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Theorem plyun0 19595
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 8847 . . . . . . 7  |-  0  e.  CC
2 snssi 3775 . . . . . . 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 3362 . . . . 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 3345 . . . . . . . 8  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  ( { 0 }  u.  { 0 } ) )
8 unidm 3331 . . . . . . . . 9  |-  ( { 0 }  u.  {
0 } )  =  { 0 }
98uneq2i 3339 . . . . . . . 8  |-  ( S  u.  ( { 0 }  u.  { 0 } ) )  =  ( S  u.  {
0 } )
107, 9eqtri 2316 . . . . . . 7  |-  ( ( S  u.  { 0 } )  u.  {
0 } )  =  ( S  u.  {
0 } )
1110oveq1i 5884 . . . . . 6  |-  ( ( ( S  u.  {
0 } )  u. 
{ 0 } )  ^m  NN0 )  =  ( ( S  u.  { 0 } )  ^m  NN0 )
1211rexeqi 2754 . . . . 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 2581 . . . 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 19593 . . 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 19593 . . 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 2293 1  |-  (Poly `  ( S  u.  { 0 } ) )  =  (Poly `  S )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    u. cun 3163    C_ wss 3165   {csn 3653    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   0cc0 8753    x. cmul 8758   NN0cn0 9981   ...cfz 10798   ^cexp 11120   sum_csu 12174  Polycply 19582
This theorem is referenced by:  elplyd  19600  ply1term  19602  ply0  19606  plyaddlem  19613  plymullem  19614  plyco  19639  plycj  19674
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-nn 9763  df-n0 9982  df-ply 19586
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