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Theorem coe11 19634
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 5525 . . 3  |-  ( F  =  G  ->  (coeff `  F )  =  (coeff `  G ) )
2 coefv0.1 . . 3  |-  A  =  (coeff `  F )
3 coeadd.2 . . 3  |-  B  =  (coeff `  G )
41, 2, 33eqtr4g 2340 . 2  |-  ( F  =  G  ->  A  =  B )
5 simp3 957 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  A  =  B )
65cnveqd 4857 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  `' A  =  `' B )
76imaeq1d 5011 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( `' A " ( CC  \  { 0 } ) )  =  ( `' B " ( CC 
\  { 0 } ) ) )
87supeq1d 7199 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' B " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) )
92dgrval 19610 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
1093ad2ant1 976 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
113dgrval 19610 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  =  sup (
( `' B "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
12113ad2ant2 977 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  G
)  =  sup (
( `' B "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
138, 10, 123eqtr4d 2325 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  (deg `  G ) )
1413oveq2d 5874 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... (deg `  G ) ) )
15 simpl3 960 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  A  =  B )
1615fveq1d 5527 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  ( A `  k )  =  ( B `  k ) )
1716oveq1d 5873 . . . . . 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 12179 . . . . 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 4107 . . . 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 2283 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
212, 20coeid 19620 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( A `  k )  x.  ( z ^
k ) ) ) )
22213ad2ant1 976 . . . 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 2283 . . . . . 6  |-  (deg `  G )  =  (deg
`  G )
243, 23coeid 19620 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  G
) ) ( ( B `  k )  x.  ( z ^
k ) ) ) )
25243ad2ant2 977 . . . 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 2325 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  F  =  G )
27263expia 1153 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  =  B  ->  F  =  G ) )
284, 27impbid2 195 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  G  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149   {csn 3640    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   0cc0 8737    x. cmul 8742    < clt 8867   NN0cn0 9965   ...cfz 10782   ^cexp 11104   sum_csu 12158  Polycply 19566  coeffccoe 19568  degcdgr 19569
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-inf2 7342  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-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-int 3863  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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