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Theorem plyun0 22760
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 9577 . . . . . . 7  |-  0  e.  CC
2 snssi 4160 . . . . . . 7  |-  ( 0  e.  CC  ->  { 0 }  C_  CC )
31, 2ax-mp 5 . . . . . 6  |-  { 0 }  C_  CC
43biantru 503 . . . . 5  |-  ( S 
C_  CC  <->  ( S  C_  CC  /\  { 0 } 
C_  CC ) )
5 unss 3664 . . . . 5  |-  ( ( S  C_  CC  /\  {
0 }  C_  CC ) 
<->  ( S  u.  {
0 } )  C_  CC )
64, 5bitr2i 250 . . . 4  |-  ( ( S  u.  { 0 } )  C_  CC  <->  S 
C_  CC )
7 unass 3647 . . . . . . . 8  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  ( { 0 }  u.  { 0 } ) )
8 unidm 3633 . . . . . . . . 9  |-  ( { 0 }  u.  {
0 } )  =  { 0 }
98uneq2i 3641 . . . . . . . 8  |-  ( S  u.  ( { 0 }  u.  { 0 } ) )  =  ( S  u.  {
0 } )
107, 9eqtri 2483 . . . . . . 7  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
1110oveq1i 6280 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  u. 
{ 0 } )  ^m  NN0 )  =  ( ( S  u.  { 0 } )  ^m  NN0 )
1211rexeqi 3056 . . . . 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 2956 . . . 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 695 . . 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 22758 . . 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 22758 . . 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 277 . 2  |-  ( f  e.  (Poly `  ( S  u.  { 0 } ) )  <->  f  e.  (Poly `  S ) )
1817eqriv 2450 1  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805    u. cun 3459    C_ wss 3461   {csn 4016    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   CCcc 9479   0cc0 9481    x. cmul 9486   NN0cn0 10791   ...cfz 11675   ^cexp 12148   sum_csu 13590  Polycply 22747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-om 6674  df-recs 7034  df-rdg 7068  df-nn 10532  df-n0 10792  df-ply 22751
This theorem is referenced by:  elplyd  22765  ply1term  22767  ply0  22771  plyaddlem  22778  plymullem  22779  plyco  22804  plycj  22840
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