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Theorem coe11 20132
Description: The coefficient function is one-to-one, so if the coefficients are equal then the functions are equal and vice-versa. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
coefv0.1  |-  A  =  (coeff `  F )
coeadd.2  |-  B  =  (coeff `  G )
Assertion
Ref Expression
coe11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  G  <->  A  =  B
) )

Proof of Theorem coe11
Dummy variables  k 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5695 . . 3  |-  ( F  =  G  ->  (coeff `  F )  =  (coeff `  G ) )
2 coefv0.1 . . 3  |-  A  =  (coeff `  F )
3 coeadd.2 . . 3  |-  B  =  (coeff `  G )
41, 2, 33eqtr4g 2469 . 2  |-  ( F  =  G  ->  A  =  B )
5 simp3 959 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  A  =  B )
65cnveqd 5015 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  `' A  =  `' B )
76imaeq1d 5169 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( `' A " ( CC  \  { 0 } ) )  =  ( `' B " ( CC 
\  { 0 } ) ) )
87supeq1d 7417 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' B " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) )
92dgrval 20108 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
1093ad2ant1 978 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
113dgrval 20108 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  =  sup (
( `' B "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
12113ad2ant2 979 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  G
)  =  sup (
( `' B "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
138, 10, 123eqtr4d 2454 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  (deg `  G ) )
1413oveq2d 6064 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... (deg `  G ) ) )
15 simpl3 962 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  A  =  B )
1615fveq1d 5697 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  ( A `  k )  =  ( B `  k ) )
1716oveq1d 6063 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( A `
 k )  x.  ( z ^ k
) )  =  ( ( B `  k
)  x.  ( z ^ k ) ) )
1814, 17sumeq12dv 12463 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( z ^ k ) )  =  sum_ k  e.  ( 0 ... (deg `  G ) ) ( ( B `  k
)  x.  ( z ^ k ) ) )
1918mpteq2dv 4264 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( z ^ k ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  G ) ) ( ( B `  k
)  x.  ( z ^ k ) ) ) )
20 eqid 2412 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
212, 20coeid 20118 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( A `  k )  x.  ( z ^
k ) ) ) )
22213ad2ant1 978 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( A `  k )  x.  ( z ^
k ) ) ) )
23 eqid 2412 . . . . . 6  |-  (deg `  G )  =  (deg
`  G )
243, 23coeid 20118 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  G
) ) ( ( B `  k )  x.  ( z ^
k ) ) ) )
25243ad2ant2 979 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  G
) ) ( ( B `  k )  x.  ( z ^
k ) ) ) )
2619, 22, 253eqtr4d 2454 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  F  =  G )
27263expia 1155 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  =  B  ->  F  =  G ) )
284, 27impbid2 196 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  G  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    \ cdif 3285   {csn 3782    e. cmpt 4234   `'ccnv 4844   "cima 4848   ` cfv 5421  (class class class)co 6048   supcsup 7411   CCcc 8952   0cc0 8954    x. cmul 8959    < clt 9084   NN0cn0 10185   ...cfz 11007   ^cexp 11345   sum_csu 12442  Polycply 20064  coeffccoe 20066  degcdgr 20067
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-0p 19523  df-ply 20068  df-coe 20070  df-dgr 20071
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