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Theorem mvmumamul1 18820
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 2467 . . . . . 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 18810 . . . . 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 5864 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  ( Y `  j )  =  ( Y `  k ) )
18 oveq1 6289 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  (
j Z (/) )  =  ( k Z (/) ) )
1917, 18eqeq12d 2489 . . . . . . . . . . . 12  |-  ( j  =  k  ->  (
( Y `  j
)  =  ( j Z (/) )  <->  ( Y `  k )  =  ( k Z (/) ) ) )
2019rspcva 3212 . . . . . . . . . . 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 6298 . . . . . . 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 4533 . . . . . 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 6298 . . . . 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 7593 . . . . . . . 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 4577 . . . . . . . . 9  |-  (/)  e.  _V
3433snid 4055 . . . . . . . 8  |-  (/)  e.  { (/)
}
3534a1i 11 . . . . . . 7  |-  ( (
ph  /\  i  e.  M )  ->  (/)  e.  { (/)
} )
3628, 2, 3, 5, 7, 9, 30, 11, 32, 14, 35mamufv 18653 . . . . . 6  |-  ( (
ph  /\  i  e.  M )  ->  (
i ( A  .X.  Z ) (/) )  =  ( R  gsumg  ( k  e.  N  |->  ( ( i A k ) ( .r
`  R ) ( k Z (/) ) ) ) ) )
3736eqcomd 2475 . . . . 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 2512 . . 3  |-  ( ( ( ph  /\  A. j  e.  N  ( Y `  j )  =  ( j Z
(/) ) )  /\  i  e.  M )  ->  ( ( A  .x.  Y ) `  i
)  =  ( i ( A  .X.  Z
) (/) ) )
4039ralrimiva 2878 . 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 1379    e. wcel 1767   A.wral 2814   (/)c0 3785   {csn 4027   <.cop 4033   <.cotp 4035    |-> cmpt 4505    X. cxp 4997   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Fincfn 7513   Basecbs 14483   .rcmulr 14549    gsumg cgsu 14689   Ringcrg 16983   maMul cmmul 18649   maVecMul cmvmul 18806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-en 7514  df-fin 7517  df-mamu 18650  df-mvmul 18807
This theorem is referenced by:  mavmumamul1  18821
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