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Theorem coe11 20176
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 5731 . . 3  |-  ( F  =  G  ->  (coeff `  F )  =  (coeff `  G ) )
2 coefv0.1 . . 3  |-  A  =  (coeff `  F )
3 coeadd.2 . . 3  |-  B  =  (coeff `  G )
41, 2, 33eqtr4g 2495 . 2  |-  ( F  =  G  ->  A  =  B )
5 simp3 960 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  A  =  B )
65cnveqd 5051 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  `' A  =  `' B )
76imaeq1d 5205 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( `' A " ( CC  \  { 0 } ) )  =  ( `' B " ( CC 
\  { 0 } ) ) )
87supeq1d 7454 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' B " ( CC 
\  { 0 } ) ) ,  NN0 ,  <  ) )
92dgrval 20152 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
1093ad2ant1 979 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
113dgrval 20152 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  =  sup (
( `' B "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
12113ad2ant2 980 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  G
)  =  sup (
( `' B "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
138, 10, 123eqtr4d 2480 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  (deg `  F
)  =  (deg `  G ) )
1413oveq2d 6100 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  ( 0 ... (deg `  F
) )  =  ( 0 ... (deg `  G ) ) )
15 simpl3 963 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  A  =  B )
1615fveq1d 5733 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  /\  k  e.  ( 0 ... (deg `  F ) ) )  ->  ( A `  k )  =  ( B `  k ) )
1716oveq1d 6099 . . . . . 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 12505 . . . . 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 4299 . . . 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 2438 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
212, 20coeid 20162 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( A `  k )  x.  ( z ^
k ) ) ) )
22213ad2ant1 979 . . . 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 2438 . . . . . 6  |-  (deg `  G )  =  (deg
`  G )
243, 23coeid 20162 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... (deg `  G
) ) ( ( B `  k )  x.  ( z ^
k ) ) ) )
25243ad2ant2 980 . . . 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 2480 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  A  =  B
)  ->  F  =  G )
27263expia 1156 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( A  =  B  ->  F  =  G ) )
284, 27impbid2 197 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  G  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319   {csn 3816    e. cmpt 4269   `'ccnv 4880   "cima 4884   ` cfv 5457  (class class class)co 6084   supcsup 7448   CCcc 8993   0cc0 8995    x. cmul 9000    < clt 9125   NN0cn0 10226   ...cfz 11048   ^cexp 11387   sum_csu 12484  Polycply 20108  coeffccoe 20110  degcdgr 20111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-0p 19565  df-ply 20112  df-coe 20114  df-dgr 20115
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