Theorem plycn 15444

Description: A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8130. (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 15416	. . 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 2229	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
5	4	cnfldtopon 15222	. . . . . . . . 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 9466	. . . . . . . . 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 9575	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> d e. ZZ )
10	7, 9	fzfigd 10661	. . . . . . . 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 6825	. . . . . . . . . . . . . 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 15415	. . . . . . . . . . . . . . 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 8147	. . . . . . . . . . . . . . 15 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> 0 e. CC )
17	16	snssd 3813	. . . . . . . . . . . . . 14 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> { 0 } C_ CC )
18	15, 17	unssd 3380	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( S u. { 0 } ) C_ CC )
19	13, 18	fssd 5486	. . . . . . . . . . . 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 10318	. . . . . . . . . . . 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 5773	. . . . . . . . . 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 14964	. . . . . . . . 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 15251	. . . . . . . . . 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 15248	. . . . . . . . . 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 6016	. . . . . . . . 9 |- ( ( u = ( a ` k ) /\ v = ( z ^ k ) ) -> ( u x. v ) = ( ( a ` k ) x. ( z ^ k ) ) )
30	11, 24, 26, 11, 11, 28, 29	cnmpt12 14969	. . . . . . . 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 15250	. . . . . . 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 2306	. . . . 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 15277	. . . . 5 |- ( CC -cn-> CC ) = ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) )
35	33, 34	eleqtrrdi 2323	. . . 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 2656	. 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 1395   e. wcel 2200   E.wrex 2509   u. cun 3195   C_ wss 3197   {csn 3666   |-> cmpt 4145   -->wf 5314   ` cfv 5318  (class class class)co 6007   e. cmpo 6009   ^m cmap 6803   CCcc 8005   0cc0 8007   x. cmul 8012   NN0cn0 9377   ...cfz 10212   ^cexp 10768   sum_csu 11872   TopOpenctopn 13281  ℂ_fldccnfld 14528  TopOnctopon 14692   Cn ccn 14867   tX ctx 14934   -cn->ccncf 15252  Polycply 15410

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127  ax-addf 8129

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-map 6805  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-z 9455  df-dec 9587  df-uz 9731  df-q 9823  df-rp 9858  df-xneg 9976  df-xadd 9977  df-fz 10213  df-fzo 10347  df-seqfrec 10678  df-exp 10769  df-ihash 11006  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-clim 11798  df-sumdc 11873  df-struct 13042  df-ndx 13043  df-slot 13044  df-base 13046  df-plusg 13131  df-mulr 13132  df-starv 13133  df-tset 13137  df-ple 13138  df-ds 13140  df-unif 13141  df-rest 13282  df-topn 13283  df-topgen 13301  df-psmet 14515  df-xmet 14516  df-met 14517  df-bl 14518  df-mopn 14519  df-fg 14521  df-metu 14522  df-cnfld 14529  df-top 14680  df-topon 14693  df-topsp 14713  df-bases 14725  df-cn 14870  df-cnp 14871  df-tx 14935  df-xms 15021  df-ms 15022  df-cncf 15253  df-ply 15412

This theorem is referenced by: (None)