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Theorem 0dgrb 21713
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 2442 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
2 eqid 2442 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
31, 2coeid 21705 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
z ^ k ) ) ) )
43adantr 465 . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (deg `  F )  =  0 )
65oveq2d 6106 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
0 ... (deg `  F
) )  =  ( 0 ... 0 ) )
76sumeq1d 13177 . . . . . . . 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 10656 . . . . . . . . . 10  |-  0  e.  ZZ
9 exp0 11868 . . . . . . . . . . . . . 14  |-  ( z  e.  CC  ->  (
z ^ 0 )  =  1 )
109adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
z ^ 0 )  =  1 )
1110oveq2d 6106 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
121coef3 21699 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
13 0nn0 10593 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
14 ffvelrn 5840 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  0  e.  NN0 )  ->  (
(coeff `  F ) `  0 )  e.  CC )
1512, 13, 14sylancl 662 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  S
)  ->  ( (coeff `  F ) `  0
)  e.  CC )
1615ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
(coeff `  F ) `  0 )  e.  CC )
1716mulid1d 9402 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
1811, 17eqtrd 2474 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
1918, 16eqeltrd 2516 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  e.  CC )
20 fveq2 5690 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
21 oveq2 6098 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
z ^ k )  =  ( z ^
0 ) )
2220, 21oveq12d 6108 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) ) )
2322fsum1 13217 . . . . . . . . . 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 663 . . . . . . . . 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 2474 . . . . . . . 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 2474 . . . . . . 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 4377 . . . . . 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 2474 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
29 fconstmpt 4881 . . . . 5  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
3028, 29syl6eqr 2492 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( CC  X.  { ( (coeff `  F ) `  0 ) } ) )
3130fveq1d 5692 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( ( CC  X.  { ( (coeff `  F ) `  0
) } ) ` 
0 ) )
32 0cn 9377 . . . . . . . 8  |-  0  e.  CC
33 fvex 5700 . . . . . . . . 9  |-  ( (coeff `  F ) `  0
)  e.  _V
3433fvconst2 5932 . . . . . . . 8  |-  ( 0  e.  CC  ->  (
( CC  X.  {
( (coeff `  F
) `  0 ) } ) `  0
)  =  ( (coeff `  F ) `  0
) )
3532, 34ax-mp 5 . . . . . . 7  |-  ( ( CC  X.  { ( (coeff `  F ) `  0 ) } ) `  0 )  =  ( (coeff `  F ) `  0
)
3631, 35syl6eq 2490 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( (coeff `  F
) `  0 )
)
3736sneqd 3888 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  { ( F `  0 ) }  =  { (
(coeff `  F ) `  0 ) } )
3837xpeq2d 4863 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( CC  X.  { ( F ` 
0 ) } )  =  ( CC  X.  { ( (coeff `  F ) `  0
) } ) )
3930, 38eqtr4d 2477 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( CC  X.  { ( F `  0 ) } ) )
4039ex 434 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( (deg `  F )  =  0  ->  F  =  ( CC  X.  { ( F `  0 ) } ) ) )
41 plyf 21665 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
42 ffvelrn 5840 . . . . 5  |-  ( ( F : CC --> CC  /\  0  e.  CC )  ->  ( F `  0
)  e.  CC )
4341, 32, 42sylancl 662 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  e.  CC )
44 0dgr 21712 . . . 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 5690 . . . 4  |-  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  (deg `  F
)  =  (deg `  ( CC  X.  { ( F `  0 ) } ) ) )
4746eqeq1d 2450 . . 3  |-  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  ( (deg `  F )  =  0  <-> 
(deg `  ( CC  X.  { ( F ` 
0 ) } ) )  =  0 ) )
4845, 47syl5ibrcom 222 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  (deg `  F
)  =  0 ) )
4940, 48impbid 191 1  |-  ( F  e.  (Poly `  S
)  ->  ( (deg `  F )  =  0  <-> 
F  =  ( CC 
X.  { ( F `
 0 ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3876    e. cmpt 4349    X. cxp 4837   -->wf 5413   ` cfv 5417  (class class class)co 6090   CCcc 9279   0cc0 9281   1c1 9282    x. cmul 9286   NN0cn0 10578   ZZcz 10645   ...cfz 11436   ^cexp 11864   sum_csu 13162  Polycply 21651  coeffccoe 21653  degcdgr 21654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-rlim 12966  df-sum 13163  df-0p 21147  df-ply 21655  df-coe 21657  df-dgr 21658
This theorem is referenced by:  dgreq0  21731  dgrcolem2  21740  dgrco  21741  plyrem  21770  fta1  21773  aaliou2  21805  dgrnznn  29490
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