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Theorem mamurid 27327
Description: Diagonal matrices are right identities. (Contributed by Stefan O'Rear, 3-Sep-2015.)
Hypotheses
Ref Expression
mamucl.b  |-  B  =  ( Base `  R
)
mamucl.r  |-  ( ph  ->  R  e.  Ring )
mamudiag.o  |-  .1.  =  ( 1r `  R )
mamudiag.z  |-  .0.  =  ( 0g `  R )
mamudiag.i  |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )
mamudiag.m  |-  ( ph  ->  M  e.  Fin )
mamulid.n  |-  ( ph  ->  N  e.  Fin )
mamurid.f  |-  F  =  ( R maMul  <. N ,  M ,  M >. )
mamurid.x  |-  ( ph  ->  X  e.  ( B  ^m  ( N  X.  M ) ) )
Assertion
Ref Expression
mamurid  |-  ( ph  ->  ( X F I )  =  X )
Distinct variable groups:    i, j, B    i, M, j    ph, i,
j    .1. , i, j    .0. , i, j
Allowed substitution hints:    R( i, j)    F( i, j)    I( i, j)    N( i, j)    X( i, j)

Proof of Theorem mamurid
Dummy variables  k  m  l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamurid.f . . . . 5  |-  F  =  ( R maMul  <. N ,  M ,  M >. )
2 mamucl.b . . . . 5  |-  B  =  ( Base `  R
)
3 eqid 2404 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamucl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  R  e.  Ring )
6 mamulid.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
76adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  N  e.  Fin )
8 mamudiag.m . . . . . 6  |-  ( ph  ->  M  e.  Fin )
98adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  M  e.  Fin )
10 mamurid.x . . . . . 6  |-  ( ph  ->  X  e.  ( B  ^m  ( N  X.  M ) ) )
1110adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  X  e.  ( B  ^m  ( N  X.  M
) ) )
12 mamudiag.o . . . . . . 7  |-  .1.  =  ( 1r `  R )
13 mamudiag.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
14 mamudiag.i . . . . . . 7  |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )
152, 4, 12, 13, 14, 8mamudiagcl 27325 . . . . . 6  |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
1615adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  I  e.  ( B  ^m  ( M  X.  M
) ) )
17 simprl 733 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
l  e.  N )
18 simprr 734 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  m  e.  M )
191, 2, 3, 5, 7, 9, 9, 11, 16, 17, 18mamufv 27313 . . . 4  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l ( X F I ) m )  =  ( R 
gsumg  ( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) ) )
20 rngmnd 15628 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
215, 20syl 16 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  R  e.  Mnd )
224ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  R  e.  Ring )
23 elmapi 6997 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( N  X.  M
) )  ->  X : ( N  X.  M ) --> B )
2410, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  X : ( N  X.  M ) --> B )
2524ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  X : ( N  X.  M ) --> B )
26 simplrl 737 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  l  e.  N )
27 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  k  e.  M )
2825, 26, 27fovrnd 6177 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
l X k )  e.  B )
29 elmapi 6997 . . . . . . . . . 10  |-  ( I  e.  ( B  ^m  ( M  X.  M
) )  ->  I : ( M  X.  M ) --> B )
3015, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  I : ( M  X.  M ) --> B )
3130ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  I : ( M  X.  M ) --> B )
32 simplrr 738 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  m  e.  M )
3331, 27, 32fovrnd 6177 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
k I m )  e.  B )
342, 3rngcl 15632 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B  /\  (
k I m )  e.  B )  -> 
( ( l X k ) ( .r
`  R ) ( k I m ) )  e.  B )
3522, 28, 33, 34syl3anc 1184 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
( l X k ) ( .r `  R ) ( k I m ) )  e.  B )
36 eqid 2404 . . . . . 6  |-  ( k  e.  M  |->  ( ( l X k ) ( .r `  R
) ( k I m ) ) )  =  ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) )
3735, 36fmptd 5852 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) : M --> B )
38 eldifi 3429 . . . . . . . . . 10  |-  ( k  e.  ( M  \  { m } )  ->  k  e.  M
)
39 equequ1 1692 . . . . . . . . . . . . 13  |-  ( i  =  k  ->  (
i  =  j  <->  k  =  j ) )
4039ifbid 3717 . . . . . . . . . . . 12  |-  ( i  =  k  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
k  =  j ,  .1.  ,  .0.  )
)
41 equequ2 1694 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
k  =  j  <->  k  =  m ) )
4241ifbid 3717 . . . . . . . . . . . 12  |-  ( j  =  m  ->  if ( k  =  j ,  .1.  ,  .0.  )  =  if (
k  =  m ,  .1.  ,  .0.  )
)
43 fvex 5701 . . . . . . . . . . . . . 14  |-  ( 1r
`  R )  e. 
