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Mirrors > Home > MPE Home > Th. List > mhpvscacl | Structured version Visualization version GIF version |
Description: Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023.) |
Ref | Expression |
---|---|
mhpvscacl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpvscacl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhpvscacl.t | ⊢ · = ( ·𝑠 ‘𝑃) |
mhpvscacl.k | ⊢ 𝐾 = (Base‘𝑅) |
mhpvscacl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mhpvscacl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mhpvscacl.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
mhpvscacl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
mhpvscacl.f | ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁)) |
Ref | Expression |
---|---|
mhpvscacl | ⊢ (𝜑 → (𝑋 · 𝐹) ∈ (𝐻‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhpvscacl.h | . 2 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
2 | mhpvscacl.p | . 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | eqid 2738 | . 2 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | eqid 2738 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2738 | . 2 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | mhpvscacl.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
7 | mhpvscacl.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | mhpvscacl.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
9 | 2 | mpllmod 21003 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod) |
10 | 6, 7, 9 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝑃 ∈ LMod) |
11 | mhpvscacl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
12 | mhpvscacl.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
13 | 11, 12 | eleqtrdi 2849 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) |
14 | 2, 6, 7 | mplsca 20997 | . . . . 5 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
15 | 14 | fveq2d 6740 | . . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
16 | 13, 15 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑃))) |
17 | mhpvscacl.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁)) | |
18 | 1, 2, 3, 6, 7, 8, 17 | mhpmpl 21108 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑃)) |
19 | eqid 2738 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
20 | mhpvscacl.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑃) | |
21 | eqid 2738 | . . . 4 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
22 | 3, 19, 20, 21 | lmodvscl 19941 | . . 3 ⊢ ((𝑃 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝐹 ∈ (Base‘𝑃)) → (𝑋 · 𝐹) ∈ (Base‘𝑃)) |
23 | 10, 16, 18, 22 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ (Base‘𝑃)) |
24 | 2, 12, 3, 5, 23 | mplelf 20984 | . . . 4 ⊢ (𝜑 → (𝑋 · 𝐹):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾) |
25 | eqid 2738 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
26 | 11 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → 𝑋 ∈ 𝐾) |
27 | 18 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → 𝐹 ∈ (Base‘𝑃)) |
28 | eldifi 4056 | . . . . . . 7 ⊢ (𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅))) → 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) | |
29 | 28 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) |
30 | 2, 20, 12, 3, 25, 5, 26, 27, 29 | mplvscaval 21000 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → ((𝑋 · 𝐹)‘𝑘) = (𝑋(.r‘𝑅)(𝐹‘𝑘))) |
31 | 2, 12, 3, 5, 18 | mplelf 20984 | . . . . . . 7 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾) |
32 | ssidd 3939 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0g‘𝑅)) ⊆ (𝐹 supp (0g‘𝑅))) | |
33 | ovexd 7267 | . . . . . . . 8 ⊢ (𝜑 → (ℕ0 ↑m 𝐼) ∈ V) | |
34 | 5, 33 | rabexd 5241 | . . . . . . 7 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V) |
35 | fvexd 6751 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
36 | 31, 32, 34, 35 | suppssr 7959 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → (𝐹‘𝑘) = (0g‘𝑅)) |
37 | 36 | oveq2d 7248 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → (𝑋(.r‘𝑅)(𝐹‘𝑘)) = (𝑋(.r‘𝑅)(0g‘𝑅))) |
38 | 12, 25, 4 | ringrz 19631 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) → (𝑋(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
39 | 7, 11, 38 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (𝑋(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
40 | 39 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → (𝑋(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
41 | 30, 37, 40 | 3eqtrd 2782 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → ((𝑋 · 𝐹)‘𝑘) = (0g‘𝑅)) |
42 | 24, 41 | suppss 7957 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝐹) supp (0g‘𝑅)) ⊆ (𝐹 supp (0g‘𝑅))) |
43 | 1, 4, 5, 6, 7, 8, 17 | mhpdeg 21109 | . . 3 ⊢ (𝜑 → (𝐹 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
44 | 42, 43 | sstrd 3926 | . 2 ⊢ (𝜑 → ((𝑋 · 𝐹) supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
45 | 1, 2, 3, 4, 5, 6, 7, 8, 23, 44 | ismhp2 21106 | 1 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ (𝐻‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 {crab 3066 Vcvv 3421 ∖ cdif 3878 ◡ccnv 5565 “ cima 5569 ‘cfv 6398 (class class class)co 7232 supp csupp 7924 ↑m cmap 8529 Fincfn 8647 ℕcn 11855 ℕ0cn0 12115 Basecbs 16785 ↾s cress 16809 .rcmulr 16828 Scalarcsca 16830 ·𝑠 cvsca 16831 0gc0g 16969 Σg cgsu 16970 Ringcrg 19587 LModclmod 19924 ℂfldccnfld 20388 mPoly cmpl 20889 mHomP cmhp 21093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-of 7488 df-om 7664 df-1st 7780 df-2nd 7781 df-supp 7925 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-er 8412 df-map 8531 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-fsupp 9011 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-nn 11856 df-2 11918 df-3 11919 df-4 11920 df-5 11921 df-6 11922 df-7 11923 df-8 11924 df-9 11925 df-n0 12116 df-z 12202 df-uz 12464 df-fz 13121 df-struct 16725 df-sets 16742 df-slot 16760 df-ndx 16770 df-base 16786 df-ress 16810 df-plusg 16840 df-mulr 16841 df-sca 16843 df-vsca 16844 df-tset 16846 df-0g 16971 df-mgm 18139 df-sgrp 18188 df-mnd 18199 df-grp 18393 df-minusg 18394 df-sbg 18395 df-subg 18565 df-mgp 19530 df-ur 19542 df-ring 19589 df-lmod 19926 df-lss 19994 df-psr 20892 df-mpl 20894 df-mhp 21097 |
This theorem is referenced by: mhplss 21119 |
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