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Theorem mp2pm2mplem1 19474
Description: Lemma 1 for mp2pm2mp 19479. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.)
Hypotheses
Ref Expression
mp2pm2mp.a  |-  A  =  ( N Mat  R )
mp2pm2mp.q  |-  Q  =  (Poly1 `  A )
mp2pm2mp.l  |-  L  =  ( Base `  Q
)
mp2pm2mp.m  |-  .x.  =  ( .s `  P )
mp2pm2mp.e  |-  E  =  (.g `  (mulGrp `  P
) )
mp2pm2mp.y  |-  Y  =  (var1 `  R )
mp2pm2mp.i  |-  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
Assertion
Ref Expression
mp2pm2mplem1  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  (
I `  O )  =  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O
) `  k )
j )  .x.  (
k E Y ) ) ) ) ) )
Distinct variable groups:    E, p    L, p    i, N, j, p    i, O, j, p    k, O, p    P, p    R, p    Y, p    .x. , p
Allowed substitution hints:    A( i, j, k, p)    P( i,
j, k)    Q( i,
j, k, p)    R( i, j, k)    .x. ( i, j, k)    E( i, j, k)    I( i, j, k, p)    L( i, j, k)    N( k)    Y( i, j, k)

Proof of Theorem mp2pm2mplem1
StepHypRef Expression
1 mp2pm2mp.i . . 3  |-  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
21a1i 11 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  I  =  ( p  e.  L  |->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p
) `  k )
j )  .x.  (
k E Y ) ) ) ) ) ) )
3 fveq2 5848 . . . . . . . . 9  |-  ( p  =  O  ->  (coe1 `  p )  =  (coe1 `  O ) )
43fveq1d 5850 . . . . . . . 8  |-  ( p  =  O  ->  (
(coe1 `  p ) `  k )  =  ( (coe1 `  O ) `  k ) )
54oveqd 6287 . . . . . . 7  |-  ( p  =  O  ->  (
i ( (coe1 `  p
) `  k )
j )  =  ( i ( (coe1 `  O
) `  k )
j ) )
65oveq1d 6285 . . . . . 6  |-  ( p  =  O  ->  (
( i ( (coe1 `  p ) `  k
) j )  .x.  ( k E Y ) )  =  ( ( i ( (coe1 `  O ) `  k
) j )  .x.  ( k E Y ) ) )
76mpteq2dv 4526 . . . . 5  |-  ( p  =  O  ->  (
k  e.  NN0  |->  ( ( i ( (coe1 `  p
) `  k )
j )  .x.  (
k E Y ) ) )  =  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O
) `  k )
j )  .x.  (
k E Y ) ) ) )
87oveq2d 6286 . . . 4  |-  ( p  =  O  ->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) )  =  ( P 
gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )
98mpt2eq3dv 6336 . . 3  |-  ( p  =  O  ->  (
i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  =  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
109adantl 464 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  /\  p  =  O )  ->  (
i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  p ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  =  ( i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) ) )
11 simp3 996 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  O  e.  L )
12 simp1 994 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  N  e.  Fin )
13 mpt2exga 6849 . . 3  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  e.  _V )
1412, 12, 13syl2anc 659 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  (
i  e.  N , 
j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O ) `  k ) j ) 
.x.  ( k E Y ) ) ) ) )  e.  _V )
152, 10, 11, 14fvmptd 5936 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  (
I `  O )  =  ( i  e.  N ,  j  e.  N  |->  ( P  gsumg  ( k  e.  NN0  |->  ( ( i ( (coe1 `  O
) `  k )
j )  .x.  (
k E Y ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Fincfn 7509   NN0cn0 10791   Basecbs 14716   .scvsca 14788    gsumg cgsu 14930  .gcmg 16255  mulGrpcmgp 17336   Ringcrg 17393  var1cv1 18410  Poly1cpl1 18411  coe1cco1 18412   Mat cmat 19076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774
This theorem is referenced by:  mp2pm2mplem3  19476
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