ply1lmod - Metamath Proof Explorer

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Theorem ply1lmod 22480

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 2762	. . 3 ⊢ (PwSer₁‘𝑅) = (PwSer₁‘𝑅)
2	1	psr1lmod 22477	. 2 ⊢ (𝑅 ∈ Ring → (PwSer₁‘𝑅) ∈ LMod)
3		eqid 2762	. . . 4 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
4		eqid 2762	. . . 4 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
5	3, 4	ply1bas 22424	. . 3 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(1_o mPoly 𝑅))
6	3, 1, 4	ply1lss 22425	. . 3 ⊢ (𝑅 ∈ Ring → (Base‘(Poly₁‘𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅)))
7	5, 6	eqeltrrid 2867	. 2 ⊢ (𝑅 ∈ Ring → (Base‘(1_o mPoly 𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅)))
8		ply1lmod.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
9	8, 1	ply1val 22423	. . 3 ⊢ 𝑃 = ((PwSer₁‘𝑅) ↾_s (Base‘(1_o mPoly 𝑅)))
10		eqid 2762	. . 3 ⊢ (LSubSp‘(PwSer₁‘𝑅)) = (LSubSp‘(PwSer₁‘𝑅))
11	9, 10	lsslmod 21148	. 2 ⊢ (((PwSer₁‘𝑅) ∈ LMod ∧ (Base‘(1_o mPoly 𝑅)) ∈ (LSubSp‘(PwSer₁‘𝑅))) → 𝑃 ∈ LMod)
12	2, 7, 11	syl2anc 596	1 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ‘cfv 6537 (class class class)co 7416 1_oc1o 8451 Basecbs 17305 Ringcrg 20376 LModclmod 21048 LSubSpclss 21119 mPoly cmpl 22125 PwSer₁cps1 22404 Poly₁cpl1 22406

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13564 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-hom 17370 df-cco 17371 df-0g 17530 df-prds 17536 df-pws 17538 df-mgm 18734 df-sgrp 18825 df-mnd 18841 df-grp 19064 df-minusg 19065 df-sbg 19066 df-subg 19250 df-cmn 19913 df-abl 19914 df-mgp 20278 df-rng 20292 df-ur 20325 df-ring 20378 df-lmod 21050 df-lss 21120 df-psr 22128 df-mpl 22130 df-opsr 22132 df-psr1 22409 df-ply1 22411

This theorem is used by: ply1ascl0 22483 ply1ascl1 22484 ply10s0 22486 ply1tmcl 22502 coe1pwmul 22509 ply1sclf 22515 ply1scl0 22520 ply1scl1 22522 ply1coefsupp 22526 ply1coe 22527 cply1coe0bi 22531 gsumsmonply1 22536 gsummoncoe1 22537 lply1binomsc 22540 evls1sca 22552 evl1scvarpw 22592 evl1gsummon 22594 evls1fpws 22598 evls1vsca 22602 asclply1subcl 22603 evls1maplmhm 22606 cpmatacl 22945 cpmatinvcl 22946 mat2pmatbas 22955 mat2pmatghm 22959 mat2pmatmul 22960 decpmatid 22999 pmatcollpwscmatlem1 23018 pm2mpcl 23026 idpm2idmp 23030 mply1topmatcllem 23032 mply1topmatcl 23034 mp2pm2mplem4 23038 mp2pm2mplem5 23039 pm2mpghmlem2 23041 pm2mpghm 23045 pm2mpmhmlem1 23047 pm2mpmhmlem2 23048 monmat2matmon 23053 chpscmat 23071 chpscmatgsumbin 23073 chpscmatgsummon 23074 deg1invg 26336 deg1pwle 26350 deg1pw 26351 ply1remlem 26395 plypf1 26442 ply1lvec 33971 ressasclcl 33983 coe1mon 33999 ply1coedeg 34001 deg1vr 34004 ply1degltlss 34008 gsummoncoe1fzo 34009 q1pvsca 34016 r1pvsca 34017 r1p0 34018 r1plmhm 34021 vietalem 34091 irngnzply1lem 34202 extdgfialglem2 34205 2sqr3minply 34292 cos9thpiminplylem6 34299 cos9thpiminply 34300 aks5lem2 43055 ply1vr1smo 49315 ply1mulgsumlem4 49321 ply1mulgsum 49322

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