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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 2821 | . 2 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | eqid 2821 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | eqid 2821 | . 2 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
5 | mplsubg.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
6 | 0fin 8740 | . . 3 ⊢ ∅ ∈ Fin | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ∈ Fin) |
8 | unfi 8779 | . . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝑥 ∪ 𝑦) ∈ Fin) | |
9 | 8 | adantl 484 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Fin ∧ 𝑦 ∈ Fin)) → (𝑥 ∪ 𝑦) ∈ Fin) |
10 | ssfi 8732 | . . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) | |
11 | 10 | adantl 484 | . 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 20210 | . 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 20209 | 1 ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 ◡ccnv 5548 “ cima 5552 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 ℕcn 11632 ℕ0cn0 11891 Basecbs 16477 0gc0g 16707 Ringcrg 19291 LSubSpclss 19697 mPwSer cmps 20125 mPoly cmpl 20127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-tset 16578 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-subg 18270 df-mgp 19234 df-ring 19293 df-lss 19698 df-psr 20130 df-mpl 20132 |
This theorem is referenced by: mpllmod 20225 mplassa 20229 mplbas2 20245 mplind 20276 ply1lss 20358 |
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