MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coe1termlem Structured version   Unicode version

Theorem coe1termlem 21700
Description: The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
coe1term.1  |-  F  =  ( z  e.  CC  |->  ( A  x.  (
z ^ N ) ) )
Assertion
Ref Expression
coe1termlem  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( (coeff `  F
)  =  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )  /\  ( A  =/=  0  ->  (deg `  F )  =  N ) ) )
Distinct variable groups:    z, n, A    n, N, z
Allowed substitution hints:    F( z, n)

Proof of Theorem coe1termlem
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 ssid 3370 . . . 4  |-  CC  C_  CC
2 coe1term.1 . . . . 5  |-  F  =  ( z  e.  CC  |->  ( A  x.  (
z ^ N ) ) )
32ply1term 21647 . . . 4  |-  ( ( CC  C_  CC  /\  A  e.  CC  /\  N  e. 
NN0 )  ->  F  e.  (Poly `  CC )
)
41, 3mp3an1 1301 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  F  e.  (Poly `  CC ) )
5 simpr 461 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  NN0 )
6 simpl 457 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  A  e.  CC )
7 0cn 9370 . . . . . 6  |-  0  e.  CC
8 ifcl 3826 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( n  =  N ,  A , 
0 )  e.  CC )
96, 7, 8sylancl 662 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  if ( n  =  N ,  A ,  0 )  e.  CC )
109adantr 465 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  n  e.  NN0 )  ->  if ( n  =  N ,  A ,  0 )  e.  CC )
11 eqid 2438 . . . 4  |-  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )  =  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )
1210, 11fmptd 5862 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) : NN0 --> CC )
13 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
14 ifcl 3826 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( k  =  N ,  A , 
0 )  e.  CC )
156, 7, 14sylancl 662 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
1615adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
17 eqeq1 2444 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  =  N  <->  k  =  N ) )
1817ifbid 3806 . . . . . . . . 9  |-  ( n  =  k  ->  if ( n  =  N ,  A ,  0 )  =  if ( k  =  N ,  A ,  0 ) )
1918, 11fvmptg 5767 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  if ( k  =  N ,  A ,  0 )  e.  CC )  ->  ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =  if ( k  =  N ,  A ,  0 ) )
2013, 16, 19syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =  if ( k  =  N ,  A ,  0 ) )
2120neeq1d 2616 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =/=  0  <->  if (
k  =  N ,  A ,  0 )  =/=  0 ) )
22 nn0re 10580 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  RR )
2322leidd 9898 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  <_  N )
2423ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  N  <_  N
)
25 iffalse 3794 . . . . . . . . 9  |-  ( -.  k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  0 )
2625necon1ai 2648 . . . . . . . 8  |-  ( if ( k  =  N ,  A ,  0 )  =/=  0  -> 
k  =  N )
2726breq1d 4297 . . . . . . 7  |-  ( if ( k  =  N ,  A ,  0 )  =/=  0  -> 
( k  <_  N  <->  N  <_  N ) )
2824, 27syl5ibrcom 222 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( if ( k  =  N ,  A ,  0 )  =/=  0  ->  k  <_  N ) )
2921, 28sylbid 215 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =/=  0  ->  k  <_  N ) )
3029ralrimiva 2794 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  A. k  e.  NN0  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =/=  0  ->  k  <_  N ) )
31 plyco0 21635 . . . . 5  |-  ( ( N  e.  NN0  /\  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) : NN0 --> CC )  ->  ( (
( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =/=  0  ->  k  <_  N ) ) )
325, 12, 31syl2anc 661 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) " ( ZZ>= `  ( N  +  1
) ) )  =  { 0 }  <->  A. k  e.  NN0  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =/=  0  ->  k  <_  N ) ) )
3330, 32mpbird 232 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) )
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 } )
342ply1termlem 21646 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k
) ) ) )
35 elfznn0 11473 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3620oveq1d 6101 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )  =  ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k
) ) )
3735, 36sylan2 474 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  (
0 ... N ) )  ->  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) )  =  ( if ( k  =  N ,  A ,  0 )  x.  ( z ^ k
) ) )
3837sumeq2dv 13172 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  sum_ k  e.  ( 0 ... N ) ( ( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( if ( k  =  N ,  A , 
0 )  x.  (
z ^ k ) ) )
3938mpteq2dv 4374 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 k )  x.  ( z ^ k
) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( if ( k  =  N ,  A , 
0 )  x.  (
z ^ k ) ) ) )
4034, 39eqtr4d 2473 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  x.  ( z ^
k ) ) ) )
414, 5, 12, 33, 40coeeq 21670 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
(coeff `  F )  =  ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) )
424adantr 465 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  F  e.  (Poly `  CC ) )
435adantr 465 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  N  e.  NN0 )
4412adantr 465 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) : NN0 --> CC )
4533adantr 465 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  ( (
n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) " ( ZZ>= `  ( N  +  1
) ) )  =  { 0 } )
4640adantr 465 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 k )  x.  ( z ^ k
) ) ) )
47 iftrue 3792 . . . . . . . 8  |-  ( n  =  N  ->  if ( n  =  N ,  A ,  0 )  =  A )
4847, 11fvmptg 5767 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  CC )  ->  ( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 N )  =  A )
4948ancoms 453 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 N )  =  A )
5049neeq1d 2616 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  N )  =/=  0  <->  A  =/=  0 ) )
5150biimpar 485 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  ( (
n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  N )  =/=  0 )
5242, 43, 44, 45, 46, 51dgreq 21687 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  (deg `  F
)  =  N )
5352ex 434 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A  =/=  0  ->  (deg `  F )  =  N ) )
5441, 53jca 532 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( (coeff `  F
)  =  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )  /\  ( A  =/=  0  ->  (deg `  F )  =  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710    C_ wss 3323   ifcif 3786   {csn 3872   class class class wbr 4287    e. cmpt 4345   "cima 4838   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    <_ cle 9411   NN0cn0 10571   ZZ>=cuz 10853   ...cfz 11429   ^cexp 11857   sum_csu 13155  Polycply 21627  coeffccoe 21629  degcdgr 21630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-0p 21123  df-ply 21631  df-coe 21633  df-dgr 21634
This theorem is referenced by:  coe1term  21701  dgr1term  21702
  Copyright terms: Public domain W3C validator