ply1lmod - Metamath Proof Explorer

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Theorem ply1lmod 21774

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 2733	. . 3 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
2	1	psr1lmod 21771	. 2 ⊢ (𝑅 ∈ Ring → (PwSer₁‘𝑅) ∈ LMod)
3		eqid 2733	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2733	. . . 4 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
5	3, 1, 4	ply1bas 21719	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(1_o mPoly 𝑅))
6	3, 1, 4	ply1lss 21720	. . 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 21718	. . 3 ⊢ 𝑃 = ((PwSer₁‘𝑅) ↾_s (Base‘(1_o mPoly 𝑅)))
10		eqid 2733	. . 3 ⊢ (LSubSp‘(PwSer₁‘𝑅)) = (LSubSp‘(PwSer₁‘𝑅))
11	9, 10	lsslmod 20571	. 2 ⊢ (((PwSer₁‘𝑅) ∈ LMod ∧ (Base‘(1_o mPoly 𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅))) → 𝑃 ∈ LMod)
12	2, 7, 11	syl2anc 585	1 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6544 (class class class)co 7409 1_oc1o 8459 Basecbs 17144 Ringcrg 20056 LModclmod 20471 LSubSpclss 20542 mPoly cmpl 21459 PwSer₁cps1 21699 Poly₁cpl1 21701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-hom 17221 df-cco 17222 df-0g 17387 df-prds 17393 df-pws 17395 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-mgp 19988 df-ur 20005 df-ring 20058 df-lmod 20473 df-lss 20543 df-psr 21462 df-mpl 21464 df-opsr 21466 df-psr1 21704 df-ply1 21706

This theorem is referenced by: ply10s0 21778 ply1tmcl 21794 coe1pwmul 21801 ply1sclf 21807 ply1scl0 21812 ply1scl0OLD 21813 ply1scl1 21815 ply1scl1OLD 21816 ply1idvr1 21817 ply1coefsupp 21819 ply1coe 21820 cply1coe0bi 21824 gsumsmonply1 21827 gsummoncoe1 21828 lply1binomsc 21831 evls1sca 21842 evl1scvarpw 21882 evl1gsummon 21884 cpmatacl 22218 cpmatinvcl 22219 mat2pmatbas 22228 mat2pmatghm 22232 mat2pmatmul 22233 decpmatid 22272 pmatcollpwscmatlem1 22291 pm2mpcl 22299 idpm2idmp 22303 mply1topmatcllem 22305 mply1topmatcl 22307 mp2pm2mplem4 22311 mp2pm2mplem5 22312 pm2mpghmlem2 22314 pm2mpghm 22318 pm2mpmhmlem1 22320 pm2mpmhmlem2 22321 monmat2matmon 22326 chpscmat 22344 chpscmatgsumbin 22346 chpscmatgsummon 22347 deg1invg 25624 deg1pwle 25637 deg1pw 25638 ply1remlem 25680 plypf1 25726 ply1lvec 32638 evls1fpws 32646 evls1vsca 32650 ply1ascl0 32652 ply1ascl1 32653 asclply1subcl 32660 coe1mon 32664 ply1degltlss 32667 gsummoncoe1fzo 32668 irngnzply1lem 32754 evls1maplmhm 32760 ply1vr1smo 47062 ply1mulgsumlem4 47070 ply1mulgsum 47071

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