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Theorem coe1termlem 19655
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 3210 . . . 4  |-  CC  C_  CC
2 coe1term.1 . . . . 5  |-  F  =  ( z  e.  CC  |->  ( A  x.  (
z ^ N ) ) )
32ply1term 19602 . . . 4  |-  ( ( CC  C_  CC  /\  A  e.  CC  /\  N  e. 
NN0 )  ->  F  e.  (Poly `  CC )
)
41, 3mp3an1 1264 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  F  e.  (Poly `  CC ) )
5 simpr 447 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  N  e.  NN0 )
6 simpl 443 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  A  e.  CC )
7 0cn 8847 . . . . . 6  |-  0  e.  CC
8 ifcl 3614 . . . . . 6  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( n  =  N ,  A , 
0 )  e.  CC )
96, 7, 8sylancl 643 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  if ( n  =  N ,  A ,  0 )  e.  CC )
109adantr 451 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  n  e.  NN0 )  ->  if ( n  =  N ,  A ,  0 )  e.  CC )
11 eqid 2296 . . . 4  |-  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )  =  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) )
1210, 11fmptd 5700 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) : NN0 --> CC )
13 simpr 447 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  k  e.  NN0 )
14 ifcl 3614 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  CC )  ->  if ( k  =  N ,  A , 
0 )  e.  CC )
156, 7, 14sylancl 643 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
1615adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  if ( k  =  N ,  A ,  0 )  e.  CC )
17 eqeq1 2302 . . . . . . . . . 10  |-  ( n  =  k  ->  (
n  =  N  <->  k  =  N ) )
1817ifbid 3596 . . . . . . . . 9  |-  ( n  =  k  ->  if ( n  =  N ,  A ,  0 )  =  if ( k  =  N ,  A ,  0 ) )
1918, 11fvmptg 5616 . . . . . . . 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 642 . . . . . . 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 2472 . . . . . 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 9990 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  RR )
2322leidd 9355 . . . . . . . 8  |-  ( N  e.  NN0  ->  N  <_  N )
2423ad2antlr 707 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  N  <_  N
)
25 iffalse 3585 . . . . . . . . 9  |-  ( -.  k  =  N  ->  if ( k  =  N ,  A ,  0 )  =  0 )
2625necon1ai 2501 . . . . . . . 8  |-  ( if ( k  =  N ,  A ,  0 )  =/=  0  -> 
k  =  N )
2726breq1d 4049 . . . . . . 7  |-  ( if ( k  =  N ,  A ,  0 )  =/=  0  -> 
( k  <_  N  <->  N  <_  N ) )
2824, 27syl5ibrcom 213 . . . . . 6  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( if ( k  =  N ,  A ,  0 )  =/=  0  ->  k  <_  N ) )
2921, 28sylbid 206 . . . . 5  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  k  e.  NN0 )  ->  ( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  k )  =/=  0  ->  k  <_  N ) )
3029ralrimiva 2639 . . . 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 19590 . . . . 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 642 . . . 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 223 . . 3  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) )
" ( ZZ>= `  ( N  +  1 ) ) )  =  {
0 } )
342ply1termlem 19601 . . . 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 10838 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
3620oveq1d 5889 . . . . . . 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 460 . . . . . 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 12192 . . . . 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 4123 . . . 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 2331 . . 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 19625 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
(coeff `  F )  =  ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) )
424adantr 451 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  F  e.  (Poly `  CC ) )
435adantr 451 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  N  e.  NN0 )
4412adantr 451 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) : NN0 --> CC )
4533adantr 451 . . . 4  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  ( (
n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) " ( ZZ>= `  ( N  +  1
) ) )  =  { 0 } )
4640adantr 451 . . . 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 3584 . . . . . . . 8  |-  ( n  =  N  ->  if ( n  =  N ,  A ,  0 )  =  A )
4847, 11fvmptg 5616 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  CC )  ->  ( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 N )  =  A )
4948ancoms 439 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  if ( n  =  N ,  A ,  0 ) ) `
 N )  =  A )
5049neeq1d 2472 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( ( ( n  e.  NN0  |->  if ( n  =  N ,  A ,  0 ) ) `  N )  =/=  0  <->  A  =/=  0 ) )
5150biimpar 471 . . . 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 19642 . . 3  |-  ( ( ( A  e.  CC  /\  N  e.  NN0 )  /\  A  =/=  0
)  ->  (deg `  F
)  =  N )
5352ex 423 . 2  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( A  =/=  0  ->  (deg `  F )  =  N ) )
5441, 53jca 518 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   ifcif 3578   {csn 3653   class class class wbr 4039    e. cmpt 4093   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884   NN0cn0 9981   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120   sum_csu 12174  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  coe1term  19656  dgr1term  19657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
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