Theorem plycn 15556

Description: A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8198. (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 15528	. . 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 2231	. . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
5	4	cnfldtopon 15334	. . . . . . . . 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 9536	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> 0 e. ZZ )
8		simprl 531	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> d e. NN0 )
9	8	nn0zd 9645	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> d e. ZZ )
10	7, 9	fzfigd 10739	. . . . . . . 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 6882	. . . . . . . . . . . . . 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 15527	. . . . . . . . . . . . . . 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 8215	. . . . . . . . . . . . . . 15 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> 0 e. CC )
17	16	snssd 3823	. . . . . . . . . . . . . 14 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> { 0 } C_ CC )
18	15, 17	unssd 3385	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` S ) /\ ( d e. NN0 /\ a e. ( ( S u. { 0 } ) ^m NN0 ) ) ) -> ( S u. { 0 } ) C_ CC )
19	13, 18	fssd 5502	. . . . . . . . . . . 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 10394	. . . . . . . . . . . 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 5791	. . . . . . . . . 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 15076	. . . . . . . . 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 15363	. . . . . . . . . 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 15360	. . . . . . . . . 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 6037	. . . . . . . . 9 |- ( ( u = ( a ` k ) /\ v = ( z ^ k ) ) -> ( u x. v ) = ( ( a ` k ) x. ( z ^ k ) ) )
30	11, 24, 26, 11, 11, 28, 29	cnmpt12 15081	. . . . . . . 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 15362	. . . . . . 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 2308	. . . . 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 15389	. . . . 5 |- ( CC -cn-> CC ) = ( ( TopOpen ` ℂfld ) Cn ( TopOpen ` ℂfld ) )
35	33, 34	eleqtrrdi 2325	. . . 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 2659	. 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 1398   e. wcel 2202   E.wrex 2512   u. cun 3199   C_ wss 3201   {csn 3673   |-> cmpt 4155   -->wf 5329   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   ^m cmap 6860   CCcc 8073   0cc0 8075   x. cmul 8080   NN0cn0 9445   ...cfz 10288   ^cexp 10846   sum_csu 11976   TopOpenctopn 13386  ℂ_fldccnfld 14635  TopOnctopon 14804   Cn ccn 14979   tX ctx 15046   -cn->ccncf 15364  Polycply 15522

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195  ax-addf 8197

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7226  df-inf 7227  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9525  df-dec 9657  df-uz 9801  df-q 9899  df-rp 9934  df-xneg 10052  df-xadd 10053  df-fz 10289  df-fzo 10423  df-seqfrec 10756  df-exp 10847  df-ihash 11084  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-sumdc 11977  df-struct 13147  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-mulr 13237  df-starv 13238  df-tset 13242  df-ple 13243  df-ds 13245  df-unif 13246  df-rest 13387  df-topn 13388  df-topgen 13406  df-psmet 14622  df-xmet 14623  df-met 14624  df-bl 14625  df-mopn 14626  df-fg 14628  df-metu 14629  df-cnfld 14636  df-top 14792  df-topon 14805  df-topsp 14825  df-bases 14837  df-cn 14982  df-cnp 14983  df-tx 15047  df-xms 15133  df-ms 15134  df-cncf 15365  df-ply 15524

This theorem is referenced by: (None)