ply1lmod - Metamath Proof Explorer

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Theorem ply1lmod 22190

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 2734	. . 3 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
2	1	psr1lmod 22187	. 2 ⊢ (𝑅 ∈ Ring → (PwSer₁‘𝑅) ∈ LMod)
3		eqid 2734	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2734	. . . 4 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
5	3, 4	ply1bas 22133	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(1_o mPoly 𝑅))
6	3, 1, 4	ply1lss 22135	. . 3 ⊢ (𝑅 ∈ Ring → (Base‘(Poly₁‘𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅)))
7	5, 6	eqeltrrid 2839	. 2 ⊢ (𝑅 ∈ Ring → (Base‘(1_o mPoly 𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅)))
8		ply1lmod.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
9	8, 1	ply1val 22132	. . 3 ⊢ 𝑃 = ((PwSer₁‘𝑅) ↾_s (Base‘(1_o mPoly 𝑅)))
10		eqid 2734	. . 3 ⊢ (LSubSp‘(PwSer₁‘𝑅)) = (LSubSp‘(PwSer₁‘𝑅))
11	9, 10	lsslmod 20909	. 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 1541 ∈ wcel 2113 ‘cfv 6490 (class class class)co 7356 1_oc1o 8388 Basecbs 17134 Ringcrg 20166 LModclmod 20809 LSubSpclss 20880 mPoly cmpl 21860 PwSer₁cps1 22113 Poly₁cpl1 22115

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-sup 9343 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-hom 17199 df-cco 17200 df-0g 17359 df-prds 17365 df-pws 17367 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-lmod 20811 df-lss 20881 df-psr 21863 df-mpl 21865 df-opsr 21867 df-psr1 22118 df-ply1 22120

This theorem is referenced by: ply1ascl0 22193 ply1ascl1 22194 ply10s0 22196 ply1tmcl 22212 coe1pwmul 22219 ply1sclf 22225 ply1scl0 22230 ply1scl0OLD 22231 ply1scl1 22233 ply1scl1OLD 22234 ply1idvr1OLD 22237 ply1coefsupp 22239 ply1coe 22240 cply1coe0bi 22244 gsumsmonply1 22249 gsummoncoe1 22250 lply1binomsc 22253 evls1sca 22265 evl1scvarpw 22305 evl1gsummon 22307 evls1fpws 22311 evls1vsca 22315 asclply1subcl 22316 evls1maplmhm 22319 cpmatacl 22658 cpmatinvcl 22659 mat2pmatbas 22668 mat2pmatghm 22672 mat2pmatmul 22673 decpmatid 22712 pmatcollpwscmatlem1 22731 pm2mpcl 22739 idpm2idmp 22743 mply1topmatcllem 22745 mply1topmatcl 22747 mp2pm2mplem4 22751 mp2pm2mplem5 22752 pm2mpghmlem2 22754 pm2mpghm 22758 pm2mpmhmlem1 22760 pm2mpmhmlem2 22761 monmat2matmon 22766 chpscmat 22784 chpscmatgsumbin 22786 chpscmatgsummon 22787 deg1invg 26065 deg1pwle 26079 deg1pw 26080 ply1remlem 26124 plypf1 26171 ply1lvec 33589 ressasclcl 33601 coe1mon 33617 ply1coedeg 33619 deg1vr 33622 ply1degltlss 33626 gsummoncoe1fzo 33627 q1pvsca 33634 r1pvsca 33635 r1p0 33636 r1plmhm 33640 vietalem 33684 irngnzply1lem 33796 extdgfialglem2 33799 2sqr3minply 33886 cos9thpiminplylem6 33893 cos9thpiminply 33894 aks5lem2 42380 ply1vr1smo 48571 ply1mulgsumlem4 48577 ply1mulgsum 48578

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