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Mirrors > Home > MPE Home > Th. List > mplsca | Structured version Visualization version GIF version |
Description: The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mplsca.p | β’ π = (πΌ mPoly π ) |
mplsca.i | β’ (π β πΌ β π) |
mplsca.r | β’ (π β π β π) |
Ref | Expression |
---|---|
mplsca | β’ (π β π = (Scalarβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . 3 β’ (πΌ mPwSer π ) = (πΌ mPwSer π ) | |
2 | mplsca.i | . . 3 β’ (π β πΌ β π) | |
3 | mplsca.r | . . 3 β’ (π β π β π) | |
4 | 1, 2, 3 | psrsca 21727 | . 2 β’ (π β π = (Scalarβ(πΌ mPwSer π ))) |
5 | fvex 6904 | . . 3 β’ (Baseβπ) β V | |
6 | mplsca.p | . . . . 5 β’ π = (πΌ mPoly π ) | |
7 | eqid 2732 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
8 | 6, 1, 7 | mplval2 21774 | . . . 4 β’ π = ((πΌ mPwSer π ) βΎs (Baseβπ)) |
9 | eqid 2732 | . . . 4 β’ (Scalarβ(πΌ mPwSer π )) = (Scalarβ(πΌ mPwSer π )) | |
10 | 8, 9 | resssca 17292 | . . 3 β’ ((Baseβπ) β V β (Scalarβ(πΌ mPwSer π )) = (Scalarβπ)) |
11 | 5, 10 | ax-mp 5 | . 2 β’ (Scalarβ(πΌ mPwSer π )) = (Scalarβπ) |
12 | 4, 11 | eqtrdi 2788 | 1 β’ (π β π = (Scalarβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3474 βcfv 6543 (class class class)co 7411 Basecbs 17148 Scalarcsca 17204 mPwSer cmps 21676 mPoly cmpl 21678 |
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-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-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-tset 17220 df-psr 21681 df-mpl 21683 |
This theorem is referenced by: mpllvec 21798 mplcoe1 21811 mplbas2 21816 mplascl 21844 mplasclf 21845 subrgascl 21846 subrgasclcl 21847 mplmon2cl 21848 mplmon2mul 21849 mplind 21850 evlslem1 21864 mpfconst 21883 mpfind 21889 mhppwdeg 21912 mhpvscacl 21916 mhplss 21917 ply1ascl 22000 pf1ind 22094 mdegvscale 25817 mdegvsca 25818 mplascl0 41428 mplascl1 41429 selvcllem2 41452 selvvvval 41459 evlselv 41461 |
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