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Theorem eqcoe1ply1eq 31008
Description: Two polynomials over the same ring are equal if they have identical coefficients. (Contributed by AV, 7-Oct-2019.)
Hypotheses
Ref Expression
eqcoe1ply1eq.p  |-  P  =  (Poly1 `  R )
eqcoe1ply1eq.b  |-  B  =  ( Base `  P
)
eqcoe1ply1eq.a  |-  A  =  (coe1 `  K )
eqcoe1ply1eq.c  |-  C  =  (coe1 `  L )
Assertion
Ref Expression
eqcoe1ply1eq  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( A. k  e.  NN0  ( A `  k )  =  ( C `  k )  ->  K  =  L ) )
Distinct variable groups:    A, k    C, k
Allowed substitution hints:    B( k)    P( k)    R( k)    K( k)    L( k)

Proof of Theorem eqcoe1ply1eq
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 fveq2 5802 . . . . . . . . . . 11  |-  ( k  =  n  ->  ( A `  k )  =  ( A `  n ) )
2 fveq2 5802 . . . . . . . . . . 11  |-  ( k  =  n  ->  ( C `  k )  =  ( C `  n ) )
31, 2eqeq12d 2476 . . . . . . . . . 10  |-  ( k  =  n  ->  (
( A `  k
)  =  ( C `
 k )  <->  ( A `  n )  =  ( C `  n ) ) )
43rspccv 3176 . . . . . . . . 9  |-  ( A. k  e.  NN0  ( A `
 k )  =  ( C `  k
)  ->  ( n  e.  NN0  ->  ( A `  n )  =  ( C `  n ) ) )
54adantl 466 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  A. k  e.  NN0  ( A `  k )  =  ( C `  k ) )  -> 
( n  e.  NN0  ->  ( A `  n
)  =  ( C `
 n ) ) )
65imp 429 . . . . . . 7  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  A. k  e.  NN0  ( A `  k )  =  ( C `  k ) )  /\  n  e. 
NN0 )  ->  ( A `  n )  =  ( C `  n ) )
7 eqcoe1ply1eq.a . . . . . . . 8  |-  A  =  (coe1 `  K )
87fveq1i 5803 . . . . . . 7  |-  ( A `
 n )  =  ( (coe1 `  K ) `  n )
9 eqcoe1ply1eq.c . . . . . . . 8  |-  C  =  (coe1 `  L )
109fveq1i 5803 . . . . . . 7  |-  ( C `
 n )  =  ( (coe1 `  L ) `  n )
116, 8, 103eqtr3g 2518 . . . . . 6  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  A. k  e.  NN0  ( A `  k )  =  ( C `  k ) )  /\  n  e. 
NN0 )  ->  (
(coe1 `  K ) `  n )  =  ( (coe1 `  L ) `  n ) )
1211oveq1d 6218 . . . . 5  |-  ( ( ( ( R  e. 
Ring  /\  K  e.  B  /\  L  e.  B
)  /\  A. k  e.  NN0  ( A `  k )  =  ( C `  k ) )  /\  n  e. 
