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Theorem dvnply2 20165
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 5695 . . . . . 6  |-  ( x  =  0  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  0 ) )
21eleq1d 2478 . . . . 5  |-  ( x  =  0  ->  (
( ( CC  D n F ) `  x
)  e.  (Poly `  S )  <->  ( ( CC  D n F ) `
 0 )  e.  (Poly `  S )
) )
32imbi2d 308 . . . 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 5695 . . . . . 6  |-  ( x  =  n  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  n ) )
54eleq1d 2478 . . . . 5  |-  ( x  =  n  ->  (
( ( CC  D n F ) `  x
)  e.  (Poly `  S )  <->  ( ( CC  D n F ) `
 n )  e.  (Poly `  S )
) )
65imbi2d 308 . . . 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 5695 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  ( n  +  1 ) ) )
87eleq1d 2478 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  D n F ) `  x
)  e.  (Poly `  S )  <->  ( ( CC  D n F ) `
 ( n  + 
1 ) )  e.  (Poly `  S )
) )
98imbi2d 308 . . . 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 5695 . . . . . 6  |-  ( x  =  N  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  N ) )
1110eleq1d 2478 . . . . 5  |-  ( x  =  N  ->  (
( ( CC  D n F ) `  x
)  e.  (Poly `  S )  <->  ( ( CC  D n F ) `
 N )  e.  (Poly `  S )
) )
1211imbi2d 308 . . . 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 3335 . . . . . 6  |-  CC  C_  CC
14 cnex 9035 . . . . . . . 8  |-  CC  e.  _V
1514a1i 11 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  CC  e.  _V )
16 plyf 20078 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
1716adantl 453 . . . . . . 7  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F : CC
--> CC )
18 fpmg 7006 . . . . . . 7  |-  ( ( CC  e.  _V  /\  CC  e.  _V  /\  F : CC --> CC )  ->  F  e.  ( CC  ^pm 
CC ) )
1915, 15, 17, 18syl3anc 1184 . . . . . 6  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F  e.  ( CC  ^pm  CC ) )
20 dvn0 19771 . . . . . 6  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  D n F ) `  0
)  =  F )
2113, 19, 20sylancr 645 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 0 )  =  F )
22 simpr 448 . . . . 5  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  F  e.  (Poly `  S ) )
2321, 22eqeltrd 2486 . . . 4  |-  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 0 )  e.  (Poly `  S )
)
24 dvply2g 20163 . . . . . . . . 9  |-  ( ( S  e.  (SubRing ` fld )  /\  (
( CC  D n F ) `  n
)  e.  (Poly `  S ) )  -> 
( CC  _D  (
( CC  D n F ) `  n
) )  e.  (Poly `  S ) )
2524ex 424 . . . . . . . 8  |-  ( S  e.  (SubRing ` fld )  ->  ( ( ( CC  D n F ) `  n
)  e.  (Poly `  S )  ->  ( CC  _D  ( ( CC  D n F ) `
 n ) )  e.  (Poly `  S
) ) )
2625ad2antrr 707 . . . . . . 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 19772 . . . . . . . . . 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 1266 . . . . . . . . 9  |-  ( ( F  e.  ( CC 
^pm  CC )  /\  n  e.  NN0 )  ->  (
( CC  D n F ) `  (
n  +  1 ) )  =  ( CC 
_D  ( ( CC  D n F ) `
 n ) ) )
2919, 28sylan 458 . . . . . . . 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 2478 . . . . . . 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 226 . . . . . 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 425 . . . . 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 24 . . . 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 10330 . . 3  |-  ( N  e.  NN0  ->  ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S )
)  ->  ( ( CC  D n F ) `
 N )  e.  (Poly `  S )
) )
3534impcom 420 . 2  |-  ( ( ( S  e.  (SubRing ` fld )  /\  F  e.  (Poly `  S ) )  /\  N  e.  NN0 )  -> 
( ( CC  D n F ) `  N
)  e.  (Poly `  S ) )
36353impa 1148 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924    C_ wss 3288   -->wf 5417   ` cfv 5421  (class class class)co 6048    ^pm cpm 6986   CCcc 8952   0cc0 8954   1c1 8955    + caddc 8957   NN0cn0 10185  SubRingcsubrg 15827  ℂfldccnfld 16666    _D cdv 19711    D ncdvn 19712  Polycply 20064
This theorem is referenced by:  dvnply  20166  taylthlem2  20251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-mulg 14778  df-subg 14904  df-cntz 15079  df-cmn 15377  df-mgp 15612  df-rng 15626  df-cring 15627  df-ur 15628  df-subrg 15829  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-0p 19523  df-limc 19714  df-dv 19715  df-dvn 19716  df-ply 20068  df-coe 20070  df-dgr 20071
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