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Theorem mamulid 18413
Description: Diagonal matrices are left identities. (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
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 2451 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamucl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Ring )
6 mamudiag.m . . . . . 6  |-  ( ph  ->  M  e.  Fin )
76adantr 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  M  e.  Fin )
8 mamulid.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
98adantr 465 . . . . 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 18411 . . . . . 6  |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
1413adantr 465 . . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  X  e.  ( B  ^m  ( M  X.  N
) ) )
17 simprl 755 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
l  e.  M )
18 simprr 756 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
k  e.  N )
191, 2, 3, 5, 7, 7, 9, 14, 16, 17, 18mamufv 18394 . . . 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 16760 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
215, 20syl 16 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Mnd )
224ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  R  e.  Ring )
23 elmapi 7334 . . . . . . . . . 10  |-  ( I  e.  ( B  ^m  ( M  X.  M
) )  ->  I : ( M  X.  M ) --> B )
2413, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  I : ( M  X.  M ) --> B )
2524ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  I : ( M  X.  M ) --> B )
26 simplrl 759 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  l  e.  M )
27 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  m  e.  M )
2825, 26, 27fovrnd 6335 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
l I m )  e.  B )
29 elmapi 7334 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  ->  X : ( M  X.  N ) --> B )
3015, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  X : ( M  X.  N ) --> B )
3130ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  X : ( M  X.  N ) --> B )
32 simplrr 760 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  k  e.  N )
3331, 27, 32fovrnd 6335 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
m X k )  e.  B )
342, 3rngcl 16764 . . . . . . 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 1219 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  e.  B )
36 eqid 2451 . . . . . 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 5966 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) : M --> B )
38263adant3 1008 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  l  e.  M )
39 simp2 989 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  m  e.  M )
402, 4, 10, 11, 12, 6dmatcomp 18412 . . . . . . . . . . 11  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l I m )  =  if ( l  =  m ,  .1.  ,  .0.  )
)
41 equcom 1734 . . . . . . . . . . . . 13  |-  ( l  =  m  <->  m  =  l )
4241a1i 11 . . . . . . . . . . . 12  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l  =  m  <-> 
m  =  l ) )
4342ifbid 3909 . . . . . . . . . . 11  |-  ( ( l  e.  M  /\  m  e.  M )  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  if ( m  =  l ,  .1.  ,  .0.  )
)
4440, 43eqtrd 2492 . . . . . . . . . 10  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l I m )  =  if ( m  =  l ,  .1.  ,  .0.  )
)
4538, 39, 44syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( l
I m )  =  if ( m  =  l ,  .1.  ,  .0.  ) )
46 ifnefalse 3899 . . . . . . . . . 10  |-  ( m  =/=  l  ->  if ( m  =  l ,  .1.  ,  .0.  )  =  .0.  )
47463ad2ant3 1011 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  if (
m  =  l ,  .1.  ,  .0.  )  =  .0.  )
4845, 47eqtrd 2492 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( l
I m )  =  .0.  )
4948oveq1d 6205 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( (
l I m ) ( .r `  R
) ( m X k ) )  =  (  .0.  ( .r
`  R ) ( m X k ) ) )
502, 3, 11rnglz 16787 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
m X k )  e.  B )  -> 
(  .0.  ( .r
`  R ) ( m X k ) )  =  .0.  )
5122, 33, 50syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (  .0.  ( .r `  R
) ( m X k ) )  =  .0.  )
52513adant3 1008 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  (  .0.  ( .r `  R ) ( m X k ) )  =  .0.  )
5349, 52eqtrd 2492 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( (
l I m ) ( .r `  R
) ( m X k ) )  =  .0.  )
5453, 7suppsssn 6824 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( m  e.  M  |->  ( ( l I m ) ( .r `  R ) ( m X k ) ) ) supp  .0.  )  C_  { l } )
552, 11, 21, 7, 17, 37, 54gsumpt 16559 . . . 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 ) )
56 oveq2 6198 . . . . . . . 8  |-  ( m  =  l  ->  (
l I m )  =  ( l I l ) )
57 oveq1 6197 . . . . . . . 8  |-  ( m  =  l  ->  (
m X k )  =  ( l X k ) )
5856, 57oveq12d 6208 . . . . . . 7  |-  ( m  =  l  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  =  ( ( l I l ) ( .r `  R ) ( l X k ) ) )
59 ovex 6215 . . . . . . 7  |-  ( ( l I l ) ( .r `  R
) ( l X k ) )  e. 
