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Theorem m2pminvval 31256
Description: The value of the inverse transformation for the transformation of a matrix over a ring  R into a polynomial matrix over the ring  R. (Contributed by AV, 12-Nov-2019.)
Hypotheses
Ref Expression
m2pminv.a  |-  A  =  ( N Mat  R )
m2pminv.k  |-  K  =  ( Base `  A
)
m2pminv.t  |-  T  =  ( N matToPolyMat  R )
m2pminv.p  |-  P  =  (Poly1 `  R )
m2pminv.c  |-  C  =  ( N Mat  P )
m2pminv.b  |-  B  =  ( Base `  C
)
m2pminv.i  |-  I  =  ( m  e.  B  |->  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x m y ) ) ` 
0 ) ) )
Assertion
Ref Expression
m2pminvval  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
I `  M )  =  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  (
x M y ) ) `  0 ) ) )
Distinct variable groups:    B, m    m, M, x, y    m, N, x, y    R, m
Allowed substitution hints:    A( x, y, m)    B( x, y)    C( x, y, m)    P( x, y, m)    R( x, y)    T( x, y, m)    I( x, y, m)    K( x, y, m)

Proof of Theorem m2pminvval
StepHypRef Expression
1 m2pminv.i . . 3  |-  I  =  ( m  e.  B  |->  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x m y ) ) ` 
0 ) ) )
21a1i 11 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  I  =  ( m  e.  B  |->  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  (
x m y ) ) `  0 ) ) ) )
3 oveq 6209 . . . . . 6  |-  ( m  =  M  ->  (
x m y )  =  ( x M y ) )
43fveq2d 5806 . . . . 5  |-  ( m  =  M  ->  (coe1 `  ( x m y ) )  =  (coe1 `  ( x M y ) ) )
54fveq1d 5804 . . . 4  |-  ( m  =  M  ->  (
(coe1 `  ( x m y ) ) ` 
0 )  =  ( (coe1 `  ( x M y ) ) ` 
0 ) )
65mpt2eq3dv 6264 . . 3  |-  ( m  =  M  ->  (
x  e.  N , 
y  e.  N  |->  ( (coe1 `  ( x m y ) ) ` 
0 ) )  =  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x M y ) ) ` 
0 ) ) )
76adantl 466 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  m  =  M )  ->  (
x  e.  N , 
y  e.  N  |->  ( (coe1 `  ( x m y ) ) ` 
0 ) )  =  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x M y ) ) ` 
0 ) ) )
8 simp3 990 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  M  e.  B )
9 simp1 988 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  N  e.  Fin )
10 eqid 2454 . . . 4  |-  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x M y ) ) `  0
) )  =  ( x  e.  N , 
y  e.  N  |->  ( (coe1 `  ( x M y ) ) ` 
0 ) )
1110mpt2exg 6761 . . 3  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  ( x M y ) ) ` 
0 ) )  e. 
_V )
129, 9, 11syl2anc 661 . 2  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
x  e.  N , 
y  e.  N  |->  ( (coe1 `  ( x M y ) ) ` 
0 ) )  e. 
_V )
132, 7, 8, 12fvmptd 5891 1  |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  ->  (
I `  M )  =  ( x  e.  N ,  y  e.  N  |->  ( (coe1 `  (
x M y ) ) `  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   Fincfn 7423   0cc0 9396   Basecbs 14295   Ringcrg 16771  Poly1cpl1 17760  coe1cco1 17761   Mat cmat 18408   matToPolyMat cmat2pmat 31215
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
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  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-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  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-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691
This theorem is referenced by:  m2pminv  31257  m2pminv2  31259
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