_V
4412, 43eqeltri 2474 . . . . . . . . . . . . 13  |-  .1.  e.  _V
45 fvex 5701 . . . . . . . . . . . . . 14  |-  ( 0g
`  R )  e. 
_V
4613, 45eqeltri 2474 . . . . . . . . . . . . 13  |-  .0.  e.  _V
4744, 46ifex 3757 . . . . . . . . . . . 12  |-  if ( k  =  m ,  .1.  ,  .0.  )  e.  _V
4840, 42, 14, 47ovmpt2 6168 . . . . . . . . . . 11  |-  ( ( k  e.  M  /\  m  e.  M )  ->  ( k I m )  =  if ( k  =  m ,  .1.  ,  .0.  )
)
4927, 32, 48syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
k I m )  =  if ( k  =  m ,  .1.  ,  .0.  ) )
5038, 49sylan2 461 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  ( M  \  { m } ) )  -> 
( k I m )  =  if ( k  =  m ,  .1.  ,  .0.  )
)
51 eldifsni 3888 . . . . . . . . . . . 12  |-  ( k  e.  ( M  \  { m } )  ->  k  =/=  m
)
5251neneqd 2583 . . . . . . . . . . 11  |-  ( k  e.  ( M  \  { m } )  ->  -.  k  =  m )
53 iffalse 3706 . . . . . . . . . . 11  |-  ( -.  k  =  m  ->  if ( k  =  m ,  .1.  ,  .0.  )  =  .0.  )
5452, 53syl 16 . . . . . . . . . 10  |-  ( k  e.  ( M  \  { m } )  ->  if ( k  =  m ,  .1.  ,  .0.  )  =  .0.  )
5554adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  ( M  \  { m } ) )  ->  if ( k  =  m ,  .1.  ,  .0.  )  =  .0.  )
5650, 55eqtrd 2436 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  ( M  \  { m } ) )  -> 
( k I m )  =  .0.  )
5756oveq2d 6056 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  ( M  \  { m } ) )  -> 
( ( l X k ) ( .r
`  R ) ( k I m ) )  =  ( ( l X k ) ( .r `  R
)  .0.  ) )
582, 3, 13rngrz 15656 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B )  -> 
( ( l X k ) ( .r
`  R )  .0.  )  =  .0.  )
5922, 28, 58syl2anc 643 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
( l X k ) ( .r `  R )  .0.  )  =  .0.  )
6038, 59sylan2 461 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  ( M  \  { m } ) )  -> 
( ( l X k ) ( .r
`  R )  .0.  )  =  .0.  )
6157, 60eqtrd 2436 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  ( M  \  { m } ) )  -> 
( ( l X k ) ( .r
`  R ) ( k I m ) )  =  .0.  )
6261suppss2 6259 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( `' ( k  e.  M  |->  ( ( l X k ) ( .r `  R
) ( k I m ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  { m }
)
632, 13, 21, 9, 18, 37, 62gsumpt 15500 . . . 4  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( R  gsumg  ( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) )  =  ( ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) ) `  m ) )
64 oveq2 6048 . . . . . . . 8  |-  ( k  =  m  ->  (
l X k )  =  ( l X m ) )
65 oveq1 6047 . . . . . . . 8  |-  ( k  =  m  ->  (
k I m )  =  ( m I m ) )
6664, 65oveq12d 6058 . . . . . . 7  |-  ( k  =  m  ->  (
( l X k ) ( .r `  R ) ( k I m ) )  =  ( ( l X m ) ( .r `  R ) ( m I m ) ) )
67 ovex 6065 . . . . . . 7  |-  ( ( l X m ) ( .r `  R
) ( m I m ) )  e. 
