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Theorem plyun0 19579
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 8831 . . . . . . 7  |-  0  e.  CC
2 snssi 3759 . . . . . . 7  |-  ( 0  e.  CC  ->  { 0 }  C_  CC )
31, 2ax-mp 8 . . . . . 6  |-  { 0 }  C_  CC
43biantru 491 . . . . 5  |-  ( S 
C_  CC  <->  ( S  C_  CC  /\  { 0 } 
C_  CC ) )
5 unss 3349 . . . . 5  |-  ( ( S  C_  CC  /\  {
0 }  C_  CC ) 
<->  ( S  u.  {
0 } )  C_  CC )
64, 5bitr2i 241 . . . 4  |-  ( ( S  u.  { 0 } )  C_  CC  <->  S 
C_  CC )
7 unass 3332 . . . . . . . 8  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  ( { 0 }  u.  { 0 } ) )
8 unidm 3318 . . . . . . . . 9  |-  ( { 0 }  u.  {
0 } )  =  { 0 }
98uneq2i 3326 . . . . . . . 8  |-  ( S  u.  ( { 0 }  u.  { 0 } ) )  =  ( S  u.  {
0 } )
107, 9eqtri 2303 . . . . . . 7  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
1110oveq1i 5868 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  u. 
{ 0 } )  ^m  NN0 )  =  ( ( S  u.  { 0 } )  ^m  NN0 )
1211rexeqi 2741 . . . . 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 2568 . . . 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 678 . . 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 19577 . . 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 19577 . . 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 268 . 2  |-  ( f  e.  (Poly `  ( S  u.  { 0 } ) )  <->  f  e.  (Poly `  S ) )
1817eqriv 2280 1  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    u. cun 3150    C_ wss 3152   {csn 3640    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   0cc0 8737    x. cmul 8742   NN0cn0 9965   ...cfz 10782   ^cexp 11104   sum_csu 12158  Polycply 19566
This theorem is referenced by:  elplyd  19584  ply1term  19586  ply0  19590  plyaddlem  19597  plymullem  19598  plyco  19623  plycj  19658
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-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
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-ral 2548  df-rex 2549  df-reu 2550  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-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-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-ov 5861  df-recs 6388  df-rdg 6423  df-nn 9747  df-n0 9966  df-ply 19570
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