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Theorem plyun0 20109
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 9077 . . . . . . 7  |-  0  e.  CC
2 snssi 3935 . . . . . . 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 3514 . . . . 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 3497 . . . . . . . 8  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  ( { 0 }  u.  { 0 } ) )
8 unidm 3483 . . . . . . . . 9  |-  ( { 0 }  u.  {
0 } )  =  { 0 }
98uneq2i 3491 . . . . . . . 8  |-  ( S  u.  ( { 0 }  u.  { 0 } ) )  =  ( S  u.  {
0 } )
107, 9eqtri 2456 . . . . . . 7  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
1110oveq1i 6084 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  u. 
{ 0 } )  ^m  NN0 )  =  ( ( S  u.  { 0 } )  ^m  NN0 )
1211rexeqi 2902 . . . . 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 2723 . . . 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 20107 . . 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 20107 . . 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 2433 1  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2699    u. cun 3311    C_ wss 3313   {csn 3807    e. cmpt 4259   ` cfv 5447  (class class class)co 6074    ^m cmap 7011   CCcc 8981   0cc0 8983    x. cmul 8988   NN0cn0 10214   ...cfz 11036   ^cexp 11375   sum_csu 12472  Polycply 20096
This theorem is referenced by:  elplyd  20114  ply1term  20116  ply0  20120  plyaddlem  20127  plymullem  20128  plyco  20153  plycj  20188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-i2m1 9051  ax-1ne0 9052  ax-rrecex 9055  ax-cnre 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-recs 6626  df-rdg 6661  df-nn 9994  df-n0 10215  df-ply 20100
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