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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 2824 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | mplval2.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
6 | 1, 2, 3, 4, 5 | mplbas 20212 | . 2 ⊢ 𝑈 = {𝑓 ∈ 𝐵 ∣ 𝑓 finSupp (0g‘𝑅)} |
7 | 6 | ssrab3 4060 | 1 ⊢ 𝑈 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊆ wss 3939 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 finSupp cfsupp 8836 Basecbs 16486 0gc0g 16716 mPwSer cmps 20134 mPoly cmpl 20136 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-1cn 10598 ax-addcl 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-nn 11642 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-psr 20139 df-mpl 20141 |
This theorem is referenced by: mplelf 20216 mplsubrglem 20222 mpladd 20225 mplmul 20226 mplvsca 20230 ressmpladd 20241 ressmplmul 20242 ressmplvsca 20243 mplbas2 20254 ply1bas 20366 ply1ass23l 44443 |
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