Theorem plycn 16005

Description: A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8308. (Revised by GG, 16-Mar-2025.)

Assertion

Ref	Expression
plycn	|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )

Proof of Theorem plycn

Dummy variables a d k z u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elply 15977	. . 3 |- ( F e. (Poly ` S ) <-> ( S C_ CC /\ E. d e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) ) )
2	1	simprbi 275	. 2 |- ( F e. (Poly ` S ) -> E. d e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) )
3		simpr 110	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) )
4		eqid 2238	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
5	4	cnfldtopon 15783	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
6	5	a1i 9	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
7		0zd 9665	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> 0 e. ZZ )
8		simprl 535	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> d e. NN0 )
9	8	nn0zd 9776	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> d e. ZZ )
10	7, 9	fzfigd 10888	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( 0 ... d ) e. Fin )
11	5	a1i 9	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... d ) ) -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
12		elmapi 6949	. . . . . . . . . . . . . 14 |- ( a e. ( ( S u. { 0 } ) ^m NN0 ) -> a : NN0 --> ( S u. { 0 } ) )
13	12	ad2antll 495	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> a : NN0 --> ( S u. { 0 } ) )
14		plybss 15976	. . . . . . . . . . . . . . 15 |- ( F e. (Poly ` S ) -> S C_ CC )
15	14	adantr 276	. . . . . . . . . . . . . 14 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> S C_ CC )
16		0cnd 8325	. . . . . . . . . . . . . . 15 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> 0 e. CC )
17	16	snssd 3860	. . . . . . . . . . . . . 14 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> { 0 } C_ CC )
18	15, 17	unssd 3405	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( S u. { 0 } ) C_ CC )
19	13, 18	fssd 5550	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> a : NN0 --> CC )
20	19	adantr 276	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... d ) ) -> a : NN0 --> CC )
21		elfznn0 10537	. . . . . . . . . . . 12 |- ( k e. ( 0 ... d ) -> k e. NN0 )
22	21	adantl 277	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... d ) ) -> k e. NN0 )
23	20, 22	ffvelcdmd 5847	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... d ) ) -> ( a ` k ) e. CC )
24	11, 11, 23	cnmptc 15525	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... d ) ) -> ( z e. CC |-> ( a ` k ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
25	4	expcn 15812	. . . . . . . . . 10 |- ( k e. NN0 -> ( z e. CC |-> ( z ^ k ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
26	22, 25	syl 14	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... d ) ) -> ( z e. CC |-> ( z ^ k ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
27	4	mpomulcn 15809	. . . . . . . . . 10 |- ( u e. CC , v e. CC |-> ( u x. v ) ) e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
28	27	a1i 9	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... d ) ) -> ( u e. CC , v e. CC |-> ( u x. v ) ) e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) ) )
29		oveq12 6097	. . . . . . . . 9 |- ( ( u = ( a ` k ) /\ v = ( z ^ k ) ) -> ( u x. v ) = ( ( a ` k ) x. ( z ^ k ) ) )
30	11, 24, 26, 11, 11, 28, 29	cnmpt12 15530	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ k e. ( 0 ... d ) ) -> ( z e. CC |-> ( ( a ` k ) x. ( z ^ k ) ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
31	4, 6, 10, 30	fsumcn 15811	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
32	31	adantr 276	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) ) -> ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
33	3, 32	eqeltrd 2315	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) ) -> F e. ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) ) )
34	4	cncfcn1 15838	. . . . 5 |- ( CC -cn-> CC ) = ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) )
35	33, 34	eleqtrrdi 2332	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) /\ F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) ) -> F e. ( CC -cn-> CC ) )
36	35	ex 115	. . 3 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) -> F e. ( CC -cn-> CC ) ) )
37	36	rexlimdvva 2676	. 2 |- ( F e. (Poly ` S ) -> ( E. d e. NN0 E. a e. ( ( S u. { 0 } ) ^m NN0 ) F = ( z e. CC |-> sum_ k e. ( 0 ... d ) ( ( a ` k ) x. ( z ^ k ) ) ) -> F e. ( CC -cn-> CC ) ) )
38	2, 37	mpd 13	1 |- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   u. cun 3218   C_ wss 3220   {csn 3709   |-> cmpt 4193   -->wf 5376   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   ^m cmap 6927   CCcc 8183   0cc0 8185   x. cmul 8190   NN0cn0 9574   ...cfz 10427   ^cexp 10995   sum_csu 12143   TopOpenctopn 13652  ℂ_fldccnfld 15001  TopOnctopon 15253   Cn ccn 15428   tX ctx 15495   -cn->ccncf 15813  Polycply 15971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305  ax-addf 8307

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6581  df-irdg 6646  df-frec 6667  df-1o 6692  df-oadd 6696  df-er 6812  df-map 6929  df-en 7029  df-dom 7030  df-fin 7031  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9654  df-dec 9788  df-uz 9937  df-q 10035  df-rp 10071  df-xneg 10190  df-xadd 10191  df-fz 10428  df-fzo 10566  df-seqfrec 10905  df-exp 10996  df-ihash 11236  df-cj 11628  df-re 11629  df-im 11630  df-rsqrt 11785  df-abs 11786  df-clim 12069  df-sumdc 12144  df-struct 13411  df-ndx 13412  df-slot 13413  df-base 13415  df-plusg 13502  df-mulr 13503  df-starv 13504  df-tset 13508  df-ple 13509  df-ds 13511  df-unif 13512  df-rest 13653  df-topn 13654  df-topgen 13672  df-psmet 14988  df-xmet 14989  df-met 14990  df-bl 14991  df-mopn 14992  df-fg 14994  df-metu 14995  df-cnfld 15002  df-top 15241  df-topon 15254  df-topsp 15274  df-bases 15286  df-cn 15431  df-cnp 15432  df-tx 15496  df-xms 15582  df-ms 15583  df-cncf 15814  df-ply 15973

This theorem is used by: (None)