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Theorem mamulid 27435
Description: Diagonal matrices are left 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 )
mamulid.f  |-  F  =  ( R maMul  <. M ,  M ,  N >. )
mamulid.x  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
Assertion
Ref Expression
mamulid  |-  ( ph  ->  ( I F X )  =  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 mamulid
Dummy variables  l  m  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamulid.f . . . . 5  |-  F  =  ( R maMul  <. M ,  M ,  N >. )
2 mamucl.b . . . . 5  |-  B  =  ( Base `  R
)
3 eqid 2436 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamucl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Ring )
6 mamudiag.m . . . . . 6  |-  ( ph  ->  M  e.  Fin )
76adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  M  e.  Fin )
8 mamulid.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
98adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  N  e.  Fin )
10 mamudiag.o . . . . . . 7  |-  .1.  =  ( 1r `  R )
11 mamudiag.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
12 mamudiag.i . . . . . . 7  |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )
132, 4, 10, 11, 12, 6mamudiagcl 27434 . . . . . 6  |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
1413adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  I  e.  ( B  ^m  ( M  X.  M
) ) )
15 mamulid.x . . . . . 6  |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )
1615adantr 452 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  X  e.  ( B  ^m  ( M  X.  N
) ) )
17 simprl 733 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
l  e.  M )
18 simprr 734 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
k  e.  N )
191, 2, 3, 5, 7, 7, 9, 14, 16, 17, 18mamufv 27422 . . . 4  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l ( I F X ) k )  =  ( R 
gsumg  ( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) ) )
20 rngmnd 15673 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
215, 20syl 16 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Mnd )
224ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  R  e.  Ring )
23 elmapi 7038 . . . . . . . . . 10  |-  ( I  e.  ( B  ^m  ( M  X.  M
) )  ->  I : ( M  X.  M ) --> B )
2413, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  I : ( M  X.  M ) --> B )
2524ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  I : ( M  X.  M ) --> B )
26 simplrl 737 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  l  e.  M )
27 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  m  e.  M )
2825, 26, 27fovrnd 6218 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
l I m )  e.  B )
29 elmapi 7038 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  ->  X : ( M  X.  N ) --> B )
3015, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  X : ( M  X.  N ) --> B )
3130ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  X : ( M  X.  N ) --> B )
32 simplrr 738 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  k  e.  N )
3331, 27, 32fovrnd 6218 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
m X k )  e.  B )
342, 3rngcl 15677 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
l I m )  e.  B  /\  (
m X k )  e.  B )  -> 
( ( l I m ) ( .r
`  R ) ( m X k ) )  e.  B )
3522, 28, 33, 34syl3anc 1184 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  e.  B )
36 eqid 2436 . . . . . 6  |-  ( m  e.  M  |->  ( ( l I m ) ( .r `  R
) ( m X k ) ) )  =  ( m  e.  M  |->  ( ( l I m ) ( .r `  R ) ( m X k ) ) )
3735, 36fmptd 5893 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) : M --> B )
38 simplrl 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
l  e.  M )
39 eldifi 3469 . . . . . . . . . . 11  |-  ( m  e.  ( M  \  { l } )  ->  m  e.  M
)
4039adantl 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  ->  m  e.  M )
41 equequ1 1696 . . . . . . . . . . . 12  |-  ( i  =  l  ->  (
i  =  j  <->  l  =  j ) )
4241ifbid 3757 . . . . . . . . . . 11  |-  ( i  =  l  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
l  =  j ,  .1.  ,  .0.  )
)
43 equequ2 1698 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
l  =  j  <->  l  =  m ) )
4443ifbid 3757 . . . . . . . . . . 11  |-  ( j  =  m  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  if (
l  =  m ,  .1.  ,  .0.  )
)
45 fvex 5742 . . . . . . . . . . . . 13  |-  ( 1r
`  R )  e. 
