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Theorem pmatcollpw1lem1 30896
Description: Lemma 1 for pmatcollpw1 30904: The matrix consisting of the coefficients in the polynomial entries of a given matrix for the same power. (Contributed by AV, 28-Sep-2019.)
Hypotheses
Ref Expression
pmatcollpw.p  |-  P  =  (Poly1 `  R )
pmatcollpw.c  |-  C  =  ( N Mat  P )
pmatcollpw.b  |-  B  =  ( Base `  C
)
pmatcollpw.f  |-  F  =  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
Assertion
Ref Expression
pmatcollpw1lem1  |-  ( ( N  e.  V  /\  M  e.  B  /\  K  e.  NN0 )  -> 
( M F K )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K
) ) )
Distinct variable groups:    B, k, m    i, K, j, k, m    i, M, j, k, m    i, N, j, k, m    k, V, m
Allowed substitution hints:    B( i, j)    C( i, j, k, m)    P( i, j, k, m)    R( i, j, k, m)    F( i, j, k, m)    V( i, j)

Proof of Theorem pmatcollpw1lem1
StepHypRef Expression
1 pmatcollpw.f . . 3  |-  F  =  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
21a1i 11 . 2  |-  ( ( N  e.  V  /\  M  e.  B  /\  K  e.  NN0 )  ->  F  =  ( m  e.  B ,  k  e. 
NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  (
i m j ) ) `  k ) ) ) )
3 oveq 6097 . . . . . . 7  |-  ( m  =  M  ->  (
i m j )  =  ( i M j ) )
43fveq2d 5695 . . . . . 6  |-  ( m  =  M  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
54adantr 465 . . . . 5  |-  ( ( m  =  M  /\  k  =  K )  ->  (coe1 `  ( i m j ) )  =  (coe1 `  ( i M j ) ) )
6 simpr 461 . . . . 5  |-  ( ( m  =  M  /\  k  =  K )  ->  k  =  K )
75, 6fveq12d 5697 . . . 4  |-  ( ( m  =  M  /\  k  =  K )  ->  ( (coe1 `  ( i m j ) ) `  k )  =  ( (coe1 `  ( i M j ) ) `  K ) )
87mpt2eq3dv 6152 . . 3  |-  ( ( m  =  M  /\  k  =  K )  ->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K ) ) )
98adantl 466 . 2  |-  ( ( ( N  e.  V  /\  M  e.  B  /\  K  e.  NN0 )  /\  ( m  =  M  /\  k  =  K ) )  -> 
( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K ) ) )
10 simp2 989 . 2  |-  ( ( N  e.  V  /\  M  e.  B  /\  K  e.  NN0 )  ->  M  e.  B )
11 simp3 990 . 2  |-  ( ( N  e.  V  /\  M  e.  B  /\  K  e.  NN0 )  ->  K  e.  NN0 )
12 simp1 988 . . 3  |-  ( ( N  e.  V  /\  M  e.  B  /\  K  e.  NN0 )  ->  N  e.  V )
13 eqid 2443 . . . 4  |-  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K
) )  =  ( i  e.  N , 
j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K ) )
1413mpt2exg 6648 . . 3  |-  ( ( N  e.  V  /\  N  e.  V )  ->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K ) )  e. 
_V )
1512, 12, 14syl2anc 661 . 2  |-  ( ( N  e.  V  /\  M  e.  B  /\  K  e.  NN0 )  -> 
( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K ) )  e. 
_V )
162, 9, 10, 11, 15ovmpt2d 6218 1  |-  ( ( N  e.  V  /\  M  e.  B  /\  K  e.  NN0 )  -> 
( M F K )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   NN0cn0 10579   Basecbs 14174  Poly1cpl1 17633  coe1cco1 17634   Mat cmat 18280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578
This theorem is referenced by:  pmatcollpw1lem2  30897  pmatcollpw1lem3  30898  pmatcollpw1id  30899  mp2pm2mplem3  30918  pmattomply1f1  30922  pmattomply1ghm  30925  pmattomply1mhmlem0  30927  pmattomply1mhmlem1  30928
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