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Theorem coe1termlem 21861
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 3486 . . . 4  |-  CC  C_  CC
2 coe1term.1 . . . . 5  |-  F  =  ( z  e.  CC  |->  ( A  x.  (
z ^ N ) ) )
32ply1term 21808 . . . 4  |-  ( ( CC  C_  CC  /\  A  e.  CC  /\  N  e. 
NN0 )  ->  F  e.  (Poly `  CC )
)
41, 3mp3an1 1302 . . 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 9492 . . . . . 6  |-  0  e.  CC
8 ifcl 3942 . . . . . 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 2454 . . . 4  |-  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )  =  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )
1210, 11fmptd 5979 . . 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 3942 . . . . . . . . . 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 2458 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  =  N  <->  k  =  N ) )
1817ifbid 3922 . . . . . . . . 9  |-  ( n  =  k  ->  if ( n  =  N ,  A ,  0 )  =  if ( k  =  N ,  A ,  0 ) )
1918, 11fvmptg 5884 . . . . . . . 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 2729 . . . . . 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 10702 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  RR )
2322leidd 10020 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  <_  N )
2423ad2antlr 726 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  N  <_  N
)
25 iffalse 3910 . . . . . . . . 9  |-  ( -.  k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  0 )
2625necon1ai 2683 . . . . . . . 8  |-  ( if ( k  =  N ,  A ,  0 )  =/=  0  -> 
k  =  N )
2726breq1d 4413 . . . . . . 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 2830 . . . 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 21796 . . . . 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 21807 . . . 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 11601 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3620oveq1d 6218 . . . . . . 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 13301 . . . . 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 4490 . . . 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 2498 . . 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 21831 . 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 3908 . . . . . . . 8  |-  ( n  =  N  ->  if ( n  =  N ,  A ,  0 )  =  A )
4847, 11fvmptg 5884 . . . . . . 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 2729 . . . . 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 21848 . . 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    C_ wss 3439   ifcif 3902   {csn 3988   class class class wbr 4403    |-> cmpt 4461   "cima 4954   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401    <_ cle 9533   NN0cn0 10693   ZZ>=cuz 10975   ...cfz 11557   ^cexp 11985   sum_csu 13284  Polycply 21788  coeffccoe 21790  degcdgr 21791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-addf 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-fzo 11669  df-fl 11762  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-rlim 13088  df-sum 13285  df-0p 21284  df-ply 21792  df-coe 21794  df-dgr 21795
This theorem is referenced by:  coe1term  21862  dgr1term  21863
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