MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coe11 Unicode version

Theorem coe11 19738
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 5608 . . 3  |-  ( F  =  G  ->  (coeff `  F )  =  (coeff `  G ) )
2 coefv0.1 . . 3  |-  A  =  (coeff `  F )
3 coeadd.2 . . 3  |-  B  =  (coeff `  G )
41, 2, 33eqtr4g 2415 . 2  |-  ( F  =  G  ->  A  =  B )
5 simp3 957 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  A  =  B )
65cnveqd 4939 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  `' A  =  `' B )
76imaeq1d 5093 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( `' A " ( CC  \  { 0 } ) )  =  ( `' B " ( CC 
\  { 0 } ) ) )
87supeq1d 7289 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' B " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) )
92dgrval 19714 . . . . . . . . 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 19714 . . . . . . . . 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 2400 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  (deg `  G ) )
1413oveq2d 5961 . . . . . 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 5610 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  ( A `  k )  =  ( B `  k ) )
1716oveq1d 5960 . . . . . 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 12276 . . . . 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 4188 . . . 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 2358 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
212, 20coeid 19724 . . . . 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 2358 . . . . . 6  |-  (deg `  G )  =  (deg
`  G )
243, 23coeid 19724 . . . . 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 2400 . . 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 1642    e. wcel 1710    \ cdif 3225   {csn 3716    e. cmpt 4158   `'ccnv 4770   "cima 4774   ` cfv 5337  (class class class)co 5945   supcsup 7283   CCcc 8825   0cc0 8827    x. cmul 8832    < clt 8957   NN0cn0 10057   ...cfz 10874   ^cexp 11197   sum_csu 12255  Polycply 19670  coeffccoe 19672  degcdgr 19673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fz 10875  df-fzo 10963  df-fl 11017  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-rlim 12059  df-sum 12256  df-0p 19129  df-ply 19674  df-coe 19676  df-dgr 19677
  Copyright terms: Public domain W3C validator