![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ply1lmod | Structured version Visualization version GIF version |
Description: Univariate polynomials form a left module. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
Ref | Expression |
---|---|
ply1lmod.p | ⊢ 𝑃 = (Poly1‘𝑅) |
Ref | Expression |
---|---|
ply1lmod | ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
2 | 1 | psr1lmod 21636 | . 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 21582 | . . 3 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(1o mPoly 𝑅)) |
6 | 3, 1, 4 | ply1lss 21583 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(Poly1‘𝑅)) ∈ (LSubSp‘(PwSer1‘𝑅))) |
7 | 5, 6 | eqeltrrid 2839 | . 2 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPoly 𝑅)) ∈ (LSubSp‘(PwSer1‘𝑅))) |
8 | ply1lmod.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
9 | 8, 1 | ply1val 21581 | . . 3 ⊢ 𝑃 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
10 | eqid 2733 | . . 3 ⊢ (LSubSp‘(PwSer1‘𝑅)) = (LSubSp‘(PwSer1‘𝑅)) | |
11 | 9, 10 | lsslmod 20436 | . 2 ⊢ (((PwSer1‘𝑅) ∈ LMod ∧ (Base‘(1o 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 6497 (class class class)co 7358 1oc1o 8406 Basecbs 17088 Ringcrg 19969 LModclmod 20336 LSubSpclss 20407 mPoly cmpl 21324 PwSer1cps1 21562 Poly1cpl1 21564 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-0g 17328 df-prds 17334 df-pws 17336 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 df-psr 21327 df-mpl 21329 df-opsr 21331 df-psr1 21567 df-ply1 21569 |
This theorem is referenced by: ply10s0 21643 ply1tmcl 21659 coe1pwmul 21666 ply1sclf 21672 ply1scl0 21677 ply1scl1 21679 ply1idvr1 21680 ply1coefsupp 21682 ply1coe 21683 cply1coe0bi 21687 gsumsmonply1 21690 gsummoncoe1 21691 lply1binomsc 21694 evls1sca 21705 evl1scvarpw 21745 evl1gsummon 21747 cpmatacl 22081 cpmatinvcl 22082 mat2pmatbas 22091 mat2pmatghm 22095 mat2pmatmul 22096 decpmatid 22135 pmatcollpwscmatlem1 22154 pm2mpcl 22162 idpm2idmp 22166 mply1topmatcllem 22168 mply1topmatcl 22170 mp2pm2mplem4 22174 mp2pm2mplem5 22175 pm2mpghmlem2 22177 pm2mpghm 22181 pm2mpmhmlem1 22183 pm2mpmhmlem2 22184 monmat2matmon 22189 chpscmat 22207 chpscmatgsumbin 22209 chpscmatgsummon 22210 deg1invg 25487 deg1pwle 25500 deg1pw 25501 ply1remlem 25543 plypf1 25589 ply1lvec 32314 evls1fpws 32320 ply1ascl0 32325 asclply1subcl 32330 coe1mon 32334 ply1degltlss 32337 gsummoncoe1fzo 32338 irngnzply1lem 32421 ply1vr1smo 46548 ply1mulgsumlem4 46556 ply1mulgsum 46557 |
Copyright terms: Public domain | W3C validator |