Theorem plycn 14998

Description: A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8002. (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 14970	. . 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 2196	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
5	4	cnfldtopon 14776	. . . . . . . . 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 9338	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> 0 e. ZZ )
8		simprl 529	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> d e. NN0 )
9	8	nn0zd 9446	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> d e. ZZ )
10	7, 9	fzfigd 10523	. . . . . . . 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 6729	. . . . . . . . . . . . . 14 |- ( a e. ( ( S u. { 0 } ) ^m NN0 ) -> a : NN0 --> ( S u. { 0 } ) )
13	12	ad2antll 491	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> a : NN0 --> ( S u. { 0 } ) )
14		plybss 14969	. . . . . . . . . . . . . . 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 8019	. . . . . . . . . . . . . . 15 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> 0 e. CC )
17	16	snssd 3767	. . . . . . . . . . . . . 14 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> { 0 } C_ CC )
18	15, 17	unssd 3339	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( S u. { 0 } ) C_ CC )
19	13, 18	fssd 5420	. . . . . . . . . . . 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 10189	. . . . . . . . . . . 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 5698	. . . . . . . . . 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 14518	. . . . . . . . 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 14805	. . . . . . . . . 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 14802	. . . . . . . . . 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 5931	. . . . . . . . 9 |- ( ( u = ( a ` k ) /\ v = ( z ^ k ) ) -> ( u x. v ) = ( ( a ` k ) x. ( z ^ k ) ) )
30	11, 24, 26, 11, 11, 28, 29	cnmpt12 14523	. . . . . . . 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 14804	. . . . . . 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 2273	. . . . 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 14831	. . . . 5 |- ( CC -cn-> CC ) = ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) )
35	33, 34	eleqtrrdi 2290	. . . 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 2622	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   E.wrex 2476   u. cun 3155   C_ wss 3157   {csn 3622   |-> cmpt 4094   -->wf 5254   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   ^m cmap 6707   CCcc 7877   0cc0 7879   x. cmul 7884   NN0cn0 9249   ...cfz 10083   ^cexp 10630   sum_csu 11518   TopOpenctopn 12911  ℂ_fldccnfld 14112  TopOnctopon 14246   Cn ccn 14421   tX ctx 14488   -cn->ccncf 14806  Polycply 14964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999  ax-addf 8001

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-map 6709  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-dec 9458  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519  df-struct 12680  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mulr 12769  df-starv 12770  df-tset 12774  df-ple 12775  df-ds 12777  df-unif 12778  df-rest 12912  df-topn 12913  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-fg 14105  df-metu 14106  df-cnfld 14113  df-top 14234  df-topon 14247  df-topsp 14267  df-bases 14279  df-cn 14424  df-cnp 14425  df-tx 14489  df-xms 14575  df-ms 14576  df-cncf 14807  df-ply 14966

This theorem is referenced by: (None)