_V
4610, 45eqeltri 2506 . . . . . . . . . . . 12  |-  .1.  e.  _V
47 fvex 5742 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  e. 
_V
4811, 47eqeltri 2506 . . . . . . . . . . . 12  |-  .0.  e.  _V
4946, 48ifex 3797 . . . . . . . . . . 11  |-  if ( l  =  m ,  .1.  ,  .0.  )  e.  _V
5042, 44, 12, 49ovmpt2 6209 . . . . . . . . . 10  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l I m )  =  if ( l  =  m ,  .1.  ,  .0.  )
)
5138, 40, 50syl2anc 643 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
( l I m )  =  if ( l  =  m ,  .1.  ,  .0.  )
)
52 eldifsni 3928 . . . . . . . . . . . . 13  |-  ( m  e.  ( M  \  { l } )  ->  m  =/=  l
)
5352necomd 2687 . . . . . . . . . . . 12  |-  ( m  e.  ( M  \  { l } )  ->  l  =/=  m
)
5453neneqd 2617 . . . . . . . . . . 11  |-  ( m  e.  ( M  \  { l } )  ->  -.  l  =  m )
55 iffalse 3746 . . . . . . . . . . 11  |-  ( -.  l  =  m  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  .0.  )
5654, 55syl 16 . . . . . . . . . 10  |-  ( m  e.  ( M  \  { l } )  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  .0.  )
5756adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  .0.  )
5851, 57eqtrd 2468 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
( l I m )  =  .0.  )
5958oveq1d 6096 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
( ( l I m ) ( .r
`  R ) ( m X k ) )  =  (  .0.  ( .r `  R
) ( m X k ) ) )
602, 3, 11rnglz 15700 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
m X k )  e.  B )  -> 
(  .0.  ( .r
`  R ) ( m X k ) )  =  .0.  )
6122, 33, 60syl2anc 643 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (  .0.  ( .r `  R
) ( m X k ) )  =  .0.  )
6239, 61sylan2 461 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
(  .0.  ( .r
`  R ) ( m X k ) )  =  .0.  )
6359, 62eqtrd 2468 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
( ( l I m ) ( .r
`  R ) ( m X k ) )  =  .0.  )
6463suppss2 6300 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( `' ( m  e.  M  |->  ( ( l I m ) ( .r `  R
) ( m X k ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  { l } )
652, 11, 21, 7, 17, 37, 64gsumpt 15545 . . . 4  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( R  gsumg  ( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) )  =  ( ( m  e.  M  |->  ( ( l I m ) ( .r `  R ) ( m X k ) ) ) `  l ) )
66 oveq2 6089 . . . . . . . 8  |-  ( m  =  l  ->  (
l I m )  =  ( l I l ) )
67 oveq1 6088 . . . . . . . 8  |-  ( m  =  l  ->  (
m X k )  =  ( l X k ) )
6866, 67oveq12d 6099 . . . . . . 7  |-  ( m  =  l  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  =  ( ( l I l ) ( .r `  R ) ( l X k ) ) )
69 ovex 6106 . . . . . . 7  |-  ( ( l I l ) ( .r `  R
) ( l X k ) )  e. 
_V
7068, 36, 69fvmpt 5806 . . . . . 6  |-  ( l  e.  M  ->  (
( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) `  l
)  =  ( ( l I l ) ( .r `  R
) ( l X k ) ) )
7170ad2antrl 709 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( m  e.  M  |->  ( ( l I m ) ( .r `  R ) ( m X k ) ) ) `  l )  =  ( ( l I l ) ( .r `  R ) ( l X k ) ) )
72 equequ2 1698 . . . . . . . . . . 11  |-  ( j  =  l  ->  (
l  =  j  <->  l  =  l ) )
7372ifbid 3757 . . . . . . . . . 10  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  if (
l  =  l ,  .1.  ,  .0.  )
)
74 equid 1688 . . . . . . . . . . 11  |-  l  =  l
75 iftrue 3745 . . . . . . . . . . 11  |-  ( l  =  l  ->  if ( l  =  l ,  .1.  ,  .0.  )  =  .1.  )
7674, 75ax-mp 8 . . . . . . . . . 10  |-  if ( l  =  l ,  .1.  ,  .0.  )  =  .1.
