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Theorem coe11 23286
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 5879 . . 3  |-  ( F  =  G  ->  (coeff `  F )  =  (coeff `  G ) )
2 coefv0.1 . . 3  |-  A  =  (coeff `  F )
3 coeadd.2 . . 3  |-  B  =  (coeff `  G )
41, 2, 33eqtr4g 2530 . 2  |-  ( F  =  G  ->  A  =  B )
5 simp3 1032 . . . . . . . . . . 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 5173 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( `' A " ( CC  \  { 0 } ) )  =  ( `' B " ( CC 
\  { 0 } ) ) )
87supeq1d 7978 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' B " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) )
92dgrval 23261 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
1093ad2ant1 1051 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
113dgrval 23261 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  =  sup (
( `' B "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
12113ad2ant2 1052 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  G
)  =  sup (
( `' B "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
138, 10, 123eqtr4d 2515 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  (deg `  G ) )
1413oveq2d 6324 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... (deg `  G ) ) )
15 simpl3 1035 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  A  =  B )
1615fveq1d 5881 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  ( A `  k )  =  ( B `  k ) )
1716oveq1d 6323 . . . . . 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 13849 . . . . 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 4483 . . . 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 2471 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
212, 20coeid 23271 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( A `  k )  x.  ( z ^
k ) ) ) )
22213ad2ant1 1051 . . . 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 2471 . . . . . 6  |-  (deg `  G )  =  (deg
`  G )
243, 23coeid 23271 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  G
) ) ( ( B `  k )  x.  ( z ^
k ) ) ) )
25243ad2ant2 1052 . . . 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 2515 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  F  =  G )
27263expia 1233 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  =  B  ->  F  =  G ) )
284, 27impbid2 209 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  G  <->  A  =  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    \ cdif 3387   {csn 3959    |-> cmpt 4454   `'ccnv 4838   "cima 4842   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   0cc0 9557    x. cmul 9562    < clt 9693   NN0cn0 10893   ...cfz 11810   ^cexp 12310   sum_csu 13829  Polycply 23217  coeffccoe 23219  degcdgr 23220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224
This theorem is referenced by: (None)
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