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Theorem mvmumamul1 18473
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 2451 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
4 mvmumamul1.r . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  R  e.  Ring )
6 mvmumamul1.m . . . . . . 7  |-  ( ph  ->  M  e.  Fin )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  M  e.  Fin )
8 mvmumamul1.n . . . . . . 7  |-  ( ph  ->  N  e.  Fin )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  N  e.  Fin )
10 mvmumamul1.a . . . . . . 7  |-  ( ph  ->  A  e.  ( B  ^m  ( M  X.  N ) ) )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  A  e.  ( B  ^m  ( M  X.  N ) ) )
12 mvmumamul1.y . . . . . . 7  |-  ( ph  ->  Y  e.  ( B  ^m  N ) )
1312adantr 465 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  Y  e.  ( B  ^m  N
) )
14 simpr 461 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  i  e.  M )
151, 2, 3, 5, 7, 9, 11, 13, 14mvmulfv 18463 . . . . 5  |-  ( (
ph  /\  i  e.  M )  ->  (
( A  .x.  Y
) `  i )  =  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r `  R
) ( Y `  k ) ) ) ) )
1615adantlr 714 . . . 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 5786 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  ( Y `  j )  =  ( Y `  k ) )
18 oveq1 6194 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
j Z (/) )  =  ( k Z (/) ) )
1917, 18eqeq12d 2472 . . . . . . . . . . . 12  |-  ( j  =  k  ->  (
( Y `  j
)  =  ( j Z (/) )  <->  ( Y `  k )  =  ( k Z (/) ) ) )
2019rspcva 3164 . . . . . . . . . . 11  |-  ( ( k  e.  N  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  -> 
( Y `  k
)  =  ( k Z (/) ) )
2120expcom 435 . . . . . . . . . 10  |-  ( A. j  e.  N  ( Y `  j )  =  ( j Z
(/) )  ->  (
k  e.  N  -> 
( Y `  k
)  =  ( k Z (/) ) ) )
2221adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z (/) ) )  ->  ( k  e.  N  ->  ( Y `  k )  =  ( k Z (/) ) ) )
2322imp 429 . . . . . . . 8  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  k  e.  N )  ->  ( Y `  k
)  =  ( k Z (/) ) )
2423oveq2d 6203 . . . . . . 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
(/) ) ) )
2524mpteq2dva 4473 . . . . . 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 (/) ) ) ) )
2625oveq2d 6203 . . . . 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 (/) ) ) ) ) )
2726adantr 465 . . . 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 (/) ) ) ) ) )
28 mvmumamul1.x . . . . . . 7  |-  .X.  =  ( R maMul  <. M ,  N ,  { (/) } >. )
29 snfi 7487 . . . . . . . 8  |-  { (/) }  e.  Fin
3029a1i 11 . . . . . . 7  |-  ( (
ph  /\  i  e.  M )  ->  { (/) }  e.  Fin )
31 mvmumamul1.z . . . . . . . 8  |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  { (/) } ) ) )
3231adantr 465 . . . . . . 7  |-  ( (
ph  /\  i  e.  M )  ->  Z  e.  ( B  ^m  ( N  X.  { (/) } ) ) )
33 0ex 4517 . . . . . . . . 9  |-  (/)  e.  _V
3433snid 4000 . . . . . . . 8  |-  (/)  e.  { (/)
}
3534a1i 11 . . . . . . 7  |-  ( (
ph  /\  i  e.  M )  ->  (/)  e.  { (/)
} )
3628, 2, 3, 5, 7, 9, 30, 11, 32, 14, 35mamufv 18391 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  (
i ( A  .X.  Z ) (/) )  =  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) ) )
3736eqcomd 2458 . . . . 5  |-  ( (
ph  /\  i  e.  M )  ->  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) )  =  ( i ( A  .X.  Z ) (/) ) )
3837adantlr 714 . . . 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 ) (/) ) )
3916, 27, 383eqtrd 2495 . . 3  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  i  e.  M )  ->  ( ( A  .x.  Y ) `  i
)  =  ( i ( A  .X.  Z
) (/) ) )
4039ralrimiva 2820 . 2  |-  ( (
ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z (/) ) )  ->  A. i  e.  M  ( ( A  .x.  Y ) `  i
)  =  ( i ( A  .X.  Z
) (/) ) )
4140ex 434 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 369    = wceq 1370    e. wcel 1758   A.wral 2793   (/)c0 3732   {csn 3972   <.cop 3978   <.cotp 3980    |-> cmpt 4445    X. cxp 4933   ` cfv 5513  (class class class)co 6187    ^m cmap 7311   Fincfn 7407   Basecbs 14273   .rcmulr 14338    gsumg cgsu 14478   Ringcrg 16748   maMul cmmul 18385   maVecMul cmvmul 18459
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-1o 7017  df-en 7408  df-fin 7411  df-mamu 18387  df-mvmul 18460
This theorem is referenced by:  mavmumamul1  18474
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