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Theorem mvmumamul1 19348
Description: The multiplication of an MxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an MxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.)
Hypotheses
Ref Expression
mvmumamul1.x  |-  .X.  =  ( R maMul  <. M ,  N ,  { (/) } >. )
mvmumamul1.t  |-  .x.  =  ( R maVecMul  <. M ,  N >. )
mvmumamul1.b  |-  B  =  ( Base `  R
)
mvmumamul1.r  |-  ( ph  ->  R  e.  Ring )
mvmumamul1.m  |-  ( ph  ->  M  e.  Fin )
mvmumamul1.n  |-  ( ph  ->  N  e.  Fin )
mvmumamul1.a  |-  ( ph  ->  A  e.  ( B  ^m  ( M  X.  N ) ) )
mvmumamul1.y  |-  ( ph  ->  Y  e.  ( B  ^m  N ) )
mvmumamul1.z  |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  { (/) } ) ) )
Assertion
Ref Expression
mvmumamul1  |-  ( ph  ->  ( A. j  e.  N  ( Y `  j )  =  ( j Z (/) )  ->  A. i  e.  M  ( ( A  .x.  Y ) `  i
)  =  ( i ( A  .X.  Z
) (/) ) ) )
Distinct variable groups:    i, j, N    i, Y, j    i, Z, j    ph, i, j
Allowed substitution hints:    A( i, j)    B( i, j)    R( i, j)    .x. ( i, j)    .X. ( i,
j)    M( i, j)

Proof of Theorem mvmumamul1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 mvmumamul1.t . . . . . 6  |-  .x.  =  ( R maVecMul  <. M ,  N >. )
2 mvmumamul1.b . . . . . 6  |-  B  =  ( Base `  R
)
3 eqid 2402 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
4 mvmumamul1.r . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
54adantr 463 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  R  e.  Ring )
6 mvmumamul1.m . . . . . . 7  |-  ( ph  ->  M  e.  Fin )
76adantr 463 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  M  e.  Fin )
8 mvmumamul1.n . . . . . . 7  |-  ( ph  ->  N  e.  Fin )
98adantr 463 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  N  e.  Fin )
10 mvmumamul1.a . . . . . . 7  |-  ( ph  ->  A  e.  ( B  ^m  ( M  X.  N ) ) )
1110adantr 463 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  A  e.  ( B  ^m  ( M  X.  N ) ) )
12 mvmumamul1.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( B  ^m  N ) )
1312adantr 463 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  Y  e.  ( B  ^m  N
) )
14 simpr 459 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  i  e.  M )
151, 2, 3, 5, 7, 9, 11, 13, 14mvmulfv 19338 . . . . 5  |-  ( (
ph  /\  i  e.  M )  ->  (
( A  .x.  Y
) `  i )  =  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r `  R
) ( Y `  k ) ) ) ) )
1615adantlr 713 . . . 4  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  i  e.  M )  ->  ( ( A  .x.  Y ) `  i
)  =  ( R 
gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( Y `  k ) ) ) ) )
17 fveq2 5849 . . . . . . . . . . . 12  |-  ( j  =  k  ->  ( Y `  j )  =  ( Y `  k ) )
18 oveq1 6285 . . . . . . . . . . . 12  |-  ( j  =  k  ->  (
j Z (/) )  =  ( k Z (/) ) )
1917, 18eqeq12d 2424 . . . . . . . . . . 11  |-  ( j  =  k  ->  (
( Y `  j
)  =  ( j Z (/) )  <->  ( Y `  k )  =  ( k Z (/) ) ) )
2019rspccv 3157 . . . . . . . . . 10  |-  ( A. j  e.  N  ( Y `  j )  =  ( j Z
(/) )  ->  (
k  e.  N  -> 
( Y `  k
)  =  ( k Z (/) ) ) )
2120adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z (/) ) )  ->  ( k  e.  N  ->  ( Y `  k )  =  ( k Z (/) ) ) )
2221imp 427 . . . . . . . 8  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  k  e.  N )  ->  ( Y `  k
)  =  ( k Z (/) ) )
2322oveq2d 6294 . . . . . . 7  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  k  e.  N )  ->  ( ( i A k ) ( .r
`  R ) ( Y `  k ) )  =  ( ( i A k ) ( .r `  R
) ( k Z
(/) ) ) )
2423mpteq2dva 4481 . . . . . 6  |-  ( (
ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z (/) ) )  ->  ( k  e.  N  |->  ( ( i A k ) ( .r `  R ) ( Y `  k
) ) )  =  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) )
2524oveq2d 6294 . . . . 5  |-  ( (
ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z (/) ) )  ->  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r `  R
) ( Y `  k ) ) ) )  =  ( R 
gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) ) )
2625adantr 463 . . . 4  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  i  e.  M )  ->  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( Y `  k ) ) ) )  =  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) ) )
27 mvmumamul1.x . . . . . . 7  |-  .X.  =  ( R maMul  <. M ,  N ,  { (/) } >. )
28 snfi 7634 . . . . . . . 8  |-  { (/) }  e.  Fin
2928a1i 11 . . . . . . 7  |-  ( (
ph  /\  i  e.  M )  ->  { (/) }  e.  Fin )
30 mvmumamul1.z . . . . . . . 8  |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  { (/) } ) ) )
3130adantr 463 . . . . . . 7  |-  ( (
ph  /\  i  e.  M )  ->  Z  e.  ( B  ^m  ( N  X.  { (/) } ) ) )
32 0ex 4526 . . . . . . . . 9  |-  (/)  e.  _V
3332snid 4000 . . . . . . . 8  |-  (/)  e.  { (/)
}
3433a1i 11 . . . . . . 7  |-  ( (
ph  /\  i  e.  M )  ->  (/)  e.  { (/)
} )
3527, 2, 3, 5, 7, 9, 29, 11, 31, 14, 34mamufv 19181 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  (
i ( A  .X.  Z ) (/) )  =  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) ) )
3635eqcomd 2410 . . . . 5  |-  ( (
ph  /\  i  e.  M )  ->  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) )  =  ( i ( A  .X.  Z ) (/) ) )
3736adantlr 713 . . . 4  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  i  e.  M )  ->  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) )  =  ( i ( A  .X.  Z ) (/) ) )
3816, 26, 373eqtrd 2447 . . 3  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  i  e.  M )  ->  ( ( A  .x.  Y ) `  i
)  =  ( i ( A  .X.  Z
) (/) ) )
3938ralrimiva 2818 . 2  |-  ( (
ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z (/) ) )  ->  A. i  e.  M  ( ( A  .x.  Y ) `  i
)  =  ( i ( A  .X.  Z
) (/) ) )
4039ex 432 1  |-  ( ph  ->  ( A. j  e.  N  ( Y `  j )  =  ( j Z (/) )  ->  A. i  e.  M  ( ( A  .x.  Y ) `  i
)  =  ( i ( A  .X.  Z
) (/) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   (/)c0 3738   {csn 3972   <.cop 3978   <.cotp 3980    |-> cmpt 4453    X. cxp 4821   ` cfv 5569  (class class class)co 6278    ^m cmap 7457   Fincfn 7554   Basecbs 14841   .rcmulr 14910    gsumg cgsu 15055   Ringcrg 17518   maMul cmmul 19177   maVecMul cmvmul 19334
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  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 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-1o 7167  df-en 7555  df-fin 7558  df-mamu 19178  df-mvmul 19335
This theorem is referenced by:  mavmumamul1  19349
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