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Theorem dvnply2 19765
Description: Polynomials have polynomials as derivatives of all orders. (Contributed by Mario Carneiro, 1-Jan-2017.)
Assertion
Ref Expression
dvnply2  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  ( ( CC  D n F ) `
 N )  e.  (Poly `  S )
)

Proof of Theorem dvnply2
Dummy variables  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5605 . . . . . 6  |-  ( x  =  0  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  0 ) )
21eleq1d 2424 . . . . 5  |-  ( x  =  0  ->  (
( ( CC  D n F ) `  x
)  e.  (Poly `  S )  <->  ( ( CC  D n F ) `
 0 )  e.  (Poly `  S )
) )
32imbi2d 307 . . . 4  |-  ( x  =  0  ->  (
( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 x )  e.  (Poly `  S )
)  <->  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 0 )  e.  (Poly `  S )
) ) )
4 fveq2 5605 . . . . . 6  |-  ( x  =  n  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  n ) )
54eleq1d 2424 . . . . 5  |-  ( x  =  n  ->  (
( ( CC  D n F ) `  x
)  e.  (Poly `  S )  <->  ( ( CC  D n F ) `
 n )  e.  (Poly `  S )
) )
65imbi2d 307 . . . 4  |-  ( x  =  n  ->  (
( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 x )  e.  (Poly `  S )
)  <->  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 n )  e.  (Poly `  S )
) ) )
7 fveq2 5605 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  ( n  +  1 ) ) )
87eleq1d 2424 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  D n F ) `  x
)  e.  (Poly `  S )  <->  ( ( CC  D n F ) `
 ( n  + 
1 ) )  e.  (Poly `  S )
) )
98imbi2d 307 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 x )  e.  (Poly `  S )
)  <->  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 ( n  + 
1 ) )  e.  (Poly `  S )
) ) )
10 fveq2 5605 . . . . . 6  |-  ( x  =  N  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  N ) )
1110eleq1d 2424 . . . . 5  |-  ( x  =  N  ->  (
( ( CC  D n F ) `  x
)  e.  (Poly `  S )  <->  ( ( CC  D n F ) `
 N )  e.  (Poly `  S )
) )
1211imbi2d 307 . . . 4  |-  ( x  =  N  ->  (
( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 x )  e.  (Poly `  S )
)  <->  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 N )  e.  (Poly `  S )
) ) )
13 ssid 3273 . . . . . 6  |-  CC  C_  CC
14 cnex 8905 . . . . . . . 8  |-  CC  e.  _V
1514a1i 10 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  CC  e.  _V )
16 plyf 19678 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
1716adantl 452 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F : CC
--> CC )
18 fpmg 6878 . . . . . . 7  |-  ( ( CC  e.  _V  /\  CC  e.  _V  /\  F : CC --> CC )  ->  F  e.  ( CC  ^pm 
CC ) )
1915, 15, 17, 18syl3anc 1182 . . . . . 6  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F  e.  ( CC  ^pm  CC ) )
20 dvn0 19371 . . . . . 6  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  D n F ) `  0
)  =  F )
2113, 19, 20sylancr 644 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 0 )  =  F )
22 simpr 447 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
2321, 22eqeltrd 2432 . . . 4  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 0 )  e.  (Poly `  S )
)
24 dvply2g 19763 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  (
( CC  D n F ) `  n
)  e.  (Poly `  S ) )  -> 
( CC  _D  (
( CC  D n F ) `  n
) )  e.  (Poly `  S ) )
2524ex 423 . . . . . . . 8  |-  ( S  e.  (SubRing ` fld )  ->  ( ( ( CC  D n F ) `  n
)  e.  (Poly `  S )  ->  ( CC  _D  ( ( CC  D n F ) `
 n ) )  e.  (Poly `  S
) ) )
2625ad2antrr 706 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( ( CC  D n F ) `
 n )  e.  (Poly `  S )  ->  ( CC  _D  (
( CC  D n F ) `  n
) )  e.  (Poly `  S ) ) )
27 dvnp1 19372 . . . . . . . . . 10  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  ( CC  _D  (
( CC  D n F ) `  n
) ) )
2813, 27mp3an1 1264 . . . . . . . . 9  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  n  e.  NN0 )  ->  (
( CC  D n F ) `  (
n  +  1 ) )  =  ( CC 
_D  ( ( CC  D n F ) `
 n ) ) )
2919, 28sylan 457 . . . . . . . 8  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( CC  D n F ) `  (
n  +  1 ) )  =  ( CC 
_D  ( ( CC  D n F ) `
 n ) ) )
3029eleq1d 2424 . . . . . . 7  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( ( CC  D n F ) `
 ( n  + 
1 ) )  e.  (Poly `  S )  <->  ( CC  _D  ( ( CC  D n F ) `  n ) )  e.  (Poly `  S ) ) )
3126, 30sylibrd 225 . . . . . 6  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  n  e.  NN0 )  -> 
( ( ( CC  D n F ) `
 n )  e.  (Poly `  S )  ->  ( ( CC  D n F ) `  (
n  +  1 ) )  e.  (Poly `  S ) ) )
3231expcom 424 . . . . 5  |-  ( n  e.  NN0  ->  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( (
( CC  D n F ) `  n
)  e.  (Poly `  S )  ->  (
( CC  D n F ) `  (
n  +  1 ) )  e.  (Poly `  S ) ) ) )
3332a2d 23 . . . 4  |-  ( n  e.  NN0  ->  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  -> 
( ( CC  D n F ) `  n
)  e.  (Poly `  S ) )  -> 
( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 ( n  + 
1 ) )  e.  (Poly `  S )
) ) )
343, 6, 9, 12, 23, 33nn0ind 10197 . . 3  |-  ( N  e.  NN0  ->  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 N )  e.  (Poly `  S )
) )
3534impcom 419 . 2  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  N  e.  NN0 )  -> 
( ( CC  D n F ) `  N
)  e.  (Poly `  S ) )
36353impa 1146 1  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  ( ( CC  D n F ) `
 N )  e.  (Poly `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   _Vcvv 2864    C_ wss 3228   -->wf 5330   ` cfv 5334  (class class class)co 5942    ^pm cpm 6858   CCcc 8822   0cc0 8824   1c1 8825    + caddc 8827   NN0cn0 10054  SubRingcsubrg 15634  ℂfldccnfld 16476    _D cdv 19311    D ncdvn 19312  Polycply 19664
This theorem is referenced by:  dvnply  19766  taylthlem2  19851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-icc 10752  df-fz 10872  df-fzo 10960  df-fl 11014  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-rlim 12053  df-sum 12250  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-hom 13323  df-cco 13324  df-rest 13420  df-topn 13421  df-topgen 13437  df-pt 13438  df-prds 13441  df-xrs 13496  df-0g 13497  df-gsum 13498  df-qtop 13503  df-imas 13504  df-xps 13506  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-submnd 14509  df-grp 14582  df-minusg 14583  df-mulg 14585  df-subg 14711  df-cntz 14886  df-cmn 15184  df-mgp 15419  df-rng 15433  df-cring 15434  df-ur 15435  df-subrg 15636  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-cnfld 16477  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lp 16968  df-perf 16969  df-cn 17057  df-cnp 17058  df-haus 17143  df-tx 17357  df-hmeo 17546  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-xms 17981  df-ms 17982  df-tms 17983  df-cncf 18479  df-0p 19123  df-limc 19314  df-dv 19315  df-dvn 19316  df-ply 19668  df-coe 19670  df-dgr 19671
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