ply1lmod - Metamath Proof Explorer

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Theorem ply1lmod 21781

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 2732	. . 3 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
2	1	psr1lmod 21778	. 2 ⊢ (𝑅 ∈ Ring → (PwSer₁‘𝑅) ∈ LMod)
3		eqid 2732	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2732	. . . 4 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
5	3, 1, 4	ply1bas 21725	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(1_o mPoly 𝑅))
6	3, 1, 4	ply1lss 21726	. . 3 ⊢ (𝑅 ∈ Ring → (Base‘(Poly₁‘𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅)))
7	5, 6	eqeltrrid 2838	. 2 ⊢ (𝑅 ∈ Ring → (Base‘(1_o mPoly 𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅)))
8		ply1lmod.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
9	8, 1	ply1val 21724	. . 3 ⊢ 𝑃 = ((PwSer₁‘𝑅) ↾_s (Base‘(1_o mPoly 𝑅)))
10		eqid 2732	. . 3 ⊢ (LSubSp‘(PwSer₁‘𝑅)) = (LSubSp‘(PwSer₁‘𝑅))
11	9, 10	lsslmod 20576	. 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 2106 ‘cfv 6543 (class class class)co 7411 1_oc1o 8461 Basecbs 17146 Ringcrg 20058 LModclmod 20475 LSubSpclss 20547 mPoly cmpl 21465 PwSer₁cps1 21705 Poly₁cpl1 21707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-fz 13487 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-hom 17223 df-cco 17224 df-0g 17389 df-prds 17395 df-pws 17397 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-sbg 18826 df-subg 19005 df-mgp 19990 df-ur 20007 df-ring 20060 df-lmod 20477 df-lss 20548 df-psr 21468 df-mpl 21470 df-opsr 21472 df-psr1 21710 df-ply1 21712

This theorem is referenced by: ply10s0 21785 ply1tmcl 21801 coe1pwmul 21808 ply1sclf 21814 ply1scl0 21819 ply1scl0OLD 21820 ply1scl1 21822 ply1scl1OLD 21823 ply1idvr1 21824 ply1coefsupp 21826 ply1coe 21827 cply1coe0bi 21831 gsumsmonply1 21834 gsummoncoe1 21835 lply1binomsc 21838 evls1sca 21849 evl1scvarpw 21889 evl1gsummon 21891 cpmatacl 22225 cpmatinvcl 22226 mat2pmatbas 22235 mat2pmatghm 22239 mat2pmatmul 22240 decpmatid 22279 pmatcollpwscmatlem1 22298 pm2mpcl 22306 idpm2idmp 22310 mply1topmatcllem 22312 mply1topmatcl 22314 mp2pm2mplem4 22318 mp2pm2mplem5 22319 pm2mpghmlem2 22321 pm2mpghm 22325 pm2mpmhmlem1 22327 pm2mpmhmlem2 22328 monmat2matmon 22333 chpscmat 22351 chpscmatgsumbin 22353 chpscmatgsummon 22354 deg1invg 25631 deg1pwle 25644 deg1pw 25645 ply1remlem 25687 plypf1 25733 ply1lvec 32683 evls1fpws 32691 evls1vsca 32695 ply1ascl0 32697 ply1ascl1 32698 asclply1subcl 32705 coe1mon 32709 ply1degltlss 32713 gsummoncoe1fzo 32714 q1pvsca 32720 r1pvsca 32721 r1p0 32722 r1plmhm 32726 irngnzply1lem 32814 evls1maplmhm 32820 ply1vr1smo 47141 ply1mulgsumlem4 47148 ply1mulgsum 47149

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