7773, 76syl6eq 2484 . . . . . . . . 9  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  .1.  )
7842, 77, 12, 46ovmpt2 6209 . . . . . . . 8  |-  ( ( l  e.  M  /\  l  e.  M )  ->  ( l I l )  =  .1.  )
7978anidms 627 . . . . . . 7  |-  ( l  e.  M  ->  (
l I l )  =  .1.  )
8079ad2antrl 709 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l I l )  =  .1.  )
8180oveq1d 6096 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( l I l ) ( .r
`  R ) ( l X k ) )  =  (  .1.  ( .r `  R
) ( l X k ) ) )
8230fovrnda 6217 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l X k )  e.  B )
832, 3, 10rnglidm 15687 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B )  -> 
(  .1.  ( .r
`  R ) ( l X k ) )  =  ( l X k ) )
845, 82, 83syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
(  .1.  ( .r
`  R ) ( l X k ) )  =  ( l X k ) )
8571, 81, 843eqtrd 2472 . . . 4  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( m  e.  M  |->  ( ( l I m ) ( .r `  R ) ( m X k ) ) ) `  l )  =  ( l X k ) )
8619, 65, 853eqtrd 2472 . . 3  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l ( I F X ) k )  =  ( l X k ) )
8786ralrimivva 2798 . 2  |-  ( ph  ->  A. l  e.  M  A. k  e.  N  ( l ( I F X ) k )  =  ( l X k ) )
882, 4, 1, 6, 6, 8, 13, 15mamucl 27433 . . . . 5  |-  ( ph  ->  ( I F X )  e.  ( B  ^m  ( M  X.  N ) ) )
89 elmapi 7038 . . . . 5  |-  ( ( I F X )  e.  ( B  ^m  ( M  X.  N
) )  ->  (
I F X ) : ( M  X.  N ) --> B )
9088, 89syl 16 . . . 4  |-  ( ph  ->  ( I F X ) : ( M  X.  N ) --> B )
91 ffn 5591 . . . 4  |-  ( ( I F X ) : ( M  X.  N ) --> B  -> 
( I F X )  Fn  ( M  X.  N ) )
9290, 91syl 16 . . 3  |-  ( ph  ->  ( I F X )  Fn  ( M  X.  N ) )
93 ffn 5591 . . . 4  |-  ( X : ( M  X.  N ) --> B  ->  X  Fn  ( M  X.  N ) )
9430, 93syl 16 . . 3  |-  ( ph  ->  X  Fn  ( M  X.  N ) )
95 eqfnov2 6177 . . 3  |-  ( ( ( I F X )  Fn  ( M  X.  N )  /\  X  Fn  ( M  X.  N ) )  -> 
( ( I F X )  =  X  <->  A. l  e.  M  A. k  e.  N  ( l ( I F X ) k )  =  ( l X k ) ) )
9692, 94, 95syl2anc 643 . 2  |-  ( ph  ->  ( ( I F X )  =  X  <->  A. l  e.  M  A. k  e.  N  ( l ( I F X ) k )  =  ( l X k ) ) )
9787, 96mpbird 224 1  |-  ( ph  ->  ( I F X )  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    \ cdif 3317   ifcif 3739   {csn 3814   <.cotp 3818    e. cmpt 4266    X. cxp 4876    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083    ^m cmap 7018   Fincfn 7109   Basecbs 13469   .rcmulr 13530   0gc0g 13723    gsumg cgsu 13724   Mndcmnd 14684   Ringcrg 15660   1rcur 15662   maMul cmmul 27416
This theorem is referenced by:  matrng  27457  mat1  27459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-mamu 27418
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