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Mirrors > Home > MPE Home > Th. List > mpllss | Structured version Visualization version GIF version |
Description: The set of polynomials is closed under scalar multiplication, i.e. it is a linear subspace of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.) |
Ref | Expression |
---|---|
mplsubg.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
mplsubg.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplsubg.u | ⊢ 𝑈 = (Base‘𝑃) |
mplsubg.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mpllss.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
Ref | Expression |
---|---|
mpllss | ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplsubg.s | . 2 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
2 | eqid 2760 | . 2 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | eqid 2760 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | eqid 2760 | . 2 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
5 | mplsubg.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
6 | 0fin 8353 | . . 3 ⊢ ∅ ∈ Fin | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ∈ Fin) |
8 | unfi 8392 | . . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin) | |
9 | 8 | adantl 473 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → (𝑥 ∪ 𝑦) ∈ Fin) |
10 | ssfi 8345 | . . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) | |
11 | 10 | adantl 473 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ Fin) |
12 | mplsubg.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
13 | mplsubg.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
14 | 1, 12, 13, 5 | mplsubglem2 19638 | . 2 ⊢ (𝜑 → 𝑈 = {𝑔 ∈ (Base‘𝑆) ∣ (𝑔 supp (0g‘𝑅)) ∈ Fin}) |
15 | mpllss.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
16 | 1, 2, 3, 4, 5, 7, 9, 11, 14, 15 | mpllsslem 19637 | 1 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 {crab 3054 ∪ cun 3713 ⊆ wss 3715 ∅c0 4058 ◡ccnv 5265 “ cima 5269 ‘cfv 6049 (class class class)co 6813 ↑𝑚 cmap 8023 Fincfn 8121 ℕcn 11212 ℕ0cn0 11484 Basecbs 16059 0gc0g 16302 Ringcrg 18747 LSubSpclss 19134 mPwSer cmps 19553 mPoly cmpl 19555 |
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-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-subg 17792 df-mgp 18690 df-ring 18749 df-lss 19135 df-psr 19558 df-mpl 19560 |
This theorem is referenced by: mpllmod 19653 mplassa 19656 mplbas2 19672 mplind 19704 ply1lss 19768 |
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