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Theorem mp2pm2mplem1 31274
Description: Lemma 1 for mp2pm2mp 31279. (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 5794 . . . . . . . . 9  |-  ( p  =  O  ->  (coe1 `  p )  =  (coe1 `  O ) )
43fveq1d 5796 . . . . . . . 8  |-  ( p  =  O  ->  (
(coe1 `  p ) `  k )  =  ( (coe1 `  O ) `  k ) )
54oveqd 6212 . . . . . . 7  |-  ( p  =  O  ->  (
i ( (coe1 `  p
) `  k )
j )  =  ( i ( (coe1 `  O
) `  k )
j ) )
65oveq1d 6210 . . . . . 6  |-  ( p  =  O  ->  (
( i ( (coe1 `  p ) `  k
) j )  .x.  ( k E Y ) )  =  ( ( i ( (coe1 `  O ) `  k
) j )  .x.  ( k E Y ) ) )
76mpteq2dv 4482 . . . . 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 6211 . . . 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 6256 . . 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 466 . 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 990 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  O  e.  L )
12 simp1 988 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  O  e.  L )  ->  N  e.  Fin )
13 mpt2exga 6754 . . 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 661 . 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 5883 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 965    = wceq 1370    e. wcel 1758   _Vcvv 3072    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   Fincfn 7415   NN0cn0 10685   Basecbs 14287   .scvsca 14356    gsumg cgsu 14493  .gcmg 15528  mulGrpcmgp 16708   Ringcrg 16763  var1cv1 17751  Poly1cpl1 17752  coe1cco1 17753   Mat cmat 18400
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683
This theorem is referenced by:  mp2pm2mplem3  31276  mp2pm2mplem5  31278
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