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Theorem 0dgrb 20157
Description: A function has degree zero iff it is a constant function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
0dgrb  |-  ( F  e.  (Poly `  S
)  ->  ( (deg `  F )  =  0  <-> 
F  =  ( CC 
X.  { ( F `
 0 ) } ) ) )

Proof of Theorem 0dgrb
Dummy variables  z 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
2 eqid 2435 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
31, 2coeid 20149 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
z ^ k ) ) ) )
43adantr 452 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
z ^ k ) ) ) )
5 simplr 732 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (deg `  F )  =  0 )
65oveq2d 6089 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
0 ... (deg `  F
) )  =  ( 0 ... 0 ) )
76sumeq1d 12487 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) )  =  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) )
8 0z 10285 . . . . . . . . . 10  |-  0  e.  ZZ
9 exp0 11378 . . . . . . . . . . . . . 14  |-  ( z  e.  CC  ->  (
z ^ 0 )  =  1 )
109adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
z ^ 0 )  =  1 )
1110oveq2d 6089 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
121coef3 20143 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
13 0nn0 10228 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
14 ffvelrn 5860 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  0  e.  NN0 )  ->  (
(coeff `  F ) `  0 )  e.  CC )
1512, 13, 14sylancl 644 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  S
)  ->  ( (coeff `  F ) `  0
)  e.  CC )
1615ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
(coeff `  F ) `  0 )  e.  CC )
1716mulid1d 9097 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
1811, 17eqtrd 2467 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
1918, 16eqeltrd 2509 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  e.  CC )
20 fveq2 5720 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
21 oveq2 6081 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
z ^ k )  =  ( z ^
0 ) )
2220, 21oveq12d 6091 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) ) )
2322fsum1 12527 . . . . . . . . . 10  |-  ( ( 0  e.  ZZ  /\  ( ( (coeff `  F ) `  0
)  x.  ( z ^ 0 ) )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) ) )
248, 19, 23sylancr 645 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( z ^ k ) )  =  ( ( (coeff `  F ) `  0
)  x.  ( z ^ 0 ) ) )
2524, 18eqtrd 2467 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... 0
) ( ( (coeff `  F ) `  k
)  x.  ( z ^ k ) )  =  ( (coeff `  F ) `  0
) )
267, 25eqtrd 2467 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) )  =  ( (coeff `  F ) `  0 ) )
2726mpteq2dva 4287 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  k )  x.  ( z ^ k
) ) )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
284, 27eqtrd 2467 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
29 fconstmpt 4913 . . . . 5  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
3028, 29syl6eqr 2485 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( CC  X.  { ( (coeff `  F ) `  0 ) } ) )
3130fveq1d 5722 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( ( CC  X.  { ( (coeff `  F ) `  0
) } ) ` 
0 ) )
32 0cn 9076 . . . . . . . 8  |-  0  e.  CC
33 fvex 5734 . . . . . . . . 9  |-  ( (coeff `  F ) `  0
)  e.  _V
3433fvconst2 5939 . . . . . . . 8  |-  ( 0  e.  CC  ->  (
( CC  X.  {
( (coeff `  F
) `  0 ) } ) `  0
)  =  ( (coeff `  F ) `  0
) )
3532, 34ax-mp 8 . . . . . . 7  |-  ( ( CC  X.  { ( (coeff `  F ) `  0 ) } ) `  0 )  =  ( (coeff `  F ) `  0
)
3631, 35syl6eq 2483 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( (coeff `  F
) `  0 )
)
3736sneqd 3819 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  { ( F `  0 ) }  =  { (
(coeff `  F ) `  0 ) } )
3837xpeq2d 4894 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( CC  X.  { ( F ` 
0 ) } )  =  ( CC  X.  { ( (coeff `  F ) `  0
) } ) )
3930, 38eqtr4d 2470 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( CC  X.  { ( F `  0 ) } ) )
4039ex 424 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( (deg `  F )  =  0  ->  F  =  ( CC  X.  { ( F `  0 ) } ) ) )
41 plyf 20109 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
42 ffvelrn 5860 . . . . 5  |-  ( ( F : CC --> CC  /\  0  e.  CC )  ->  ( F `  0
)  e.  CC )
4341, 32, 42sylancl 644 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  e.  CC )
44 0dgr 20156 . . . 4  |-  ( ( F `  0 )  e.  CC  ->  (deg `  ( CC  X.  {
( F `  0
) } ) )  =  0 )
4543, 44syl 16 . . 3  |-  ( F  e.  (Poly `  S
)  ->  (deg `  ( CC  X.  { ( F `
 0 ) } ) )  =  0 )
46 fveq2 5720 . . . 4  |-  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  (deg `  F
)  =  (deg `  ( CC  X.  { ( F `  0 ) } ) ) )
4746eqeq1d 2443 . . 3  |-  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  ( (deg `  F )  =  0  <-> 
(deg `  ( CC  X.  { ( F ` 
0 ) } ) )  =  0 ) )
4845, 47syl5ibrcom 214 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  (deg `  F
)  =  0 ) )
4940, 48impbid 184 1  |-  ( F  e.  (Poly `  S
)  ->  ( (deg `  F )  =  0  <-> 
F  =  ( CC 
X.  { ( F `
 0 ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806    e. cmpt 4258    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987   NN0cn0 10213   ZZcz 10274   ...cfz 11035   ^cexp 11374   sum_csu 12471  Polycply 20095  coeffccoe 20097  degcdgr 20098
This theorem is referenced by:  dgreq0  20175  dgrcolem2  20184  dgrco  20185  plyrem  20214  fta1  20217  aaliou2  20249  dgrnznn  27298
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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