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Mirrors > Home > MPE Home > Th. List > ply1basf | Structured version Visualization version GIF version |
Description: Univariate polynomial base set elements are functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
ply1rcl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1rcl.b | ⊢ 𝐵 = (Base‘𝑃) |
ply1basf.k | ⊢ 𝐾 = (Base‘𝑅) |
Ref | Expression |
---|---|
ply1basf | ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1o)⟶𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . 2 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
2 | ply1basf.k | . 2 ⊢ 𝐾 = (Base‘𝑅) | |
3 | eqid 2825 | . 2 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
4 | psr1baslem 19915 | . 2 ⊢ (ℕ0 ↑𝑚 1o) = {𝑎 ∈ (ℕ0 ↑𝑚 1o) ∣ (◡𝑎 “ ℕ) ∈ Fin} | |
5 | id 22 | . . 3 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ 𝐵) | |
6 | ply1rcl.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
7 | eqid 2825 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
8 | ply1rcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
9 | 6, 7, 8 | ply1bas 19925 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
10 | 5, 9 | syl6eleq 2916 | . 2 ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (Base‘(1o mPoly 𝑅))) |
11 | 1, 2, 3, 4, 10 | mplelf 19794 | 1 ⊢ (𝐹 ∈ 𝐵 → 𝐹:(ℕ0 ↑𝑚 1o)⟶𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 1oc1o 7819 ↑𝑚 cmap 8122 ℕ0cn0 11618 Basecbs 16222 mPoly cmpl 19714 PwSer1cps1 19905 Poly1cpl1 19907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-tset 16324 df-ple 16325 df-psr 19717 df-mpl 19719 df-opsr 19721 df-psr1 19910 df-ply1 19912 |
This theorem is referenced by: 00ply1bas 19970 coe1add 19994 |
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