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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 2724 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | mplval2.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
6 | 1, 2, 3, 4, 5 | mplbas 21861 | . 2 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp (0g‘𝑅)} |
7 | 6 | ssrab3 4073 | 1 ⊢ 𝑈 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ⊆ wss 3941 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 finSupp cfsupp 9358 Basecbs 17145 0gc0g 17386 mPwSer cmps 21768 mPoly cmpl 21770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-1cn 11165 ax-addcl 11167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-nn 12211 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-psr 21773 df-mpl 21775 |
This theorem is referenced by: mplelf 21869 mplsubrglem 21875 mpladd 21880 mplneg 21881 mplmul 21882 mplvsca 21886 ressmpladd 21896 ressmplmul 21897 ressmplvsca 21898 mplbas2 21909 psdmplcl 22015 ply1bas 22039 ply1ass23l 22070 |
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