‘𝑅)𝐺)) |
13 | 12 | oveq1d 6828 | . . . . 5 ⊢ (𝜑 → ((𝐹 · 𝐺) supp (0g‘𝑅)) = ((({𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {𝐹}) ∘𝑓 (.r‘𝑅)𝐺) supp (0g‘𝑅))) |
14 | eqid 2760 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | ovex 6841 | . . . . . . . 8 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
16 | 15 | rabex 4964 | . . . . . . 7 ⊢ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ∈ V |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ∈ V) |
18 | mdegaddle.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
19 | 1, 3, 4, 6, 11 | mplelf 19635 | . . . . . 6 ⊢ (𝜑 → 𝐺:{𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
20 | 7, 3, 5, 14, 17, 18, 9, 19 | rrgsupp 19493 | . . . . 5 ⊢ (𝜑 → ((({𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {𝐹}) ∘𝑓 (.r‘𝑅)𝐺) supp (0g‘𝑅)) = (𝐺 supp (0g‘𝑅))) |
21 | 13, 20 | eqtrd 2794 | . . . 4 ⊢ (𝜑 → ((𝐹 · 𝐺) supp (0g‘𝑅)) = (𝐺 supp (0g‘𝑅))) |
22 | 21 | imaeq2d 5624 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ((𝐹 · 𝐺) supp (0g‘𝑅))) = ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ (𝐺 supp (0g‘𝑅)))) |
23 | 22 | supeq1d 8517 | . 2 ⊢ (𝜑 → sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ((𝐹 · 𝐺) supp (0g‘𝑅))), ℝ*, < ) = sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ (𝐺 supp (0g‘𝑅))), ℝ*, < )) |
24 | mdegaddle.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
25 | 1 | mpllmod 19653 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod) |
26 | 24, 18, 25 | syl2anc 696 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ LMod) |
27 | 1, 24, 18 | mplsca 19647 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑌)) |
28 | 27 | fveq2d 6356 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑌))) |
29 | 10, 28 | eleqtrd 2841 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Scalar‘𝑌))) |
30 | eqid 2760 | . . . . 5 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌) | |
31 | eqid 2760 | . . . . 5 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) | |
32 | 4, 30, 2, 31 | lmodvscl 19082 | . . . 4 ⊢ ((𝑌 ∈ LMod ∧ 𝐹 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝐺 ∈ 𝐵) → (𝐹 · 𝐺) ∈ 𝐵) |
33 | 26, 29, 11, 32 | syl3anc 1477 | . . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) |
34 | mdegaddle.d | . . . 4 ⊢ 𝐷 = (𝐼 mDeg 𝑅) | |
35 | eqid 2760 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) | |
36 | 34, 1, 4, 14, 6, 35 | mdegval 24022 | . . 3 ⊢ ((𝐹 · 𝐺) ∈ 𝐵 → (𝐷‘(𝐹 · 𝐺)) = sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ((𝐹 · 𝐺) supp (0g‘𝑅))), ℝ*, < )) |
37 | 33, 36 | syl 17 | . 2 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ((𝐹 · 𝐺) supp (0g‘𝑅))), ℝ*, < )) |
38 | 34, 1, 4, 14, 6, 35 | mdegval 24022 | . . 3 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) = sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ (𝐺 supp (0g‘𝑅))), ℝ*, < )) |
39 | 11, 38 | syl 17 | . 2 ⊢ (𝜑 → (𝐷‘𝐺) = sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ (𝐺 supp (0g‘𝑅))), ℝ*, < )) |
40 | 23, 37, 39 | 3eqtr4d 2804 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 {crab 3054 Vcvv 3340 {csn 4321 ↦ cmpt 4881 × cxp 5264 ◡ccnv 5265 “ cima 5269 ‘cfv 6049 (class class class)co 6813 ∘𝑓 cof 7060 supp csupp 7463 ↑𝑚 cmap 8023 Fincfn 8121 supcsup 8511 ℝ*cxr 10265 < clt 10266 ℕcn 11212 ℕ0cn0 11484 Basecbs 16059 .rcmulr 16144 Scalarcsca 16146 ·𝑠 cvsca 16147 0gc0g 16302 Σg cgsu 16303 Ringcrg 18747 LModclmod 19065 RLRegcrlreg 19481 mPoly cmpl 19555 ℂfldccnfld 19948 mDeg cmdg 24012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-sup 8513 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-tset 16162 df-0g 16304 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-grp 17626 df-minusg 17627 df-sbg 17628 df-subg 17792 df-mgp 18690 df-ur 18702 df-ring 18749 df-lmod 19067 df-lss 19135 df-rlreg 19485 df-psr 19558 df-mpl 19560 df-mdeg 24014 |
This theorem is referenced by: deg1vsca 24064 |
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