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Theorem mamulid 18710
Description: The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
Hypotheses
Ref Expression
mamumat1cl.b  |-  B  =  ( Base `  R
)
mamumat1cl.r  |-  ( ph  ->  R  e.  Ring )
mamumat1cl.o  |-  .1.  =  ( 1r `  R )
mamumat1cl.z  |-  .0.  =  ( 0g `  R )
mamumat1cl.i  |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )
mamumat1cl.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    .0. , i, j    .1. , 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  k 
l  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamulid.f . . . . 5  |-  F  =  ( R maMul  <. M ,  M ,  N >. )
2 mamumat1cl.b . . . . 5  |-  B  =  ( Base `  R
)
3 eqid 2467 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamumat1cl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Ring )
6 mamumat1cl.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 mamumat1cl.o . . . . . . 7  |-  .1.  =  ( 1r `  R )
11 mamumat1cl.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
12 mamumat1cl.i . . . . . . 7  |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )
132, 4, 10, 11, 12, 6mamumat1cl 18708 . . . . . 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 18656 . . . 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 16995 . . . . . 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 7437 . . . . . . . . . 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 6429 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
l I m )  e.  B )
29 elmapi 7437 . . . . . . . . . 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 6429 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
m X k )  e.  B )
342, 3rngcl 16999 . . . . . . 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 1228 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  e.  B )
36 eqid 2467 . . . . . 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 6043 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) : M --> B )
38263adant3 1016 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  l  e.  M )
39 simp2 997 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  m  e.  M )
402, 4, 10, 11, 12, 6mat1comp 18709 . . . . . . . . . . 11  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l I m )  =  if ( l  =  m ,  .1.  ,  .0.  )
)
41 equcom 1743 . . . . . . . . . . . . 13  |-  ( l  =  m  <->  m  =  l )
4241a1i 11 . . . . . . . . . . . 12  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l  =  m  <-> 
m  =  l ) )
4342ifbid 3961 . . . . . . . . . . 11  |-  ( ( l  e.  M  /\  m  e.  M )  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  if ( m  =  l ,  .1.  ,  .0.  )
)
4440, 43eqtrd 2508 . . . . . . . . . 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 3951 . . . . . . . . . 10  |-  ( m  =/=  l  ->  if ( m  =  l ,  .1.  ,  .0.  )  =  .0.  )
47463ad2ant3 1019 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  if (
m  =  l ,  .1.  ,  .0.  )  =  .0.  )
4845, 47eqtrd 2508 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( l
I m )  =  .0.  )
4948oveq1d 6297 . . . . . . 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 17022 . . . . . . . . 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 1016 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  (  .0.  ( .r `  R ) ( m X k ) )  =  .0.  )
5349, 52eqtrd 2508 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( (
l I m ) ( .r `  R
) ( m X k ) )  =  .0.  )
5453, 7suppsssn 6932 . . . . 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 16779 . . . 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 6290 . . . . . . . 8  |-  ( m  =  l  ->  (
l I m )  =  ( l I l ) )
57 oveq1 6289 . . . . . . . 8  |-  ( m  =  l  ->  (
m X k )  =  ( l X k ) )
5856, 57oveq12d 6300 . . . . . . 7  |-  ( m  =  l  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  =  ( ( l I l ) ( .r `  R ) ( l X k ) ) )
59 ovex 6307 . . . . . . 7  |-  ( ( l I l ) ( .r `  R
) ( l X k ) )  e. 
_V
6058, 36, 59fvmpt 5948 . . . . . 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 1747 . . . . . . . . . 10  |-  ( i  =  l  ->  (
i  =  j  <->  l  =  j ) )
6362ifbid 3961 . . . . . . . . 9  |-  ( i  =  l  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
l  =  j ,  .1.  ,  .0.  )
)
64 equequ2 1748 . . . . . . . . . . 11  |-  ( j  =  l  ->  (
l  =  j  <->  l  =  l ) )
6564ifbid 3961 . . . . . . . . . 10  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  if (
l  =  l ,  .1.  ,  .0.  )
)
66 equid 1740 . . . . . . . . . . 11  |-  l  =  l
6766iftruei 3946 . . . . . . . . . 10  |-  if ( l  =  l ,  .1.  ,  .0.  )  =  .1.
6865, 67syl6eq 2524 . . . . . . . . 9  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  .1.  )
69 fvex 5874 . . . . . . . . . 10  |-  ( 1r
`  R )  e. 
_V
7010, 69eqeltri 2551 . . . . . . . . 9  |-  .1.  e.  _V
7163, 68, 12, 70ovmpt2 6420 . . . . . . . 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 6297 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( l I l ) ( .r
`  R ) ( l X k ) )  =  (  .1.  ( .r `  R
) ( l X k ) ) )
7530fovrnda 6428 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l X k )  e.  B )
762, 3, 10rnglidm 17009 . . . . . 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 2512 . . . 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 2512 . . 3  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l ( I F X ) k )  =  ( l X k ) )
8079ralrimivva 2885 . 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 18670 . . . . 5  |-  ( ph  ->  ( I F X )  e.  ( B  ^m  ( M  X.  N ) ) )
82 elmapi 7437 . . . . 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 5729 . . . 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 5729 . . . 4  |-  ( X : ( M  X.  N ) --> B  ->  X  Fn  ( M  X.  N ) )
8730, 86syl 16 . . 3  |-  ( ph  ->  X  Fn  ( M  X.  N ) )
88 eqfnov2 6391 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113   ifcif 3939   <.cotp 4035    |-> cmpt 4505    X. cxp 4997    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284    ^m cmap 7417   Fincfn 7513   Basecbs 14486   .rcmulr 14552   0gc0g 14691    gsumg cgsu 14692   Mndcmnd 15722   1rcur 16943   Ringcrg 16986   maMul cmmul 18652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-gsum 14694  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-mamu 18653
This theorem is referenced by:  matrng  18712  mat1  18716
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