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Theorem 0dgrb 22406
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 2467 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
2 eqid 2467 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
31, 2coeid 22398 . . . . . . 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 6300 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
0 ... (deg `  F
) )  =  ( 0 ... 0 ) )
76sumeq1d 13486 . . . . . . . 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 10875 . . . . . . . . . 10  |-  0  e.  ZZ
9 exp0 12138 . . . . . . . . . . . . . 14  |-  ( z  e.  CC  ->  (
z ^ 0 )  =  1 )
109adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
z ^ 0 )  =  1 )
1110oveq2d 6300 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
121coef3 22392 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
13 0nn0 10810 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
14 ffvelrn 6019 . . . . . . . . . . . . . . 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 9613 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
1811, 17eqtrd 2508 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
1918, 16eqeltrd 2555 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  e.  CC )
20 fveq2 5866 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
21 oveq2 6292 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
z ^ k )  =  ( z ^
0 ) )
2220, 21oveq12d 6302 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) ) )
2322fsum1 13527 . . . . . . . . . 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 2508 . . . . . . . 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 2508 . . . . . . 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 4533 . . . . . 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 2508 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
29 fconstmpt 5043 . . . . 5  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
3028, 29syl6eqr 2526 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( CC  X.  { ( (coeff `  F ) `  0 ) } ) )
3130fveq1d 5868 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( ( CC  X.  { ( (coeff `  F ) `  0
) } ) ` 
0 ) )
32 0cn 9588 . . . . . . . 8  |-  0  e.  CC
33 fvex 5876 . . . . . . . . 9  |-  ( (coeff `  F ) `  0
)  e.  _V
3433fvconst2 6116 . . . . . . . 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 2524 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( (coeff `  F
) `  0 )
)
3736sneqd 4039 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  { ( F `  0 ) }  =  { (
(coeff `  F ) `  0 ) } )
3837xpeq2d 5023 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( CC  X.  { ( F ` 
0 ) } )  =  ( CC  X.  { ( (coeff `  F ) `  0
) } ) )
3930, 38eqtr4d 2511 . . 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 22358 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
42 ffvelrn 6019 . . . . 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 22405 . . . 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 5866 . . . 4  |-  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  (deg `  F
)  =  (deg `  ( CC  X.  { ( F `  0 ) } ) ) )
4746eqeq1d 2469 . . 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 1379    e. wcel 1767   {csn 4027    |-> cmpt 4505    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6284   CCcc 9490   0cc0 9492   1c1 9493    x. cmul 9497   NN0cn0 10795   ZZcz 10864   ...cfz 11672   ^cexp 12134   sum_csu 13471  Polycply 22344  coeffccoe 22346  degcdgr 22347
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-0p 21840  df-ply 22348  df-coe 22350  df-dgr 22351
This theorem is referenced by:  dgreq0  22424  dgrcolem2  22433  dgrco  22434  plyrem  22463  fta1  22466  aaliou2  22498  dgrnznn  30717
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