Theorem plycn 15912

Description: A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8302. (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 15884	. . 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 15690	. . . . . . . . 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 9660	. . . . . . . . 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 9770	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> d e. ZZ )
10	7, 9	fzfigd 10881	. . . . . . . 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 6944	. . . . . . . . . . . . . 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 15883	. . . . . . . . . . . . . . 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 8319	. . . . . . . . . . . . . . 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 5547	. . . . . . . . . . . 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 10531	. . . . . . . . . . . 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 5844	. . . . . . . . . 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 15432	. . . . . . . . 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 15719	. . . . . . . . . 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 15716	. . . . . . . . . 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 6094	. . . . . . . . 9 |- ( ( u = ( a ` k ) /\ v = ( z ^ k ) ) -> ( u x. v ) = ( ( a ` k ) x. ( z ^ k ) ) )
30	11, 24, 26, 11, 11, 28, 29	cnmpt12 15437	. . . . . . . 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 15718	. . . . . . 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 15745	. . . . 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 4192   -->wf 5373   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   ^m cmap 6922   CCcc 8177   0cc0 8179   x. cmul 8184   NN0cn0 9567   ...cfz 10421   ^cexp 10988   sum_csu 12135   TopOpenctopn 13643  ℂ_fldccnfld 14942  TopOnctopon 15160   Cn ccn 15335   tX ctx 15402   -cn->ccncf 15720  Polycply 15878

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-addf 8301

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-7 9370  df-8 9371  df-9 9372  df-n0 9568  df-z 9649  df-dec 9782  df-uz 9931  df-q 10029  df-rp 10065  df-xneg 10184  df-xadd 10185  df-fz 10422  df-fzo 10560  df-seqfrec 10898  df-exp 10989  df-ihash 11229  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-clim 12061  df-sumdc 12136  df-struct 13403  df-ndx 13404  df-slot 13405  df-base 13407  df-plusg 13493  df-mulr 13494  df-starv 13495  df-tset 13499  df-ple 13500  df-ds 13502  df-unif 13503  df-rest 13644  df-topn 13645  df-topgen 13663  df-psmet 14929  df-xmet 14930  df-met 14931  df-bl 14932  df-mopn 14933  df-fg 14935  df-metu 14936  df-cnfld 14943  df-top 15148  df-topon 15161  df-topsp 15181  df-bases 15193  df-cn 15338  df-cnp 15339  df-tx 15403  df-xms 15489  df-ms 15490  df-cncf 15721  df-ply 15880

This theorem is used by: (None)