| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ply1sca | Structured version Visualization version GIF version | ||
| Description: Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| Ref | Expression |
|---|---|
| ply1lmod.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| Ref | Expression |
|---|---|
| ply1sca | ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘𝑃)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
| 2 | 1 | psr1sca 22251 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘(PwSer1‘𝑅))) |
| 3 | fvex 6919 | . . 3 ⊢ (Base‘(1o mPoly 𝑅)) ∈ V | |
| 4 | ply1lmod.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 5 | 4, 1 | ply1val 22195 | . . . 4 ⊢ 𝑃 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
| 6 | eqid 2737 | . . . 4 ⊢ (Scalar‘(PwSer1‘𝑅)) = (Scalar‘(PwSer1‘𝑅)) | |
| 7 | 5, 6 | resssca 17387 | . . 3 ⊢ ((Base‘(1o mPoly 𝑅)) ∈ V → (Scalar‘(PwSer1‘𝑅)) = (Scalar‘𝑃)) |
| 8 | 3, 7 | ax-mp 5 | . 2 ⊢ (Scalar‘(PwSer1‘𝑅)) = (Scalar‘𝑃) |
| 9 | 2, 8 | eqtrdi 2793 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 = (Scalar‘𝑃)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 Basecbs 17247 Scalarcsca 17300 mPoly cmpl 21926 PwSer1cps1 22176 Poly1cpl1 22178 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-ple 17317 df-psr 21929 df-opsr 21933 df-psr1 22181 df-ply1 22183 |
| This theorem is referenced by: ply1sca2 22255 ply1ascl0 22256 ply1ascl1 22257 ply10s0 22259 ply1ascl 22261 coe1pwmul 22282 ply1scl0 22293 ply1scl1 22296 ply1idvr1OLD 22299 ply1coefsupp 22301 ply1coe 22302 cply1coe0bi 22306 ply1chr 22310 gsumsmonply1 22311 gsummoncoe1 22312 lply1binomsc 22315 ply1fermltlchr 22316 evls1sca 22327 evl1vsd 22348 evl1scvarpw 22367 evl1gsummon 22369 evls1fpws 22373 evls1vsca 22377 asclply1subcl 22378 evls1maplmhm 22381 cpmatacl 22722 cpmatinvcl 22723 mat2pmatbas 22732 mat2pmatghm 22736 mat2pmatmul 22737 mat2pmatlin 22741 decpmatid 22776 pmatcollpw2lem 22783 monmatcollpw 22785 pmatcollpwlem 22786 pmatcollpwscmatlem1 22795 pm2mpcl 22803 idpm2idmp 22807 mply1topmatcllem 22809 mply1topmatcl 22811 mp2pm2mplem4 22815 mp2pm2mplem5 22816 pm2mpghmlem2 22818 pm2mpghm 22822 pm2mpmhmlem1 22824 pm2mpmhmlem2 22825 monmat2matmon 22830 chpscmat 22848 chpscmatgsumbin 22850 chpscmatgsummon 22851 deg1pwle 26159 deg1pw 26160 ply1remlem 26204 fta1blem 26210 plypf1 26251 ply1lvec 33585 ressasclcl 33596 ply1asclunit 33599 coe1mon 33610 deg1vr 33614 ply1degltlss 33617 gsummoncoe1fzo 33618 q1pvsca 33624 r1pvsca 33625 r1p0 33626 r1plmhm 33630 ply1degltdimlem 33673 irngnzply1lem 33740 algextdeglem8 33765 2sqr3minply 33791 aks5lem2 42188 ply1asclzrhval 42189 ply1vr1smo 48299 ply1sclrmsm 48300 ply1mulgsumlem4 48306 ply1mulgsum 48307 |
| Copyright terms: Public domain | W3C validator |