MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2matmul Structured version   Unicode version

Theorem mpt2matmul 19402
Description: Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.)
Hypotheses
Ref Expression
mpt2matmul.a  |-  A  =  ( N Mat  R )
mpt2matmul.b  |-  B  =  ( Base `  R
)
mpt2matmul.m  |-  .X.  =  ( .r `  A )
mpt2matmul.t  |-  .x.  =  ( .r `  R )
mpt2matmul.r  |-  ( ph  ->  R  e.  V )
mpt2matmul.n  |-  ( ph  ->  N  e.  Fin )
mpt2matmul.x  |-  X  =  ( i  e.  N ,  j  e.  N  |->  C )
mpt2matmul.y  |-  Y  =  ( i  e.  N ,  j  e.  N  |->  E )
mpt2matmul.c  |-  ( (
ph  /\  i  e.  N  /\  j  e.  N
)  ->  C  e.  B )
mpt2matmul.e  |-  ( (
ph  /\  i  e.  N  /\  j  e.  N
)  ->  E  e.  B )
mpt2matmul.d  |-  ( (
ph  /\  ( k  =  i  /\  m  =  j ) )  ->  D  =  C )
mpt2matmul.f  |-  ( (
ph  /\  ( m  =  i  /\  l  =  j ) )  ->  F  =  E )
mpt2matmul.1  |-  ( (
ph  /\  k  e.  N  /\  m  e.  N
)  ->  D  e.  U )
mpt2matmul.2  |-  ( (
ph  /\  m  e.  N  /\  l  e.  N
)  ->  F  e.  W )
Assertion
Ref Expression
mpt2matmul  |-  ( ph  ->  ( X  .X.  Y
)  =  ( k  e.  N ,  l  e.  N  |->  ( R 
gsumg  ( m  e.  N  |->  ( D  .x.  F
) ) ) ) )
Distinct variable groups:    D, i,
j    i, F, j    i, N, j, k, l, m    R, i, j, k, l, m    k, X, l, m    k, Y, l, m    ph, i, j, k, l, m    .x. , k, l
Allowed substitution hints:    A( i, j, k, m, l)    B( i, j, k, m, l)    C( i, j, k, m, l)    D( k, m, l)    .x. ( i, j, m)    .X. ( i,
j, k, m, l)    U( i, j, k, m, l)    E( i, j, k, m, l)    F( k, m, l)    V( i, j, k, m, l)    W( i, j, k, m, l)    X( i, j)    Y( i, j)

Proof of Theorem mpt2matmul
StepHypRef Expression
1 mpt2matmul.n . . 3  |-  ( ph  ->  N  e.  Fin )
2 mpt2matmul.r . . 3  |-  ( ph  ->  R  e.  V )
3 mpt2matmul.a . . . . . . 7  |-  A  =  ( N Mat  R )
4 eqid 2429 . . . . . . 7  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
53, 4matmulr 19394 . . . . . 6  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
6 mpt2matmul.m . . . . . 6  |-  .X.  =  ( .r `  A )
75, 6syl6eqr 2488 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( R maMul  <. N ,  N ,  N >. )  =  .X.  )
87oveqd 6322 . . . 4  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( X ( R maMul  <. N ,  N ,  N >. ) Y )  =  ( X  .X.  Y ) )
98eqcomd 2437 . . 3  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( X  .X.  Y
)  =  ( X ( R maMul  <. N ,  N ,  N >. ) Y ) )
101, 2, 9syl2anc 665 . 2  |-  ( ph  ->  ( X  .X.  Y
)  =  ( X ( R maMul  <. N ,  N ,  N >. ) Y ) )
11 eqid 2429 . . 3  |-  ( Base `  R )  =  (
Base `  R )
12 mpt2matmul.t . . 3  |-  .x.  =  ( .r `  R )
13 mpt2matmul.x . . . . 5  |-  X  =  ( i  e.  N ,  j  e.  N  |->  C )
14 eqid 2429 . . . . . 6  |-  ( Base `  A )  =  (
Base `  A )
15 mpt2matmul.c . . . . . . 7  |-  ( (
ph  /\  i  e.  N  /\  j  e.  N
)  ->  C  e.  B )
16 mpt2matmul.b . . . . . . 7  |-  B  =  ( Base `  R
)
1715, 16syl6eleq 2527 . . . . . 6  |-  ( (
ph  /\  i  e.  N  /\  j  e.  N
)  ->  C  e.  ( Base `  R )
)
183, 11, 14, 1, 2, 17matbas2d 19379 . . . . 5  |-  ( ph  ->  ( i  e.  N ,  j  e.  N  |->  C )  e.  (
