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Mirrors > Home > MPE Home > Th. List > mplelf | Structured version Visualization version GIF version |
Description: A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
mplelf.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplelf.k | ⊢ 𝐾 = (Base‘𝑅) |
mplelf.b | ⊢ 𝐵 = (Base‘𝑃) |
mplelf.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
mplelf.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
mplelf | ⊢ (𝜑 → 𝑋:𝐷⟶𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2771 | . 2 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | mplelf.k | . 2 ⊢ 𝐾 = (Base‘𝑅) | |
3 | mplelf.d | . 2 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
4 | eqid 2771 | . 2 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
5 | mplelf.p | . . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
6 | mplelf.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
7 | 5, 1, 6, 4 | mplbasss 19938 | . . 3 ⊢ 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)) |
8 | mplelf.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | 7, 8 | sseldi 3849 | . 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐼 mPwSer 𝑅))) |
10 | 1, 2, 3, 4, 9 | psrelbas 19885 | 1 ⊢ (𝜑 → 𝑋:𝐷⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 {crab 3085 ◡ccnv 5402 “ cima 5406 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 ↑𝑚 cmap 8204 Fincfn 8304 ℕcn 11437 ℕ0cn0 11705 Basecbs 16337 mPwSer cmps 19857 mPoly cmpl 19859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-tset 16438 df-psr 19862 df-mpl 19864 |
This theorem is referenced by: mplsubrglem 19945 mplvscaval 19954 mplmonmul 19970 mplcoe1 19971 mplbas2 19976 mplcoe4 20008 evlslem2 20017 evlslem6 20018 evlslem1 20020 mhpaddcl 20056 mhpinvcl 20057 mhpvscacl 20059 ply1basf 20088 mdegfval 24374 mdegleb 24376 mdegldg 24378 mdegaddle 24386 mdegvsca 24388 mdegle0 24389 mdegmullem 24390 |
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