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Theorem coe1termlem 20168
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 3359 . . . 4  |-  CC  C_  CC
2 coe1term.1 . . . . 5  |-  F  =  ( z  e.  CC  |->  ( A  x.  (
z ^ N ) ) )
32ply1term 20115 . . . 4  |-  ( ( CC  C_  CC  /\  A  e.  CC  /\  N  e. 
NN0 )  ->  F  e.  (Poly `  CC )
)
41, 3mp3an1 1266 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  F  e.  (Poly `  CC ) )
5 simpr 448 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  NN0 )
6 simpl 444 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  A  e.  CC )
7 0cn 9076 . . . . . 6  |-  0  e.  CC
8 ifcl 3767 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( n  =  N ,  A , 
0 )  e.  CC )
96, 7, 8sylancl 644 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  if ( n  =  N ,  A ,  0 )  e.  CC )
109adantr 452 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  n  e.  NN0 )  ->  if ( n  =  N ,  A ,  0 )  e.  CC )
11 eqid 2435 . . . 4  |-  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )  =  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )
1210, 11fmptd 5885 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) : NN0 --> CC )
13 simpr 448 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
14 ifcl 3767 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( k  =  N ,  A , 
0 )  e.  CC )
156, 7, 14sylancl 644 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
1615adantr 452 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
17 eqeq1 2441 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  =  N  <->  k  =  N ) )
1817ifbid 3749 . . . . . . . . 9  |-  ( n  =  k  ->  if ( n  =  N ,  A ,  0 )  =  if ( k  =  N ,  A ,  0 ) )
1918, 11fvmptg 5796 . . . . . . . 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 643 . . . . . . 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 2611 . . . . . 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 10222 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  RR )
2322leidd 9585 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  <_  N )
2423ad2antlr 708 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  N  <_  N
)
25 iffalse 3738 . . . . . . . . 9  |-  ( -.  k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  0 )
2625necon1ai 2640 . . . . . . . 8  |-  ( if ( k  =  N ,  A ,  0 )  =/=  0  -> 
k  =  N )
2726breq1d 4214 . . . . . . 7  |-  ( if ( k  =  N ,  A ,  0 )  =/=  0  -> 
( k  <_  N  <->  N  <_  N ) )
2824, 27syl5ibrcom 214 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( if ( k  =  N ,  A ,  0 )  =/=  0  ->  k  <_  N ) )
2921, 28sylbid 207 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =/=  0  ->  k  <_  N ) )
3029ralrimiva 2781 . . . 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 20103 . . . . 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 643 . . . 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 224 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) )
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 } )
342ply1termlem 20114 . . . 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 11075 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3620oveq1d 6088 . . . . . . 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 461 . . . . . 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 12489 . . . . 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 4288 . . . 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 2470 . . 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 20138 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
(coeff `  F )  =  ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) )
424adantr 452 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  F  e.  (Poly `  CC ) )
435adantr 452 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  N  e.  NN0 )
4412adantr 452 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) : NN0 --> CC )
4533adantr 452 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  ( (
n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) " ( ZZ>= `  ( N  +  1
) ) )  =  { 0 } )
4640adantr 452 . . . 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 3737 . . . . . . . 8  |-  ( n  =  N  ->  if ( n  =  N ,  A ,  0 )  =  A )
4847, 11fvmptg 5796 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  CC )  ->  ( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 N )  =  A )
4948ancoms 440 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 N )  =  A )
5049neeq1d 2611 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  N )  =/=  0  <->  A  =/=  0 ) )
5150biimpar 472 . . . 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 20155 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  (deg `  F
)  =  N )
5352ex 424 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A  =/=  0  ->  (deg `  F )  =  N ) )
5441, 53jca 519 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    C_ wss 3312   ifcif 3731   {csn 3806   class class class wbr 4204    e. cmpt 4258   "cima 4873   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    <_ cle 9113   NN0cn0 10213   ZZ>=cuz 10480   ...cfz 11035   ^cexp 11374   sum_csu 12471  Polycply 20095  coeffccoe 20097  degcdgr 20098
This theorem is referenced by:  coe1term  20169  dgr1term  20170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
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