Base `  A )
)
1913, 18syl5eqel 2521 . . . 4  |-  ( ph  ->  X  e.  ( Base `  A ) )
203, 11matbas2 19377 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
211, 2, 20syl2anc 665 . . . 4  |-  ( ph  ->  ( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
2219, 21eleqtrrd 2520 . . 3  |-  ( ph  ->  X  e.  ( (
Base `  R )  ^m  ( N  X.  N
) ) )
23 mpt2matmul.y . . . . 5  |-  Y  =  ( i  e.  N ,  j  e.  N  |->  E )
24 mpt2matmul.e . . . . . . 7  |-  ( (
ph  /\  i  e.  N  /\  j  e.  N
)  ->  E  e.  B )
2524, 16syl6eleq 2527 . . . . . 6  |-  ( (
ph  /\  i  e.  N  /\  j  e.  N
)  ->  E  e.  ( Base `  R )
)
263, 11, 14, 1, 2, 25matbas2d 19379 . . . . 5  |-  ( ph  ->  ( i  e.  N ,  j  e.  N  |->  E )  e.  (
Base `  A )
)
2723, 26syl5eqel 2521 . . . 4  |-  ( ph  ->  Y  e.  ( Base `  A ) )
2827, 21eleqtrrd 2520 . . 3  |-  ( ph  ->  Y  e.  ( (
Base `  R )  ^m  ( N  X.  N
) ) )
294, 11, 12, 2, 1, 1, 1, 22, 28mamuval 19342 . 2  |-  ( ph  ->  ( X ( R maMul  <. N ,  N ,  N >. ) Y )  =  ( k  e.  N ,  l  e.  N  |->  ( R  gsumg  ( m  e.  N  |->  ( ( k X m ) 
.x.  ( m Y l ) ) ) ) ) )
3013a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  X  =  ( i  e.  N ,  j  e.  N  |->  C ) )
31 equcom 1846 . . . . . . . . . . . . . 14  |-  ( i  =  k  <->  k  =  i )
32 equcom 1846 . . . . . . . . . . . . . 14  |-  ( j  =  m  <->  m  =  j )
3331, 32anbi12i 701 . . . . . . . . . . . . 13  |-  ( ( i  =  k  /\  j  =  m )  <->  ( k  =  i  /\  m  =  j )
)
34 mpt2matmul.d . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  =  i  /\  m  =  j ) )  ->  D  =  C )
3533, 34sylan2b 477 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  =  k  /\  j  =  m ) )  ->  D  =  C )
3635eqcomd 2437 . . . . . . . . . . 11  |-  ( (
ph  /\  ( i  =  k  /\  j  =  m ) )  ->  C  =  D )
3736ex 435 . . . . . . . . . 10  |-  ( ph  ->  ( ( i  =  k  /\  j  =  m )  ->  C  =  D ) )
38373ad2ant1 1026 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  N  /\  l  e.  N
)  ->  ( (
i  =  k  /\  j  =  m )  ->  C  =  D ) )
3938adantr 466 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  (
( i  =  k  /\  j  =  m )  ->  C  =  D ) )
4039imp 430 . . . . . . 7  |-  ( ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N
)  /\  ( i  =  k  /\  j  =  m ) )  ->  C  =  D )
41 simpl2 1009 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  k  e.  N )
42 simpr 462 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  m  e.  N )
43 simpl1 1008 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  ph )
44 mpt2matmul.1 . . . . . . . 8  |-  ( (
ph  /\  k  e.  N  /\  m  e.  N
)  ->  D  e.  U )
4543, 41, 42, 44syl3anc 1264 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  D  e.  U )
4630, 40, 41, 42, 45ovmpt2d 6438 . . . . . 6  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  (
k X m )  =  D )
4723a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  Y  =  ( i  e.  N ,  j  e.  N  |->  E ) )
48 equcom 1846 . . . . . . . . . . . . . . 15  |-  ( i  =  m  <->  m  =  i )
