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Theorem pmatcollpw1lem3 31211
Description: Lemma 3 for pmatcollpw1 31218: An entry of the matrix consisting of the coefficients in the polynomial entries of a given matrix is the corresponding coefficient in the polynomial entry of the given matrix. (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
pmatcollpw1lem3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( I ( M F K ) J )  =  ( (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    B, i, j    R, i, j   
i, I, j    i, J, j
Allowed substitution hints:    C( i, j, k, m)    P( i,
j, k, m)    R( k, m)    F( i, j, k, m)    I( k, m)    J( k, m)

Proof of Theorem pmatcollpw1lem3
StepHypRef Expression
1 simp1l 1012 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  N  e.  Fin )
2 simpl 457 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  NN0 )  ->  M  e.  B )
323ad2ant2 1010 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  M  e.  B )
4 simpr 461 . . . . 5  |-  ( ( M  e.  B  /\  K  e.  NN0 )  ->  K  e.  NN0 )
543ad2ant2 1010 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  K  e.  NN0 )
6 pmatcollpw.p . . . . 5  |-  P  =  (Poly1 `  R )
7 pmatcollpw.c . . . . 5  |-  C  =  ( N Mat  P )
8 pmatcollpw.b . . . . 5  |-  B  =  ( Base `  C
)
9 pmatcollpw.f . . . . 5  |-  F  =  ( m  e.  B ,  k  e.  NN0  |->  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i m j ) ) `  k ) ) )
106, 7, 8, 9pmatcollpw1lem1 31209 . . . 4  |-  ( ( N  e.  Fin  /\  M  e.  B  /\  K  e.  NN0 )  -> 
( M F K )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K
) ) )
111, 3, 5, 10syl3anc 1219 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( M F K )  =  ( i  e.  N ,  j  e.  N  |->  ( (coe1 `  ( i M j ) ) `  K
) ) )
12 oveq12 6185 . . . . . 6  |-  ( ( i  =  I  /\  j  =  J )  ->  ( i M j )  =  ( I M J ) )
1312fveq2d 5779 . . . . 5  |-  ( ( i  =  I  /\  j  =  J )  ->  (coe1 `  ( i M j ) )  =  (coe1 `  ( I M J ) ) )
1413fveq1d 5777 . . . 4  |-  ( ( i  =  I  /\  j  =  J )  ->  ( (coe1 `  ( i M j ) ) `  K )  =  ( (coe1 `  ( I M J ) ) `  K ) )
1514adantl 466 . . 3  |-  ( ( ( ( N  e. 
Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  (
I  e.  N  /\  J  e.  N )
)  /\  ( i  =  I  /\  j  =  J ) )  -> 
( (coe1 `  ( i M j ) ) `  K )  =  ( (coe1 `  ( I M J ) ) `  K ) )
16 simpl 457 . . . 4  |-  ( ( I  e.  N  /\  J  e.  N )  ->  I  e.  N )
17163ad2ant3 1011 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  I  e.  N )
18 simpr 461 . . . 4  |-  ( ( I  e.  N  /\  J  e.  N )  ->  J  e.  N )
19183ad2ant3 1011 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  J  e.  N )
20 fvex 5785 . . . 4  |-  ( (coe1 `  ( I M J ) ) `  K
)  e.  _V
2120a1i 11 . . 3  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( (coe1 `  ( I M J ) ) `  K )  e.  _V )
2211, 15, 17, 19, 21ovmpt2d 6304 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( I ( M F K ) J )  =  ( (coe1 `  ( I M J ) ) `  K
) )
2322idi 2 1  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( M  e.  B  /\  K  e.  NN0 )  /\  ( I  e.  N  /\  J  e.  N ) )  -> 
( I ( M F K ) J )  =  ( (coe1 `  ( I M J ) ) `  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757   _Vcvv 3054   ` cfv 5502  (class class class)co 6176    |-> cmpt2 6178   Fincfn 7396   NN0cn0 10666   Basecbs 14262   Ringcrg 16737  Poly1cpl1 17726  coe1cco1 17727   Mat cmat 18375
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664
This theorem is referenced by:  pmatcollpw1dstlem1  31213  pmatcollpw1dst  31214  pmatcollpwfsupp  31215  pmatcollpw1lem4  31216  pmatcollpw1lem5  31217  pmatcollpwlem  31219  pmatcollpwscmatlem3  31233
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