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Mirrors > Home > MPE Home > Th. List > mplbasss | Structured version Visualization version GIF version |
Description: The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
mplval2.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplval2.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
mplval2.u | ⊢ 𝑈 = (Base‘𝑃) |
mplbasss.b | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
mplbasss | ⊢ 𝑈 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplval2.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
2 | mplval2.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
3 | mplbasss.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
4 | eqid 2778 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | mplval2.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
6 | 1, 2, 3, 4, 5 | mplbas 19926 | . 2 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp (0g‘𝑅)} |
7 | 6 | ssrab3 3949 | 1 ⊢ 𝑈 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ⊆ wss 3831 class class class wbr 4930 ‘cfv 6190 (class class class)co 6978 finSupp cfsupp 8630 Basecbs 16342 0gc0g 16572 mPwSer cmps 19848 mPoly cmpl 19850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-1cn 10395 ax-addcl 10397 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-nn 11442 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-psr 19853 df-mpl 19855 |
This theorem is referenced by: mplelf 19930 mplsubrglem 19936 mpladd 19939 mplmul 19940 mplvsca 19944 ressmpladd 19954 ressmplmul 19955 ressmplvsca 19956 mplbas2 19967 ply1bas 20069 ply1ass23l 43804 |
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