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Theorem coe1termlem 22522
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 3528 . . . 4  |-  CC  C_  CC
2 coe1term.1 . . . . 5  |-  F  =  ( z  e.  CC  |->  ( A  x.  (
z ^ N ) ) )
32ply1term 22469 . . . 4  |-  ( ( CC  C_  CC  /\  A  e.  CC  /\  N  e. 
NN0 )  ->  F  e.  (Poly `  CC )
)
41, 3mp3an1 1311 . . 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 9600 . . . . . 6  |-  0  e.  CC
8 ifcl 3987 . . . . . 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 2467 . . . 4  |-  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )  =  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )
1210, 11fmptd 6056 . . 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 3987 . . . . . . . . . 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 2471 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  =  N  <->  k  =  N ) )
1817ifbid 3967 . . . . . . . . 9  |-  ( n  =  k  ->  if ( n  =  N ,  A ,  0 )  =  if ( k  =  N ,  A ,  0 ) )
1918, 11fvmptg 5955 . . . . . . . 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 2744 . . . . . 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 10816 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  RR )
2322leidd 10131 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  <_  N )
2423ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  N  <_  N
)
25 iffalse 3954 . . . . . . . . 9  |-  ( -.  k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  0 )
2625necon1ai 2698 . . . . . . . 8  |-  ( if ( k  =  N ,  A ,  0 )  =/=  0  -> 
k  =  N )
2726breq1d 4463 . . . . . . 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 2881 . . . 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 22457 . . . . 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 22468 . . . 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 11782 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3620oveq1d 6310 . . . . . . 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 13505 . . . . 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 4540 . . . 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 2511 . . 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 22492 . 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 3951 . . . . . . . 8  |-  ( n  =  N  ->  if ( n  =  N ,  A ,  0 )  =  A )
4847, 11fvmptg 5955 . . . . . . 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 2744 . . . . 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 22509 . . 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2817    C_ wss 3481   ifcif 3945   {csn 4033   class class class wbr 4453    |-> cmpt 4511   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    <_ cle 9641   NN0cn0 10807   ZZ>=cuz 11094   ...cfz 11684   ^cexp 12146   sum_csu 13488  Polycply 22449  coeffccoe 22451  degcdgr 22452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-0p 21945  df-ply 22453  df-coe 22455  df-dgr 22456
This theorem is referenced by:  coe1term  22523  dgr1term  22524
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