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Theorem mat2pmatval 19515
Description: The result of a matrix transformation. (Contributed by AV, 31-Jul-2019.)
Hypotheses
Ref Expression
mat2pmatfval.t  |-  T  =  ( N matToPolyMat  R )
mat2pmatfval.a  |-  A  =  ( N Mat  R )
mat2pmatfval.b  |-  B  =  ( Base `  A
)
mat2pmatfval.p  |-  P  =  (Poly1 `  R )
mat2pmatfval.s  |-  S  =  (algSc `  P )
Assertion
Ref Expression
mat2pmatval  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  ( T `  M
)  =  ( x  e.  N ,  y  e.  N  |->  ( S `
 ( x M y ) ) ) )
Distinct variable groups:    x, y, N    x, R, y    x, M, y
Allowed substitution hints:    A( x, y)    B( x, y)    P( x, y)    S( x, y)    T( x, y)    V( x, y)

Proof of Theorem mat2pmatval
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 mat2pmatfval.t . . . 4  |-  T  =  ( N matToPolyMat  R )
2 mat2pmatfval.a . . . 4  |-  A  =  ( N Mat  R )
3 mat2pmatfval.b . . . 4  |-  B  =  ( Base `  A
)
4 mat2pmatfval.p . . . 4  |-  P  =  (Poly1 `  R )
5 mat2pmatfval.s . . . 4  |-  S  =  (algSc `  P )
61, 2, 3, 4, 5mat2pmatfval 19514 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  T  =  ( m  e.  B  |->  ( x  e.  N ,  y  e.  N  |->  ( S `
 ( x m y ) ) ) ) )
763adant3 1017 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  T  =  ( m  e.  B  |->  ( x  e.  N ,  y  e.  N  |->  ( S `
 ( x m y ) ) ) ) )
8 oveq 6283 . . . . 5  |-  ( m  =  M  ->  (
x m y )  =  ( x M y ) )
98fveq2d 5852 . . . 4  |-  ( m  =  M  ->  ( S `  ( x m y ) )  =  ( S `  ( x M y ) ) )
109mpt2eq3dv 6343 . . 3  |-  ( m  =  M  ->  (
x  e.  N , 
y  e.  N  |->  ( S `  ( x m y ) ) )  =  ( x  e.  N ,  y  e.  N  |->  ( S `
 ( x M y ) ) ) )
1110adantl 464 . 2  |-  ( ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  /\  m  =  M
)  ->  ( x  e.  N ,  y  e.  N  |->  ( S `  ( x m y ) ) )  =  ( x  e.  N ,  y  e.  N  |->  ( S `  (
x M y ) ) ) )
12 simp3 999 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  M  e.  B )
13 simp1 997 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  N  e.  Fin )
14 mpt2exga 6859 . . 3  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( x  e.  N ,  y  e.  N  |->  ( S `  (
x M y ) ) )  e.  _V )
1513, 13, 14syl2anc 659 . 2  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  ( x  e.  N ,  y  e.  N  |->  ( S `  (
x M y ) ) )  e.  _V )
167, 11, 12, 15fvmptd 5937 1  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  ( T `  M
)  =  ( x  e.  N ,  y  e.  N  |->  ( S `
 ( x M y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3058    |-> cmpt 4452   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   Fincfn 7553   Basecbs 14839  algSccascl 18278  Poly1cpl1 18534   Mat cmat 19199   matToPolyMat cmat2pmat 19495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-mat2pmat 19498
This theorem is referenced by:  mat2pmatvalel  19516  mat2pmatbas  19517  mat2pmatghm  19521  mat2pmatmul  19522  d0mat2pmat  19529  d1mat2pmat  19530  m2cpminvid2  19546  pmatcollpwlem  19571  pmatcollpwscmatlem2  19581
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