_V
6866, 36, 67fvmpt 5765 . . . . . 6  |-  ( m  e.  M  ->  (
( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) `  m
)  =  ( ( l X m ) ( .r `  R
) ( m I m ) ) )
6968ad2antll 710 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) ) `  m )  =  ( ( l X m ) ( .r `  R ) ( m I m ) ) )
70 equequ1 1692 . . . . . . . . . 10  |-  ( i  =  m  ->  (
i  =  j  <->  m  =  j ) )
7170ifbid 3717 . . . . . . . . 9  |-  ( i  =  m  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
m  =  j ,  .1.  ,  .0.  )
)
72 equequ2 1694 . . . . . . . . . . 11  |-  ( j  =  m  ->  (
m  =  j  <->  m  =  m ) )
7372ifbid 3717 . . . . . . . . . 10  |-  ( j  =  m  ->  if ( m  =  j ,  .1.  ,  .0.  )  =  if ( m  =  m ,  .1.  ,  .0.  ) )
74 eqid 2404 . . . . . . . . . . 11  |-  m  =  m
75 iftrue 3705 . . . . . . . . . . 11  |-  ( m  =  m  ->  if ( m  =  m ,  .1.  ,  .0.  )  =  .1.  )
7674, 75ax-mp 8 . . . . . . . . . 10  |-  if ( m  =  m ,  .1.  ,  .0.  )  =  .1.
7773, 76syl6eq 2452 . . . . . . . . 9  |-  ( j  =  m  ->  if ( m  =  j ,  .1.  ,  .0.  )  =  .1.  )
7871, 77, 14, 44ovmpt2 6168 . . . . . . . 8  |-  ( ( m  e.  M  /\  m  e.  M )  ->  ( m I m )  =  .1.  )
7978anidms 627 . . . . . . 7  |-  ( m  e.  M  ->  (
m I m )  =  .1.  )
8079oveq2d 6056 . . . . . 6  |-  ( m  e.  M  ->  (
( l X m ) ( .r `  R ) ( m I m ) )  =  ( ( l X m ) ( .r `  R )  .1.  ) )
8180ad2antll 710 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( l X m ) ( .r
`  R ) ( m I m ) )  =  ( ( l X m ) ( .r `  R
)  .1.  ) )
8224fovrnda 6176 . . . . . 6  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l X m )  e.  B )
832, 3, 12rngridm 15643 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
l X m )  e.  B )  -> 
( ( l X m ) ( .r
`  R )  .1.  )  =  ( l X m ) )
845, 82, 83syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( l X m ) ( .r
`  R )  .1.  )  =  ( l X m ) )
8569, 81, 843eqtrd 2440 . . . 4  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) ) `  m )  =  ( l X m ) )
8619, 63, 853eqtrd 2440 . . 3  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l ( X F I ) m )  =  ( l X m ) )
8786ralrimivva 2758 . 2  |-  ( ph  ->  A. l  e.  N  A. m  e.  M  ( l ( X F I ) m )  =  ( l X m ) )
882, 4, 1, 6, 8, 8, 10, 15mamucl 27324 . . . . 5  |-  ( ph  ->  ( X F I )  e.  ( B  ^m  ( N  X.  M ) ) )
89 elmapi 6997 . . . . 5  |-  ( ( X F I )  e.  ( B  ^m  ( N  X.  M
) )  ->  ( X F I ) : ( N  X.  M
) --> B )
9088, 89syl 16 . . . 4  |-  ( ph  ->  ( X F I ) : ( N  X.  M ) --> B )
91 ffn 5550 . . . 4  |-  ( ( X F I ) : ( N  X.  M ) --> B  -> 
( X F I )  Fn  ( N  X.  M ) )
9290, 91syl 16 . . 3  |-  ( ph  ->  ( X F I )  Fn  ( N  X.  M ) )
93 ffn 5550 . . . 4  |-  ( X : ( N  X.  M ) --> B  ->  X  Fn  ( N  X.  M ) )
9424, 93syl 16 . . 3  |-  ( ph  ->  X  Fn  ( N  X.  M ) )
95 eqfnov2 6136 . . 3  |-  ( ( ( X F I )  Fn  ( N  X.  M )  /\  X  Fn  ( N  X.  M ) )  -> 
( ( X F I )  =  X  <->  A. l  e.  N  A. m  e.  M  ( l ( X F I ) m )  =  ( l X m ) ) )
9692, 94, 95syl2anc 643 . 2  |-  ( ph  ->  ( ( X F I )  =  X  <->  A. l  e.  N  A. m  e.  M  ( l ( X F I ) m )  =  ( l X m ) ) )
9787, 96mpbird 224 1  |-  ( ph  ->  ( X F I )  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277   ifcif 3699   {csn 3774   <.cotp 3778    e. cmpt 4226    X. cxp 4835    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    ^m cmap 6977   Fincfn 7068   Basecbs 13424   .rcmulr 13485   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639   Ringcrg 15615   1rcur 15617   maMul cmmul 27307
This theorem is referenced by:  matrng  27348  mat1  27350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-mamu 27309
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