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Mirrors > Home > MPE Home > Th. List > plycpn | Structured version Visualization version GIF version |
Description: Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
plycpn | ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ∩ ran (Cn‘ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyf 24491 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
2 | 1 | adantr 473 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → 𝐹:ℂ⟶ℂ) |
3 | cnex 10416 | . . . . . . 7 ⊢ ℂ ∈ V | |
4 | 3, 3 | fpm 8239 | . . . . . 6 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 ∈ (ℂ ↑pm ℂ)) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (ℂ ↑pm ℂ)) |
6 | dvnply 24580 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘ℂ)) | |
7 | plycn 24554 | . . . . . . 7 ⊢ (((ℂ D𝑛 𝐹)‘𝑛) ∈ (Poly‘ℂ) → ((ℂ D𝑛 𝐹)‘𝑛) ∈ (ℂ–cn→ℂ)) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑛) ∈ (ℂ–cn→ℂ)) |
9 | 2 | fdmd 6353 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → dom 𝐹 = ℂ) |
10 | 9 | oveq1d 6991 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → (dom 𝐹–cn→ℂ) = (ℂ–cn→ℂ)) |
11 | 8, 10 | eleqtrrd 2869 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ)) |
12 | ssidd 3880 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ℂ ⊆ ℂ) | |
13 | elcpn 24234 | . . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝑛 ∈ ℕ0) → (𝐹 ∈ ((Cn‘ℂ)‘𝑛) ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ ((ℂ D𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ)))) | |
14 | 12, 13 | sylan 572 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → (𝐹 ∈ ((Cn‘ℂ)‘𝑛) ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ ((ℂ D𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ)))) |
15 | 5, 11, 14 | mpbir2and 700 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ ((Cn‘ℂ)‘𝑛)) |
16 | 15 | ralrimiva 3132 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑛 ∈ ℕ0 𝐹 ∈ ((Cn‘ℂ)‘𝑛)) |
17 | ssid 3879 | . . . 4 ⊢ ℂ ⊆ ℂ | |
18 | fncpn 24233 | . . . 4 ⊢ (ℂ ⊆ ℂ → (Cn‘ℂ) Fn ℕ0) | |
19 | eleq2 2854 | . . . . 5 ⊢ (𝑥 = ((Cn‘ℂ)‘𝑛) → (𝐹 ∈ 𝑥 ↔ 𝐹 ∈ ((Cn‘ℂ)‘𝑛))) | |
20 | 19 | ralrn 6679 | . . . 4 ⊢ ((Cn‘ℂ) Fn ℕ0 → (∀𝑥 ∈ ran (Cn‘ℂ)𝐹 ∈ 𝑥 ↔ ∀𝑛 ∈ ℕ0 𝐹 ∈ ((Cn‘ℂ)‘𝑛))) |
21 | 17, 18, 20 | mp2b 10 | . . 3 ⊢ (∀𝑥 ∈ ran (Cn‘ℂ)𝐹 ∈ 𝑥 ↔ ∀𝑛 ∈ ℕ0 𝐹 ∈ ((Cn‘ℂ)‘𝑛)) |
22 | 16, 21 | sylibr 226 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑥 ∈ ran (Cn‘ℂ)𝐹 ∈ 𝑥) |
23 | elintg 4757 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 ∈ ∩ ran (Cn‘ℂ) ↔ ∀𝑥 ∈ ran (Cn‘ℂ)𝐹 ∈ 𝑥)) | |
24 | 22, 23 | mpbird 249 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ∩ ran (Cn‘ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∈ wcel 2050 ∀wral 3088 ⊆ wss 3829 ∩ cint 4749 dom cdm 5407 ran crn 5408 Fn wfn 6183 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ↑pm cpm 8207 ℂcc 10333 ℕ0cn0 11707 –cn→ccncf 23187 D𝑛 cdvn 24165 Cnccpn 24166 Polycply 24477 |
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-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 ax-mulf 10415 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-fi 8670 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-cda 9388 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-icc 12561 df-fz 12709 df-fzo 12850 df-fl 12977 df-seq 13185 df-exp 13245 df-hash 13506 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-clim 14706 df-rlim 14707 df-sum 14904 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-starv 16436 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-hom 16445 df-cco 16446 df-rest 16552 df-topn 16553 df-0g 16571 df-gsum 16572 df-topgen 16573 df-pt 16574 df-prds 16577 df-xrs 16631 df-qtop 16636 df-imas 16637 df-xps 16639 df-mre 16715 df-mrc 16716 df-acs 16718 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-mulg 18012 df-subg 18060 df-cntz 18218 df-cmn 18668 df-mgp 18963 df-ur 18975 df-ring 19022 df-cring 19023 df-subrg 19256 df-psmet 20239 df-xmet 20240 df-met 20241 df-bl 20242 df-mopn 20243 df-fbas 20244 df-fg 20245 df-cnfld 20248 df-top 21206 df-topon 21223 df-topsp 21245 df-bases 21258 df-cld 21331 df-ntr 21332 df-cls 21333 df-nei 21410 df-lp 21448 df-perf 21449 df-cn 21539 df-cnp 21540 df-haus 21627 df-tx 21874 df-hmeo 22067 df-fil 22158 df-fm 22250 df-flim 22251 df-flf 22252 df-xms 22633 df-ms 22634 df-tms 22635 df-cncf 23189 df-0p 23974 df-limc 24167 df-dv 24168 df-dvn 24169 df-cpn 24170 df-ply 24481 df-coe 24483 df-dgr 24484 |
This theorem is referenced by: aalioulem3 24626 |
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