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Theorem mamulid 27458
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 2283 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamucl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 451 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Ring )
6 mamudiag.m . . . . . 6  |-  ( ph  ->  M  e.  Fin )
76adantr 451 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  M  e.  Fin )
8 mamulid.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
98adantr 451 . . . . 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 27457 . . . . . 6  |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
1413adantr 451 . . . . 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 451 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  X  e.  ( B  ^m  ( M  X.  N
) ) )
17 simprl 732 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
l  e.  M )
18 simprr 733 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
k  e.  N )
191, 2, 3, 5, 7, 7, 9, 14, 16, 17, 18mamufv 27445 . . . 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 15350 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
215, 20syl 15 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Mnd )
224ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  R  e.  Ring )
23 elmapi 6792 . . . . . . . . . 10  |-  ( I  e.  ( B  ^m  ( M  X.  M
) )  ->  I : ( M  X.  M ) --> B )
2413, 23syl 15 . . . . . . . . 9  |-  ( ph  ->  I : ( M  X.  M ) --> B )
2524ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  I : ( M  X.  M ) --> B )
26 simplrl 736 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  l  e.  M )
27 simpr 447 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  m  e.  M )
28 fovrn 5990 . . . . . . . 8  |-  ( ( I : ( M  X.  M ) --> B  /\  l  e.  M  /\  m  e.  M
)  ->  ( l
I m )  e.  B )
2925, 26, 27, 28syl3anc 1182 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
l I m )  e.  B )
30 elmapi 6792 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  ->  X : ( M  X.  N ) --> B )
3115, 30syl 15 . . . . . . . . 9  |-  ( ph  ->  X : ( M  X.  N ) --> B )
3231ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  X : ( M  X.  N ) --> B )
33 simplrr 737 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  k  e.  N )
34 fovrn 5990 . . . . . . . 8  |-  ( ( X : ( M  X.  N ) --> B  /\  m  e.  M  /\  k  e.  N
)  ->  ( m X k )  e.  B )
3532, 27, 33, 34syl3anc 1182 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
m X k )  e.  B )
362, 3rngcl 15354 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
l I m )  e.  B  /\  (
m X k )  e.  B )  -> 
( ( l I m ) ( .r
`  R ) ( m X k ) )  e.  B )
3722, 29, 35, 36syl3anc 1182 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  e.  B )
38 eqid 2283 . . . . . 6  |-  ( m  e.  M  |->  ( ( l I m ) ( .r `  R
) ( m X k ) ) )  =  ( m  e.  M  |->  ( ( l I m ) ( .r `  R ) ( m X k ) ) )
3937, 38fmptd 5684 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) : M --> B )
40 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
l  e.  M )
41 eldifi 3298 . . . . . . . . . . 11  |-  ( m  e.  ( M  \  { l } )  ->  m  e.  M
)
4241adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  ->  m  e.  M )
43 equequ1 1648 . . . . . . . . . . . 12  |-  ( i  =  l  ->  (
i  =  j  <->  l  =  j ) )
4443ifbid 3583 . . . . . . . . . . 11  |-  ( i  =  l  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
l  =  j ,  .1.  ,  .0.  )
)
45 equequ2 1649 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
l  =  j  <->  l  =  m ) )
4645ifbid 3583 . . . . . . . . . . 11  |-  ( j  =  m  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  if (
l  =  m ,  .1.  ,  .0.  )
)
47 fvex 5539 . . . . . . . . . . . . 13  |-  ( 1r
`  R )  e. 
_V
4810, 47eqeltri 2353 . . . . . . . . . . . 12  |-  .1.  e.  _V
49 fvex 5539 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  e. 
_V
5011, 49eqeltri 2353 . . . . . . . . . . . 12  |-  .0.  e.  _V
5148, 50ifex 3623 . . . . . . . . . . 11  |-  if ( l  =  m ,  .1.  ,  .0.  )  e.  _V
5244, 46, 12, 51ovmpt2 5983 . . . . . . . . . 10  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l I m )  =  if ( l  =  m ,  .1.  ,  .0.  )
)
5340, 42, 52syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
( l I m )  =  if ( l  =  m ,  .1.  ,  .0.  )
)
54 eldifsni 3750 . . . . . . . . . . . . 13  |-  ( m  e.  ( M  \  { l } )  ->  m  =/=  l
)
5554necomd 2529 . . . . . . . . . . . 12  |-  ( m  e.  ( M  \  { l } )  ->  l  =/=  m
)
5655neneqd 2462 . . . . . . . . . . 11  |-  ( m  e.  ( M  \  { l } )  ->  -.  l  =  m )
57 iffalse 3572 . . . . . . . . . . 11  |-  ( -.  l  =  m  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  .0.  )
5856, 57syl 15 . . . . . . . . . 10  |-  ( m  e.  ( M  \  { l } )  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  .0.  )
5958adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  .0.  )
6053, 59eqtrd 2315 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
( l I m )  =  .0.  )
6160oveq1d 5873 . . . . . . 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 ) ) )
622, 3, 11rnglz 15377 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
m X k )  e.  B )  -> 
(  .0.  ( .r
`  R ) ( m X k ) )  =  .0.  )
6322, 35, 62syl2anc 642 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (  .0.  ( .r `  R
) ( m X k ) )  =  .0.  )
6441, 63sylan2 460 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
(  .0.  ( .r
`  R ) ( m X k ) )  =  .0.  )
6561, 64eqtrd 2315 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  ( M  \  { l } ) )  -> 
( ( l I m ) ( .r
`  R ) ( m X k ) )  =  .0.  )
6665suppss2 6073 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( `' ( m  e.  M  |->  ( ( l I m ) ( .r `  R
) ( m X k ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  { l } )
672, 11, 21, 7, 17, 39, 66gsumpt 15222 . . . 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 ) )
68 oveq2 5866 . . . . . . . 8  |-  ( m  =  l  ->  (
l I m )  =  ( l I l ) )
69 oveq1 5865 . . . . . . . 8  |-  ( m  =  l  ->  (
m X k )  =  ( l X k ) )
7068, 69oveq12d 5876 . . . . . . 7  |-  ( m  =  l  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  =  ( ( l I l ) ( .r `  R ) ( l X k ) ) )
71 ovex 5883 . . . . . . 7  |-  ( ( l I l ) ( .r `  R
) ( l X k ) )  e. 