4948biimpi 197 . . . . . . . . . . . . . 14  |-  ( i  =  m  ->  m  =  i )
50 equcom 1846 . . . . . . . . . . . . . . 15  |-  ( j  =  l  <->  l  =  j )
5150biimpi 197 . . . . . . . . . . . . . 14  |-  ( j  =  l  ->  l  =  j )
5249, 51anim12i 568 . . . . . . . . . . . . 13  |-  ( ( i  =  m  /\  j  =  l )  ->  ( m  =  i  /\  l  =  j ) )
53 mpt2matmul.f . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( m  =  i  /\  l  =  j ) )  ->  F  =  E )
5452, 53sylan2 476 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( i  =  m  /\  j  =  l ) )  ->  F  =  E )
5554ex 435 . . . . . . . . . . 11  |-  ( ph  ->  ( ( i  =  m  /\  j  =  l )  ->  F  =  E ) )
56553ad2ant1 1026 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  N  /\  l  e.  N
)  ->  ( (
i  =  m  /\  j  =  l )  ->  F  =  E ) )
5756adantr 466 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  (
( i  =  m  /\  j  =  l )  ->  F  =  E ) )
5857imp 430 . . . . . . . 8  |-  ( ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N
)  /\  ( i  =  m  /\  j  =  l ) )  ->  F  =  E )
5958eqcomd 2437 . . . . . . 7  |-  ( ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N
)  /\  ( i  =  m  /\  j  =  l ) )  ->  E  =  F )
60 simpl3 1010 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  l  e.  N )
61 mpt2matmul.2 . . . . . . . 8  |-  ( (
ph  /\  m  e.  N  /\  l  e.  N
)  ->  F  e.  W )
6243, 42, 60, 61syl3anc 1264 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  F  e.  W )
6347, 59, 42, 60, 62ovmpt2d 6438 . . . . . 6  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  (
m Y l )  =  F )
6446, 63oveq12d 6323 . . . . 5  |-  ( ( ( ph  /\  k  e.  N  /\  l  e.  N )  /\  m  e.  N )  ->  (
( k X m )  .x.  ( m Y l ) )  =  ( D  .x.  F ) )
6564mpteq2dva 4512 . . . 4  |-  ( (
ph  /\  k  e.  N  /\  l  e.  N
)  ->  ( m  e.  N  |->  ( ( k X m ) 
.x.  ( m Y l ) ) )  =  ( m  e.  N  |->  ( D  .x.  F ) ) )
6665oveq2d 6321 . . 3  |-  ( (
ph  /\  k  e.  N  /\  l  e.  N
)  ->  ( R  gsumg  ( m  e.  N  |->  ( ( k X m )  .x.  ( m Y l ) ) ) )  =  ( R  gsumg  ( m  e.  N  |->  ( D  .x.  F
) ) ) )
6766mpt2eq3dva 6369 . 2  |-  ( ph  ->  ( k  e.  N ,  l  e.  N  |->  ( R  gsumg  ( m  e.  N  |->  ( ( k X m )  .x.  (
m Y l ) ) ) ) )  =  ( k  e.  N ,  l  e.  N  |->  ( R  gsumg  ( m  e.  N  |->  ( D 
.x.  F ) ) ) ) )
6810, 29, 673eqtrd 2474 1  |-  ( ph  ->  ( X  .X.  Y
)  =  ( k  e.  N ,  l  e.  N  |->  ( R 
gsumg  ( m  e.  N  |->  ( D  .x.  F
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   <.cotp 4010    |-> cmpt 4484    X. cxp 4852   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307    ^m cmap 7480   Fincfn 7577   Basecbs 15084   .rcmulr 15153    gsumg cgsu 15298   maMul cmmul 19339   Mat cmat 19363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-hom 15176  df-cco 15177  df-0g 15299  df-prds 15305  df-pws 15307  df-sra 18330  df-rgmod 18331  df-dsmm 19226  df-frlm 19241  df-mamu 19340  df-mat 19364
This theorem is referenced by:  mat2pmatmul  19686
  Copyright terms: Public domain W3C validator