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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 | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplsca.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mplsca.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
Ref | Expression |
---|---|
mplsca | ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | mplsca.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
3 | mplsca.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
4 | 1, 2, 3 | psrsca 20769 | . 2 ⊢ (𝜑 → 𝑅 = (Scalar‘(𝐼 mPwSer 𝑅))) |
5 | fvex 6688 | . . 3 ⊢ (Base‘𝑃) ∈ V | |
6 | mplsca.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
7 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
8 | 6, 1, 7 | mplval2 20813 | . . . 4 ⊢ 𝑃 = ((𝐼 mPwSer 𝑅) ↾s (Base‘𝑃)) |
9 | eqid 2738 | . . . 4 ⊢ (Scalar‘(𝐼 mPwSer 𝑅)) = (Scalar‘(𝐼 mPwSer 𝑅)) | |
10 | 8, 9 | resssca 16754 | . . 3 ⊢ ((Base‘𝑃) ∈ V → (Scalar‘(𝐼 mPwSer 𝑅)) = (Scalar‘𝑃)) |
11 | 5, 10 | ax-mp 5 | . 2 ⊢ (Scalar‘(𝐼 mPwSer 𝑅)) = (Scalar‘𝑃) |
12 | 4, 11 | eqtrdi 2789 | 1 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 Vcvv 3398 ‘cfv 6340 (class class class)co 7171 Basecbs 16587 Scalarcsca 16672 mPwSer cmps 20718 mPoly cmpl 20720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-of 7426 df-om 7601 df-1st 7715 df-2nd 7716 df-supp 7858 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-1o 8132 df-er 8321 df-map 8440 df-en 8557 df-dom 8558 df-sdom 8559 df-fin 8560 df-fsupp 8908 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-2 11780 df-3 11781 df-4 11782 df-5 11783 df-6 11784 df-7 11785 df-8 11786 df-9 11787 df-n0 11978 df-z 12064 df-uz 12326 df-fz 12983 df-struct 16589 df-ndx 16590 df-slot 16591 df-base 16593 df-sets 16594 df-ress 16595 df-plusg 16682 df-mulr 16683 df-sca 16685 df-vsca 16686 df-tset 16688 df-psr 20723 df-mpl 20725 |
This theorem is referenced by: mpllvec 20836 mplcoe1 20849 mplbas2 20854 mplascl 20877 mplasclf 20878 subrgascl 20879 subrgasclcl 20880 mplmon2cl 20881 mplmon2mul 20882 mplind 20883 evlslem1 20897 mpfconst 20916 mpfind 20922 mhppwdeg 20945 mhpvscacl 20949 mhplss 20950 ply1ascl 21034 pf1ind 21126 mdegvscale 24828 mdegvsca 24829 selvval2lem2 39799 selvval2lem4 39802 mhphf 39856 |
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