_V
6058, 36, 59fvmpt 5873 . . . . . 6  |-  ( l  e.  M  ->  (
( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) `  l
)  =  ( ( l I l ) ( .r `  R
) ( l X k ) ) )
6160ad2antrl 727 . . . . 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 ) ) )
62 equequ1 1738 . . . . . . . . . 10  |-  ( i  =  l  ->  (
i  =  j  <->  l  =  j ) )
6362ifbid 3909 . . . . . . . . 9  |-  ( i  =  l  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
l  =  j ,  .1.  ,  .0.  )
)
64 equequ2 1739 . . . . . . . . . . 11  |-  ( j  =  l  ->  (
l  =  j  <->  l  =  l ) )
6564ifbid 3909 . . . . . . . . . 10  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  if (
l  =  l ,  .1.  ,  .0.  )
)
66 equid 1731 . . . . . . . . . . 11  |-  l  =  l
6766iftruei 3896 . . . . . . . . . 10  |-  if ( l  =  l ,  .1.  ,  .0.  )  =  .1.
6865, 67syl6eq 2508 . . . . . . . . 9  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  .1.  )
69 fvex 5799 . . . . . . . . . 10  |-  ( 1r
`  R )  e. 
_V
7010, 69eqeltri 2535 . . . . . . . . 9  |-  .1.  e.  _V
7163, 68, 12, 70ovmpt2 6326 . . . . . . . 8  |-  ( ( l  e.  M  /\  l  e.  M )  ->  ( l I l )  =  .1.  )
7271anidms 645 . . . . . . 7  |-  ( l  e.  M  ->  (
l I l )  =  .1.  )
7372ad2antrl 727 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l I l )  =  .1.  )
7473oveq1d 6205 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( l I l ) ( .r
`  R ) ( l X k ) )  =  (  .1.  ( .r `  R
) ( l X k ) ) )
7530fovrnda 6334 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l X k )  e.  B )
762, 3, 10rnglidm 16774 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B )  -> 
(  .1.  ( .r
`  R ) ( l X k ) )  =  ( l X k ) )
775, 75, 76syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
(  .1.  ( .r
`  R ) ( l X k ) )  =  ( l X k ) )
7861, 74, 773eqtrd 2496 . . . 4  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( m  e.  M  |->  ( ( l I m ) ( .r `  R ) ( m X k ) ) ) `  l )  =  ( l X k ) )
7919, 55, 783eqtrd 2496 . . 3  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l ( I F X ) k )  =  ( l X k ) )
8079ralrimivva 2904 . 2  |-  ( ph  ->  A. l  e.  M  A. k  e.  N  ( l ( I F X ) k )  =  ( l X k ) )
812, 4, 1, 6, 6, 8, 13, 15mamucl 18410 . . . . 5  |-  ( ph  ->  ( I F X )  e.  ( B  ^m  ( M  X.  N ) ) )
82 elmapi 7334 . . . . 5  |-  ( ( I F X )  e.  ( B  ^m  ( M  X.  N
) )  ->  (
I F X ) : ( M  X.  N ) --> B )
8381, 82syl 16 . . . 4  |-  ( ph  ->  ( I F X ) : ( M  X.  N ) --> B )
84 ffn 5657 . . . 4  |-  ( ( I F X ) : ( M  X.  N ) --> B  -> 
( I F X )  Fn  ( M  X.  N ) )
8583, 84syl 16 . . 3  |-  ( ph  ->  ( I F X )  Fn  ( M  X.  N ) )
86 ffn 5657 . . . 4  |-  ( X : ( M  X.  N ) --> B  ->  X  Fn  ( M  X.  N ) )
8730, 86syl 16 . . 3  |-  ( ph  ->  X  Fn  ( M  X.  N ) )
88 eqfnov2 6297 . . 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 ) ) )
8985, 87, 88syl2anc 661 . 2  |-  ( ph  ->  ( ( I F X )  =  X  <->  A. l  e.  M  A. k  e.  N  ( l ( I F X ) k )  =  ( l X k ) ) )
9080, 89mpbird 232 1  |-  ( ph  ->  ( I F X )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   _Vcvv 3068   ifcif 3889   <.cotp 3983    |-> cmpt 4448    X. cxp 4936    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190    |-> cmpt2 6192    ^m cmap 7314   Fincfn 7410   Basecbs 14276   .rcmulr 14341   0gc0g 14480    gsumg cgsu 14481   Mndcmnd 15511   1rcur 16708   Ringcrg 16751   maMul cmmul 18388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-ot 3984  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-0g 14482  df-gsum 14483  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-grp 15647  df-minusg 15648  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-mamu 18390
This theorem is referenced by:  matrng  18440  mat1  18445
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