ply1lmod - Metamath Proof Explorer

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Theorem ply1lmod 22136

Description: Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.)

**Hypothesis**
Ref	Expression
ply1lmod.p	⊢ 𝑃 = (Poly₁‘𝑅)

**Assertion**
Ref	Expression
ply1lmod	⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)

**Proof of Theorem ply1lmod**
Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
2	1	psr1lmod 22133	. 2 ⊢ (𝑅 ∈ Ring → (PwSer₁‘𝑅) ∈ LMod)
3		eqid 2729	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2729	. . . 4 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
5	3, 4	ply1bas 22079	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(1_o mPoly 𝑅))
6	3, 1, 4	ply1lss 22081	. . 3 ⊢ (𝑅 ∈ Ring → (Base‘(Poly₁‘𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅)))
7	5, 6	eqeltrrid 2833	. 2 ⊢ (𝑅 ∈ Ring → (Base‘(1_o mPoly 𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅)))
8		ply1lmod.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
9	8, 1	ply1val 22078	. . 3 ⊢ 𝑃 = ((PwSer₁‘𝑅) ↾_s (Base‘(1_o mPoly 𝑅)))
10		eqid 2729	. . 3 ⊢ (LSubSp‘(PwSer₁‘𝑅)) = (LSubSp‘(PwSer₁‘𝑅))
11	9, 10	lsslmod 20866	. 2 ⊢ (((PwSer₁‘𝑅) ∈ LMod ∧ (Base‘(1_o mPoly 𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅))) → 𝑃 ∈ LMod)
12	2, 7, 11	syl2anc 584	1 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ‘cfv 6511 (class class class)co 7387 1_oc1o 8427 Basecbs 17179 Ringcrg 20142 LModclmod 20766 LSubSpclss 20837 mPoly cmpl 21815 PwSer₁cps1 22059 Poly₁cpl1 22061

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-hom 17244 df-cco 17245 df-0g 17404 df-prds 17410 df-pws 17412 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-lmod 20768 df-lss 20838 df-psr 21818 df-mpl 21820 df-opsr 21822 df-psr1 22064 df-ply1 22066

This theorem is referenced by: ply1ascl0 22139 ply1ascl1 22140 ply10s0 22142 ply1tmcl 22158 coe1pwmul 22165 ply1sclf 22171 ply1scl0 22176 ply1scl0OLD 22177 ply1scl1 22179 ply1scl1OLD 22180 ply1idvr1OLD 22182 ply1coefsupp 22184 ply1coe 22185 cply1coe0bi 22189 gsumsmonply1 22194 gsummoncoe1 22195 lply1binomsc 22198 evls1sca 22210 evl1scvarpw 22250 evl1gsummon 22252 evls1fpws 22256 evls1vsca 22260 asclply1subcl 22261 evls1maplmhm 22264 cpmatacl 22603 cpmatinvcl 22604 mat2pmatbas 22613 mat2pmatghm 22617 mat2pmatmul 22618 decpmatid 22657 pmatcollpwscmatlem1 22676 pm2mpcl 22684 idpm2idmp 22688 mply1topmatcllem 22690 mply1topmatcl 22692 mp2pm2mplem4 22696 mp2pm2mplem5 22697 pm2mpghmlem2 22699 pm2mpghm 22703 pm2mpmhmlem1 22705 pm2mpmhmlem2 22706 monmat2matmon 22711 chpscmat 22729 chpscmatgsumbin 22731 chpscmatgsummon 22732 deg1invg 26011 deg1pwle 26025 deg1pw 26026 ply1remlem 26070 plypf1 26117 ply1lvec 33528 ressasclcl 33540 coe1mon 33554 deg1vr 33558 ply1degltlss 33562 gsummoncoe1fzo 33563 q1pvsca 33569 r1pvsca 33570 r1p0 33571 r1plmhm 33575 irngnzply1lem 33685 2sqr3minply 33770 cos9thpiminplylem6 33777 cos9thpiminply 33778 aks5lem2 42175 ply1vr1smo 48371 ply1mulgsumlem4 48378 ply1mulgsum 48379

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