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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 2610 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | mdegvsca.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑌) | |
5 | eqid 2610 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | eqid 2610 | . . . . . . 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 19113 | . . . . . . . 8 ⊢ 𝐸 ⊆ (Base‘𝑅) |
9 | mdegvsca.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝐸) | |
10 | 8, 9 | sseldi 3566 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑅)) |
11 | mdegvsca.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
12 | 1, 2, 3, 4, 5, 6, 10, 11 | mplvsca 19268 | . . . . . 6 ⊢ (𝜑 → (𝐹 · 𝐺) = (({𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {𝐹}) ∘𝑓 (.r‘𝑅)𝐺)) |
13 | 12 | oveq1d 6564 | . . . . 5 ⊢ (𝜑 → ((𝐹 · 𝐺) supp (0g‘𝑅)) = ((({𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {𝐹}) ∘𝑓 (.r‘𝑅)𝐺) supp (0g‘𝑅))) |
14 | eqid 2610 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | ovex 6577 | . . . . . . . 8 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V | |
16 | 15 | rabex 4740 | . . . . . . 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 19254 | . . . . . 6 ⊢ (𝜑 → 𝐺:{𝑥 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
20 | 7, 3, 5, 14, 17, 18, 9, 19 | rrgsupp 19112 | . . . . 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 2644 | . . . 4 ⊢ (𝜑 → ((𝐹 · 𝐺) supp (0g‘𝑅)) = (𝐺 supp (0g‘𝑅))) |
22 | 21 | imaeq2d 5385 | . . 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 8235 | . 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 19272 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod) |
26 | 24, 18, 25 | syl2anc 691 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ LMod) |
27 | 1, 24, 18 | mplsca 19266 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑌)) |
28 | 27 | fveq2d 6107 | . . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑌))) |
29 | 10, 28 | eleqtrd 2690 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Scalar‘𝑌))) |
30 | eqid 2610 | . . . . 5 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌) | |
31 | eqid 2610 | . . . . 5 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌)) | |
32 | 4, 30, 2, 31 | lmodvscl 18703 | . . . 4 ⊢ ((𝑌 ∈ LMod ∧ 𝐹 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝐺 ∈ 𝐵) → (𝐹 · 𝐺) ∈ 𝐵) |
33 | 26, 29, 11, 32 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) |
34 | mdegaddle.d | . . . 4 ⊢ 𝐷 = (𝐼 mDeg 𝑅) | |
35 | eqid 2610 | . . . 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 23627 | . . 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 23627 | . . 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 2654 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 · 𝐺)) = (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 {csn 4125 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 supp csupp 7182 ↑𝑚 cmap 7744 Fincfn 7841 supcsup 8229 ℝ*cxr 9952 < clt 9953 ℕcn 10897 ℕ0cn0 11169 Basecbs 15695 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 Σg cgsu 15924 Ringcrg 18370 LModclmod 18686 RLRegcrlreg 19100 mPoly cmpl 19174 ℂfldccnfld 19567 mDeg cmdg 23617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lss 18754 df-rlreg 19104 df-psr 19177 df-mpl 19179 df-mdeg 23619 |
This theorem is referenced by: deg1vsca 23669 |
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