_V
7270, 38, 71fvmpt 5602 . . . . . 6  |-  ( l  e.  M  ->  (
( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) `  l
)  =  ( ( l I l ) ( .r `  R
) ( l X k ) ) )
7372ad2antrl 708 . . . . 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 ) ) )
74 equequ2 1649 . . . . . . . . . . 11  |-  ( j  =  l  ->  (
l  =  j  <->  l  =  l ) )
7574ifbid 3583 . . . . . . . . . 10  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  if (
l  =  l ,  .1.  ,  .0.  )
)
76 equid 1644 . . . . . . . . . . 11  |-  l  =  l
77 iftrue 3571 . . . . . . . . . . 11  |-  ( l  =  l  ->  if ( l  =  l ,  .1.  ,  .0.  )  =  .1.  )
7876, 77ax-mp 8 . . . . . . . . . 10  |-  if ( l  =  l ,  .1.  ,  .0.  )  =  .1.
7975, 78syl6eq 2331 . . . . . . . . 9  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  .1.  )
8044, 79, 12, 48ovmpt2 5983 . . . . . . . 8  |-  ( ( l  e.  M  /\  l  e.  M )  ->  ( l I l )  =  .1.  )
8180anidms 626 . . . . . . 7  |-  ( l  e.  M  ->  (
l I l )  =  .1.  )
8281ad2antrl 708 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l I l )  =  .1.  )
8382oveq1d 5873 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( l I l ) ( .r
`  R ) ( l X k ) )  =  (  .1.  ( .r `  R
) ( l X k ) ) )
84 fovrn 5990 . . . . . . . 8  |-  ( ( X : ( M  X.  N ) --> B  /\  l  e.  M  /\  k  e.  N
)  ->  ( l X k )  e.  B )
85843expb 1152 . . . . . . 7  |-  ( ( X : ( M  X.  N ) --> B  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l X k )  e.  B )
8631, 85sylan 457 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l X k )  e.  B )
872, 3, 10rnglidm 15364 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B )  -> 
(  .1.  ( .r
`  R ) ( l X k ) )  =  ( l X k ) )
885, 86, 87syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
(  .1.  ( .r
`  R ) ( l X k ) )  =  ( l X k ) )
8973, 83, 883eqtrd 2319 . . . 4  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( m  e.  M  |->  ( ( l I m ) ( .r `  R ) ( m X k ) ) ) `  l )  =  ( l X k ) )
9019, 67, 893eqtrd 2319 . . 3  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l ( I F X ) k )  =  ( l X k ) )
9190ralrimivva 2635 . 2  |-  ( ph  ->  A. l  e.  M  A. k  e.  N  ( l ( I F X ) k )  =  ( l X k ) )
922, 4, 1, 6, 6, 8, 13, 15mamucl 27456 . . . . 5  |-  ( ph  ->  ( I F X )  e.  ( B  ^m  ( M  X.  N ) ) )
93 elmapi 6792 . . . . 5  |-  ( ( I F X )  e.  ( B  ^m  ( M  X.  N
) )  ->  (
I F X ) : ( M  X.  N ) --> B )
9492, 93syl 15 . . . 4  |-  ( ph  ->  ( I F X ) : ( M  X.  N ) --> B )
95 ffn 5389 . . . 4  |-  ( ( I F X ) : ( M  X.  N ) --> B  -> 
( I F X )  Fn  ( M  X.  N ) )
9694, 95syl 15 . . 3  |-  ( ph  ->  ( I F X )  Fn  ( M  X.  N ) )
97 ffn 5389 . . . 4  |-  ( X : ( M  X.  N ) --> B  ->  X  Fn  ( M  X.  N ) )
9831, 97syl 15 . . 3  |-  ( ph  ->  X  Fn  ( M  X.  N ) )
99 eqfnov2 5951 . . 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 ) ) )
10096, 98, 99syl2anc 642 . 2  |-  ( ph  ->  ( ( I F X )  =  X  <->  A. l  e.  M  A. k  e.  N  ( l ( I F X ) k )  =  ( l X k ) ) )
10191, 100mpbird 223 1  |-  ( ph  ->  ( I F X )  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149   ifcif 3565   {csn 3640   <.cotp 3644    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ^m cmap 6772   Fincfn 6863   Basecbs 13148   .rcmulr 13209   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361   Ringcrg 15337   1rcur 15339   maMul cmmul 27439
This theorem is referenced by:  matrng  27480  mat1  27482
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-mamu 27441
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