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Theorem 0dgrb 20034
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 2389 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
2 eqid 2389 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
31, 2coeid 20026 . . . . . . 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 6038 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
0 ... (deg `  F
) )  =  ( 0 ... 0 ) )
76sumeq1d 12424 . . . . . . . 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 10227 . . . . . . . . . 10  |-  0  e.  ZZ
9 exp0 11315 . . . . . . . . . . . . . 14  |-  ( z  e.  CC  ->  (
z ^ 0 )  =  1 )
109adantl 453 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
z ^ 0 )  =  1 )
1110oveq2d 6038 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
121coef3 20020 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
13 0nn0 10170 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
14 ffvelrn 5809 . . . . . . . . . . . . . . 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 9040 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
1811, 17eqtrd 2421 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
1918, 16eqeltrd 2463 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  e.  CC )
20 fveq2 5670 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
21 oveq2 6030 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
z ^ k )  =  ( z ^
0 ) )
2220, 21oveq12d 6040 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) ) )
2322fsum1 12464 . . . . . . . . . 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 2421 . . . . . . . 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 2421 . . . . . . 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 4238 . . . . . 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 2421 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
29 fconstmpt 4863 . . . . 5  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
3028, 29syl6eqr 2439 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( CC  X.  { ( (coeff `  F ) `  0 ) } ) )
3130fveq1d 5672 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( ( CC  X.  { ( (coeff `  F ) `  0
) } ) ` 
0 ) )
32 0cn 9019 . . . . . . . 8  |-  0  e.  CC
33 fvex 5684 . . . . . . . . 9  |-  ( (coeff `  F ) `  0
)  e.  _V
3433fvconst2 5888 . . . . . . . 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 2437 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( (coeff `  F
) `  0 )
)
3736sneqd 3772 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  { ( F `  0 ) }  =  { (
(coeff `  F ) `  0 ) } )
3837xpeq2d 4844 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( CC  X.  { ( F ` 
0 ) } )  =  ( CC  X.  { ( (coeff `  F ) `  0
) } ) )
3930, 38eqtr4d 2424 . . 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 19986 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
42 ffvelrn 5809 . . . . 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 20033 . . . 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 5670 . . . 4  |-  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  (deg `  F
)  =  (deg `  ( CC  X.  { ( F `  0 ) } ) ) )
4746eqeq1d 2397 . . 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 1649    e. wcel 1717   {csn 3759    e. cmpt 4209    X. cxp 4818   -->wf 5392   ` cfv 5396  (class class class)co 6022   CCcc 8923   0cc0 8925   1c1 8926    x. cmul 8930   NN0cn0 10155   ZZcz 10216   ...cfz 10977   ^cexp 11311   sum_csu 12408  Polycply 19972  coeffccoe 19974  degcdgr 19975
This theorem is referenced by:  dgreq0  20052  dgrcolem2  20061  dgrco  20062  plyrem  20091  fta1  20094  aaliou2  20126  dgrnznn  27011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-fz 10978  df-fzo 11068  df-fl 11131  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-rlim 12212  df-sum 12409  df-0p 19431  df-ply 19976  df-coe 19978  df-dgr 19979
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