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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.) Remove closure hypotheses. (Revised by SN, 4-Sep-2025.) |
| Ref | Expression |
|---|---|
| mhpvscacl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
| mhpvscacl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
| mhpvscacl.t | ⊢ · = ( ·𝑠 ‘𝑃) |
| mhpvscacl.k | ⊢ 𝐾 = (Base‘𝑅) |
| mhpvscacl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
| 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 2737 | . 2 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 4 | eqid 2737 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 5 | eqid 2737 | . 2 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
| 6 | mhpvscacl.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁)) | |
| 7 | 1, 6 | mhprcl 22147 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
| 8 | eqid 2737 | . . 3 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
| 9 | mhpvscacl.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑃) | |
| 10 | eqid 2737 | . . 3 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃)) | |
| 11 | reldmmhp 22141 | . . . . 5 ⊢ Rel dom mHomP | |
| 12 | 11, 1, 6 | elfvov1 7473 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) |
| 13 | mhpvscacl.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
| 14 | 2, 12, 13 | mpllmodd 22044 | . . 3 ⊢ (𝜑 → 𝑃 ∈ LMod) |
| 15 | mhpvscacl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
| 16 | mhpvscacl.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
| 17 | 15, 16 | eleqtrdi 2851 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅)) |
| 18 | 2, 12, 13 | mplsca 22033 | . . . . 5 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
| 19 | 18 | fveq2d 6910 | . . . 4 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
| 20 | 17, 19 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑃))) |
| 21 | 1, 2, 3, 6 | mhpmpl 22148 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑃)) |
| 22 | 3, 8, 9, 10, 14, 20, 21 | lmodvscld 20877 | . 2 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ (Base‘𝑃)) |
| 23 | 2, 16, 3, 5, 22 | mplelf 22018 | . . . 4 ⊢ (𝜑 → (𝑋 · 𝐹):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾) |
| 24 | eqid 2737 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 25 | 15 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → 𝑋 ∈ 𝐾) |
| 26 | 21 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → 𝐹 ∈ (Base‘𝑃)) |
| 27 | eldifi 4131 | . . . . . . 7 ⊢ (𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅))) → 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) | |
| 28 | 27 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → 𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) |
| 29 | 2, 9, 16, 3, 24, 5, 25, 26, 28 | mplvscaval 22036 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → ((𝑋 · 𝐹)‘𝑘) = (𝑋(.r‘𝑅)(𝐹‘𝑘))) |
| 30 | 2, 16, 3, 5, 21 | mplelf 22018 | . . . . . . 7 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶𝐾) |
| 31 | ssidd 4007 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0g‘𝑅)) ⊆ (𝐹 supp (0g‘𝑅))) | |
| 32 | fvexd 6921 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
| 33 | 30, 31, 6, 32 | suppssrg 8221 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → (𝐹‘𝑘) = (0g‘𝑅)) |
| 34 | 33 | oveq2d 7447 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → (𝑋(.r‘𝑅)(𝐹‘𝑘)) = (𝑋(.r‘𝑅)(0g‘𝑅))) |
| 35 | 16, 24, 4, 13, 15 | ringrzd 20293 | . . . . . 6 ⊢ (𝜑 → (𝑋(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
| 36 | 35 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → (𝑋(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅)) |
| 37 | 29, 34, 36 | 3eqtrd 2781 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ (𝐹 supp (0g‘𝑅)))) → ((𝑋 · 𝐹)‘𝑘) = (0g‘𝑅)) |
| 38 | 23, 37 | suppss 8219 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝐹) supp (0g‘𝑅)) ⊆ (𝐹 supp (0g‘𝑅))) |
| 39 | 1, 4, 5, 6 | mhpdeg 22149 | . . 3 ⊢ (𝜑 → (𝐹 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
| 40 | 38, 39 | sstrd 3994 | . 2 ⊢ (𝜑 → ((𝑋 · 𝐹) supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
| 41 | 1, 2, 3, 4, 5, 7, 22, 40 | ismhp2 22145 | 1 ⊢ (𝜑 → (𝑋 · 𝐹) ∈ (𝐻‘𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∖ cdif 3948 ◡ccnv 5684 “ cima 5688 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 ↑m cmap 8866 Fincfn 8985 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 Σg cgsu 17485 Ringcrg 20230 ℂfldccnfld 21364 mPoly cmpl 21926 mHomP cmhp 22133 |
| 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 |
| 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-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-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-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-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 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-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-lmod 20860 df-lss 20930 df-psr 21929 df-mpl 21931 df-mhp 22140 |
| This theorem is referenced by: mhplss 22159 |
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