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Theorem 0dgrb 19628
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 2283 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
2 eqid 2283 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
31, 2coeid 19620 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( (coeff `  F ) `  k )  x.  (
z ^ k ) ) ) )
43adantr 451 . . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (deg `  F )  =  0 )
65oveq2d 5874 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
0 ... (deg `  F
) )  =  ( 0 ... 0 ) )
76sumeq1d 12174 . . . . . . . 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 10035 . . . . . . . . . 10  |-  0  e.  ZZ
9 exp0 11108 . . . . . . . . . . . . . 14  |-  ( z  e.  CC  ->  (
z ^ 0 )  =  1 )
109adantl 452 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
z ^ 0 )  =  1 )
1110oveq2d 5874 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( ( (coeff `  F
) `  0 )  x.  1 ) )
121coef3 19614 . . . . . . . . . . . . . . 15  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
13 0nn0 9980 . . . . . . . . . . . . . . 15  |-  0  e.  NN0
14 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( (coeff `  F ) : NN0 --> CC  /\  0  e.  NN0 )  ->  (
(coeff `  F ) `  0 )  e.  CC )
1512, 13, 14sylancl 643 . . . . . . . . . . . . . 14  |-  ( F  e.  (Poly `  S
)  ->  ( (coeff `  F ) `  0
)  e.  CC )
1615ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
(coeff `  F ) `  0 )  e.  CC )
1716mulid1d 8852 . . . . . . . . . . . 12  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  1 )  =  ( (coeff `  F ) `  0 ) )
1811, 17eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  =  ( (coeff `  F ) `  0 ) )
1918, 16eqeltrd 2357 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  /\  z  e.  CC )  ->  (
( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) )  e.  CC )
20 fveq2 5525 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
(coeff `  F ) `  k )  =  ( (coeff `  F ) `  0 ) )
21 oveq2 5866 . . . . . . . . . . . 12  |-  ( k  =  0  ->  (
z ^ k )  =  ( z ^
0 ) )
2220, 21oveq12d 5876 . . . . . . . . . . 11  |-  ( k  =  0  ->  (
( (coeff `  F
) `  k )  x.  ( z ^ k
) )  =  ( ( (coeff `  F
) `  0 )  x.  ( z ^ 0 ) ) )
2322fsum1 12214 . . . . . . . . . 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 644 . . . . . . . . 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 2315 . . . . . . . 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 2315 . . . . . . 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 4106 . . . . . 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 2315 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) ) )
29 fconstmpt 4732 . . . . 5  |-  ( CC 
X.  { ( (coeff `  F ) `  0
) } )  =  ( z  e.  CC  |->  ( (coeff `  F ) `  0 ) )
3028, 29syl6eqr 2333 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( CC  X.  { ( (coeff `  F ) `  0 ) } ) )
3130fveq1d 5527 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( ( CC  X.  { ( (coeff `  F ) `  0
) } ) ` 
0 ) )
32 0cn 8831 . . . . . . . 8  |-  0  e.  CC
33 fvex 5539 . . . . . . . . 9  |-  ( (coeff `  F ) `  0
)  e.  _V
3433fvconst2 5729 . . . . . . . 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 2331 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( F `  0 )  =  ( (coeff `  F
) `  0 )
)
3736sneqd 3653 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  { ( F `  0 ) }  =  { (
(coeff `  F ) `  0 ) } )
3837xpeq2d 4713 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  ( CC  X.  { ( F ` 
0 ) } )  =  ( CC  X.  { ( (coeff `  F ) `  0
) } ) )
3930, 38eqtr4d 2318 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  (deg `  F )  =  0 )  ->  F  =  ( CC  X.  { ( F `  0 ) } ) )
4039ex 423 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( (deg `  F )  =  0  ->  F  =  ( CC  X.  { ( F `  0 ) } ) ) )
41 plyf 19580 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
42 ffvelrn 5663 . . . . 5  |-  ( ( F : CC --> CC  /\  0  e.  CC )  ->  ( F `  0
)  e.  CC )
4341, 32, 42sylancl 643 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  e.  CC )
44 0dgr 19627 . . . 4  |-  ( ( F `  0 )  e.  CC  ->  (deg `  ( CC  X.  {
( F `  0
) } ) )  =  0 )
4543, 44syl 15 . . 3  |-  ( F  e.  (Poly `  S
)  ->  (deg `  ( CC  X.  { ( F `
 0 ) } ) )  =  0 )
46 fveq2 5525 . . . 4  |-  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  (deg `  F
)  =  (deg `  ( CC  X.  { ( F `  0 ) } ) ) )
4746eqeq1d 2291 . . 3  |-  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  ( (deg `  F )  =  0  <-> 
(deg `  ( CC  X.  { ( F ` 
0 ) } ) )  =  0 ) )
4845, 47syl5ibrcom 213 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  ( CC  X.  { ( F ` 
0 ) } )  ->  (deg `  F
)  =  0 ) )
4940, 48impbid 183 1  |-  ( F  e.  (Poly `  S
)  ->  ( (deg `  F )  =  0  <-> 
F  =  ( CC 
X.  { ( F `
 0 ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    e. cmpt 4077    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   NN0cn0 9965   ZZcz 10024   ...cfz 10782   ^cexp 11104   sum_csu 12158  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  dgreq0  19646  dgrcolem2  19655  dgrco  19656  plyrem  19685  fta1  19688  aaliou2  19720  dgrnznn  27340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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