MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plyun0 Unicode version

Theorem plyun0 19983
Description: The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
Assertion
Ref Expression
plyun0  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )

Proof of Theorem plyun0
Dummy variables  k 
a  n  z  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0cn 9017 . . . . . . 7  |-  0  e.  CC
2 snssi 3885 . . . . . . 7  |-  ( 0  e.  CC  ->  { 0 }  C_  CC )
31, 2ax-mp 8 . . . . . 6  |-  { 0 }  C_  CC
43biantru 492 . . . . 5  |-  ( S 
C_  CC  <->  ( S  C_  CC  /\  { 0 } 
C_  CC ) )
5 unss 3464 . . . . 5  |-  ( ( S  C_  CC  /\  {
0 }  C_  CC ) 
<->  ( S  u.  {
0 } )  C_  CC )
64, 5bitr2i 242 . . . 4  |-  ( ( S  u.  { 0 } )  C_  CC  <->  S 
C_  CC )
7 unass 3447 . . . . . . . 8  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  ( { 0 }  u.  { 0 } ) )
8 unidm 3433 . . . . . . . . 9  |-  ( { 0 }  u.  {
0 } )  =  { 0 }
98uneq2i 3441 . . . . . . . 8  |-  ( S  u.  ( { 0 }  u.  { 0 } ) )  =  ( S  u.  {
0 } )
107, 9eqtri 2407 . . . . . . 7  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
1110oveq1i 6030 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  u. 
{ 0 } )  ^m  NN0 )  =  ( ( S  u.  { 0 } )  ^m  NN0 )
1211rexeqi 2852 . . . . 5  |-  ( E. a  e.  ( ( ( S  u.  {
0 } )  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) )  <->  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
1312rexbii 2674 . . . 4  |-  ( E. n  e.  NN0  E. a  e.  ( ( ( S  u.  { 0 } )  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) )  <->  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) )
146, 13anbi12i 679 . . 3  |-  ( ( ( S  u.  {
0 } )  C_  CC  /\  E. n  e. 
NN0  E. a  e.  ( ( ( S  u.  { 0 } )  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) )  <->  ( S  C_  CC  /\  E. n  e.  NN0  E. a  e.  ( ( S  u.  { 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
15 elply 19981 . . 3  |-  ( f  e.  (Poly `  ( S  u.  { 0 } ) )  <->  ( ( S  u.  { 0 } )  C_  CC  /\ 
E. n  e.  NN0  E. a  e.  ( ( ( S  u.  {
0 } )  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) ) )
16 elply 19981 . . 3  |-  ( f  e.  (Poly `  S
)  <->  ( S  C_  CC  /\  E. n  e. 
NN0  E. a  e.  ( ( S  u.  {
0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k
)  x.  ( z ^ k ) ) ) ) )
1714, 15, 163bitr4i 269 . 2  |-  ( f  e.  (Poly `  ( S  u.  { 0 } ) )  <->  f  e.  (Poly `  S ) )
1817eqriv 2384 1  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2650    u. cun 3261    C_ wss 3263   {csn 3757    e. cmpt 4207   ` cfv 5394  (class class class)co 6020    ^m cmap 6954   CCcc 8921   0cc0 8923    x. cmul 8928   NN0cn0 10153   ...cfz 10975   ^cexp 11309   sum_csu 12406  Polycply 19970
This theorem is referenced by:  elplyd  19988  ply1term  19990  ply0  19994  plyaddlem  20001  plymullem  20002  plyco  20027  plycj  20062
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-i2m1 8991  ax-1ne0 8992  ax-rrecex 8995  ax-cnre 8996
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-recs 6569  df-rdg 6604  df-nn 9933  df-n0 10154  df-ply 19974
  Copyright terms: Public domain W3C validator