NN0 )  ->  (
( (coe1 `  K ) `  n ) ( .s
`  P ) ( n (.g `  (mulGrp `  P
) ) (var1 `  R
) ) )  =  ( ( (coe1 `  L
) `  n )
( .s `  P
) ( n (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )
1312mpteq2dva 4489 . . . 4  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  A. k  e.  NN0  ( A `  k )  =  ( C `  k ) )  -> 
( n  e.  NN0  |->  ( ( (coe1 `  K
) `  n )
( .s `  P
) ( n (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) )  =  ( n  e.  NN0  |->  ( ( (coe1 `  L ) `  n ) ( .s
`  P ) ( n (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) )
1413oveq2d 6219 . . 3  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  A. k  e.  NN0  ( A `  k )  =  ( C `  k ) )  -> 
( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  K
) `  n )
( .s `  P
) ( n (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )  =  ( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  L
) `  n )
( .s `  P
) ( n (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) )
15 eqcoe1ply1eq.p . . . . . . 7  |-  P  =  (Poly1 `  R )
16 eqid 2454 . . . . . . 7  |-  (var1 `  R
)  =  (var1 `  R
)
17 eqcoe1ply1eq.b . . . . . . 7  |-  B  =  ( Base `  P
)
18 eqid 2454 . . . . . . 7  |-  ( .s
`  P )  =  ( .s `  P
)
19 eqid 2454 . . . . . . 7  |-  (mulGrp `  P )  =  (mulGrp `  P )
20 eqid 2454 . . . . . . 7  |-  (.g `  (mulGrp `  P ) )  =  (.g `  (mulGrp `  P
) )
21 eqid 2454 . . . . . . 7  |-  (coe1 `  K
)  =  (coe1 `  K
)
2215, 16, 17, 18, 19, 20, 21ply1coe 17874 . . . . . 6  |-  ( ( R  e.  Ring  /\  K  e.  B )  ->  K  =  ( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  K ) `  n ) ( .s
`  P ) ( n (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) )
23223adant3 1008 . . . . 5  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  K  =  ( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  K ) `  n ) ( .s
`  P ) ( n (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) )
24 eqid 2454 . . . . . . 7  |-  (coe1 `  L
)  =  (coe1 `  L
)
2515, 16, 17, 18, 19, 20, 24ply1coe 17874 . . . . . 6  |-  ( ( R  e.  Ring  /\  L  e.  B )  ->  L  =  ( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  L ) `  n ) ( .s
`  P ) ( n (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) )
26253adant2 1007 . . . . 5  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  L  =  ( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  L ) `  n ) ( .s
`  P ) ( n (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) ) )
2723, 26eqeq12d 2476 . . . 4  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( K  =  L  <->  ( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  K ) `  n ) ( .s
`  P ) ( n (.g `  (mulGrp `  P
) ) (var1 `  R
) ) ) ) )  =  ( P 
gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  L
) `  n )
( .s `  P
) ( n (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) )
2827adantr 465 . . 3  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  A. k  e.  NN0  ( A `  k )  =  ( C `  k ) )  -> 
( K  =  L  <-> 
( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  K
) `  n )
( .s `  P
) ( n (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) )  =  ( P  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  L
) `  n )
( .s `  P
) ( n (.g `  (mulGrp `  P )
) (var1 `  R ) ) ) ) ) ) )
2914, 28mpbird 232 . 2  |-  ( ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  /\  A. k  e.  NN0  ( A `  k )  =  ( C `  k ) )  ->  K  =  L )
3029ex 434 1  |-  ( ( R  e.  Ring  /\  K  e.  B  /\  L  e.  B )  ->  ( A. k  e.  NN0  ( A `  k )  =  ( C `  k )  ->  K  =  L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   NN0cn0 10693   Basecbs 14295   .scvsca 14364    gsumg cgsu 14501  .gcmg 15536  mulGrpcmgp 16716   Ringcrg 16771  var1cv1 17759  Poly1cpl1 17760  coe1cco1 17761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-ofr 6434  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-sca 14376  df-vsca 14377  df-tset 14379  df-ple 14380  df-0g 14502  df-gsum 14503  df-mre 14646  df-mrc 14647  df-acs 14649  df-mnd 15537  df-mhm 15586  df-submnd 15587  df-grp 15667  df-minusg 15668  df-sbg 15669  df-mulg 15670  df-subg 15800  df-ghm 15867  df-cntz 15957  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-srg 16733  df-rng 16773  df-subrg 16989  df-lmod 17076  df-lss 17140  df-psr 17549  df-mvr 17550  df-mpl 17551  df-opsr 17553  df-psr1 17763  df-vr1 17764  df-ply1 17765  df-coe1 17766
This theorem is referenced by:  ply1coe1eq  31009  cply1coe0bi  31024  mp2pm2mp  31318
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