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Theorem d1mat2pmat 19410
Description: The transformation of a matrix of dimenson 1. (Contributed by AV, 4-Aug-2019.)
Hypotheses
Ref Expression
d1mat2pmat.t  |-  T  =  ( N matToPolyMat  R )
d1mat2pmat.b  |-  B  =  ( Base `  ( N Mat  R ) )
d1mat2pmat.p  |-  P  =  (Poly1 `  R )
d1mat2pmat.s  |-  S  =  (algSc `  P )
Assertion
Ref Expression
d1mat2pmat  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  ( T `  M )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. } )

Proof of Theorem d1mat2pmat
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snfi 7589 . . . . . 6  |-  { A }  e.  Fin
2 eleq1 2526 . . . . . 6  |-  ( N  =  { A }  ->  ( N  e.  Fin  <->  { A }  e.  Fin ) )
31, 2mpbiri 233 . . . . 5  |-  ( N  =  { A }  ->  N  e.  Fin )
43adantr 463 . . . 4  |-  ( ( N  =  { A }  /\  A  e.  V
)  ->  N  e.  Fin )
543ad2ant2 1016 . . 3  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  N  e.  Fin )
6 simp1 994 . . 3  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  R  e.  V )
7 simp3 996 . . 3  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  M  e.  B )
8 d1mat2pmat.t . . . 4  |-  T  =  ( N matToPolyMat  R )
9 eqid 2454 . . . 4  |-  ( N Mat 
R )  =  ( N Mat  R )
10 d1mat2pmat.b . . . 4  |-  B  =  ( Base `  ( N Mat  R ) )
11 d1mat2pmat.p . . . 4  |-  P  =  (Poly1 `  R )
12 d1mat2pmat.s . . . 4  |-  S  =  (algSc `  P )
138, 9, 10, 11, 12mat2pmatval 19395 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  B )  ->  ( T `  M
)  =  ( i  e.  N ,  j  e.  N  |->  ( S `
 ( i M j ) ) ) )
145, 6, 7, 13syl3anc 1226 . 2  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  ( T `  M )  =  ( i  e.  N ,  j  e.  N  |->  ( S `  ( i M j ) ) ) )
15 id 22 . . . . . . 7  |-  ( A  e.  V  ->  A  e.  V )
16 fvex 5858 . . . . . . . 8  |-  ( S `
 ( A M A ) )  e. 
_V
1716a1i 11 . . . . . . 7  |-  ( A  e.  V  ->  ( S `  ( A M A ) )  e. 
_V )
1815, 15, 173jca 1174 . . . . . 6  |-  ( A  e.  V  ->  ( A  e.  V  /\  A  e.  V  /\  ( S `  ( A M A ) )  e.  _V ) )
1918adantl 464 . . . . 5  |-  ( ( N  =  { A }  /\  A  e.  V
)  ->  ( A  e.  V  /\  A  e.  V  /\  ( S `
 ( A M A ) )  e. 
_V ) )
20193ad2ant2 1016 . . . 4  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  ( A  e.  V  /\  A  e.  V  /\  ( S `  ( A M A ) )  e.  _V ) )
21 eqid 2454 . . . . 5  |-  ( i  e.  { A } ,  j  e.  { A }  |->  ( S `  ( i M j ) ) )  =  ( i  e.  { A } ,  j  e. 
{ A }  |->  ( S `  ( i M j ) ) )
22 oveq1 6277 . . . . . 6  |-  ( i  =  A  ->  (
i M j )  =  ( A M j ) )
2322fveq2d 5852 . . . . 5  |-  ( i  =  A  ->  ( S `  ( i M j ) )  =  ( S `  ( A M j ) ) )
24 oveq2 6278 . . . . . 6  |-  ( j  =  A  ->  ( A M j )  =  ( A M A ) )
2524fveq2d 5852 . . . . 5  |-  ( j  =  A  ->  ( S `  ( A M j ) )  =  ( S `  ( A M A ) ) )
2621, 23, 25mpt2sn 6864 . . . 4  |-  ( ( A  e.  V  /\  A  e.  V  /\  ( S `  ( A M A ) )  e.  _V )  -> 
( i  e.  { A } ,  j  e. 
{ A }  |->  ( S `  ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) )
>. } )
2720, 26syl 16 . . 3  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  (
i  e.  { A } ,  j  e.  { A }  |->  ( S `
 ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `
 ( A M A ) ) >. } )
28 mpt2eq12 6330 . . . . . . 7  |-  ( ( N  =  { A }  /\  N  =  { A } )  ->  (
i  e.  N , 
j  e.  N  |->  ( S `  ( i M j ) ) )  =  ( i  e.  { A } ,  j  e.  { A }  |->  ( S `  ( i M j ) ) ) )
2928eqeq1d 2456 . . . . . 6  |-  ( ( N  =  { A }  /\  N  =  { A } )  ->  (
( i  e.  N ,  j  e.  N  |->  ( S `  (
i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. }  <->  ( i  e.  { A } , 
j  e.  { A }  |->  ( S `  ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. } ) )
3029anidms 643 . . . . 5  |-  ( N  =  { A }  ->  ( ( i  e.  N ,  j  e.  N  |->  ( S `  ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. }  <->  ( i  e.  { A } , 
j  e.  { A }  |->  ( S `  ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. } ) )
3130adantr 463 . . . 4  |-  ( ( N  =  { A }  /\  A  e.  V
)  ->  ( (
i  e.  N , 
j  e.  N  |->  ( S `  ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) )
>. }  <->  ( i  e. 
{ A } , 
j  e.  { A }  |->  ( S `  ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. } ) )
32313ad2ant2 1016 . . 3  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  (
( i  e.  N ,  j  e.  N  |->  ( S `  (
i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. }  <->  ( i  e.  { A } , 
j  e.  { A }  |->  ( S `  ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. } ) )
3327, 32mpbird 232 . 2  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  (
i  e.  N , 
j  e.  N  |->  ( S `  ( i M j ) ) )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) )
>. } )
3414, 33eqtrd 2495 1  |-  ( ( R  e.  V  /\  ( N  =  { A }  /\  A  e.  V )  /\  M  e.  B )  ->  ( T `  M )  =  { <. <. A ,  A >. ,  ( S `  ( A M A ) ) >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016   <.cop 4022   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Fincfn 7509   Basecbs 14719  algSccascl 18158  Poly1cpl1 18414   Mat cmat 19079   matToPolyMat cmat2pmat 19375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-1o 7122  df-en 7510  df-fin 7513  df-mat2pmat 19378
This theorem is referenced by: (None)
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