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Theorem mamulid 19235
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 2402 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamumat1cl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 463 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Ring )
6 mamumat1cl.m . . . . . 6  |-  ( ph  ->  M  e.  Fin )
76adantr 463 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  M  e.  Fin )
8 mamulid.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
98adantr 463 . . . . 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 19233 . . . . . 6  |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
1413adantr 463 . . . . 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 463 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  X  e.  ( B  ^m  ( M  X.  N
) ) )
17 simprl 756 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
l  e.  M )
18 simprr 758 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
k  e.  N )
191, 2, 3, 5, 7, 7, 9, 14, 16, 17, 18mamufv 19181 . . . 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 ringmnd 17527 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
215, 20syl 17 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  ->  R  e.  Mnd )
224ad2antrr 724 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  R  e.  Ring )
23 elmapi 7478 . . . . . . . . . 10  |-  ( I  e.  ( B  ^m  ( M  X.  M
) )  ->  I : ( M  X.  M ) --> B )
2413, 23syl 17 . . . . . . . . 9  |-  ( ph  ->  I : ( M  X.  M ) --> B )
2524ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  I : ( M  X.  M ) --> B )
26 simplrl 762 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  l  e.  M )
27 simpr 459 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  m  e.  M )
2825, 26, 27fovrnd 6428 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
l I m )  e.  B )
29 elmapi 7478 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( M  X.  N
) )  ->  X : ( M  X.  N ) --> B )
3015, 29syl 17 . . . . . . . . 9  |-  ( ph  ->  X : ( M  X.  N ) --> B )
3130ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  X : ( M  X.  N ) --> B )
32 simplrr 763 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  k  e.  N )
3331, 27, 32fovrnd 6428 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
m X k )  e.  B )
342, 3ringcl 17532 . . . . . . 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 1230 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  e.  B )
36 eqid 2402 . . . . . 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 6033 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( m  e.  M  |->  ( ( l I m ) ( .r
`  R ) ( m X k ) ) ) : M --> B )
38263adant3 1017 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  l  e.  M )
39 simp2 998 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  m  e.  M )
402, 4, 10, 11, 12, 6mat1comp 19234 . . . . . . . . . . 11  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l I m )  =  if ( l  =  m ,  .1.  ,  .0.  )
)
41 equcom 1818 . . . . . . . . . . . . 13  |-  ( l  =  m  <->  m  =  l )
4241a1i 11 . . . . . . . . . . . 12  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l  =  m  <-> 
m  =  l ) )
4342ifbid 3907 . . . . . . . . . . 11  |-  ( ( l  e.  M  /\  m  e.  M )  ->  if ( l  =  m ,  .1.  ,  .0.  )  =  if ( m  =  l ,  .1.  ,  .0.  )
)
4440, 43eqtrd 2443 . . . . . . . . . 10  |-  ( ( l  e.  M  /\  m  e.  M )  ->  ( l I m )  =  if ( m  =  l ,  .1.  ,  .0.  )
)
4538, 39, 44syl2anc 659 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( l
I m )  =  if ( m  =  l ,  .1.  ,  .0.  ) )
46 ifnefalse 3897 . . . . . . . . . 10  |-  ( m  =/=  l  ->  if ( m  =  l ,  .1.  ,  .0.  )  =  .0.  )
47463ad2ant3 1020 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  if (
m  =  l ,  .1.  ,  .0.  )  =  .0.  )
4845, 47eqtrd 2443 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( l
I m )  =  .0.  )
4948oveq1d 6293 . . . . . . 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, 11ringlz 17555 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
m X k )  e.  B )  -> 
(  .0.  ( .r
`  R ) ( m X k ) )  =  .0.  )
5122, 33, 50syl2anc 659 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M )  ->  (  .0.  ( .r `  R
) ( m X k ) )  =  .0.  )
52513adant3 1017 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  (  .0.  ( .r `  R ) ( m X k ) )  =  .0.  )
5349, 52eqtrd 2443 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  M  /\  k  e.  N )
)  /\  m  e.  M  /\  m  =/=  l
)  ->  ( (
l I m ) ( .r `  R
) ( m X k ) )  =  .0.  )
5453, 7suppsssn 6938 . . . . 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 17309 . . . 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 6286 . . . . . . . 8  |-  ( m  =  l  ->  (
l I m )  =  ( l I l ) )
57 oveq1 6285 . . . . . . . 8  |-  ( m  =  l  ->  (
m X k )  =  ( l X k ) )
5856, 57oveq12d 6296 . . . . . . 7  |-  ( m  =  l  ->  (
( l I m ) ( .r `  R ) ( m X k ) )  =  ( ( l I l ) ( .r `  R ) ( l X k ) ) )
59 ovex 6306 . . . . . . 7  |-  ( ( l I l ) ( .r `  R
) ( l X k ) )  e. 
_V
6058, 36, 59fvmpt 5932 . . . . . 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 726 . . . . 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 1822 . . . . . . . . . 10  |-  ( i  =  l  ->  (
i  =  j  <->  l  =  j ) )
6362ifbid 3907 . . . . . . . . 9  |-  ( i  =  l  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
l  =  j ,  .1.  ,  .0.  )
)
64 equequ2 1823 . . . . . . . . . . 11  |-  ( j  =  l  ->  (
l  =  j  <->  l  =  l ) )
6564ifbid 3907 . . . . . . . . . 10  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  if (
l  =  l ,  .1.  ,  .0.  )
)
66 equid 1815 . . . . . . . . . . 11  |-  l  =  l
6766iftruei 3892 . . . . . . . . . 10  |-  if ( l  =  l ,  .1.  ,  .0.  )  =  .1.
6865, 67syl6eq 2459 . . . . . . . . 9  |-  ( j  =  l  ->  if ( l  =  j ,  .1.  ,  .0.  )  =  .1.  )
69 fvex 5859 . . . . . . . . . 10  |-  ( 1r
`  R )  e. 
_V
7010, 69eqeltri 2486 . . . . . . . . 9  |-  .1.  e.  _V
7163, 68, 12, 70ovmpt2 6419 . . . . . . . 8  |-  ( ( l  e.  M  /\  l  e.  M )  ->  ( l I l )  =  .1.  )
7271anidms 643 . . . . . . 7  |-  ( l  e.  M  ->  (
l I l )  =  .1.  )
7372ad2antrl 726 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l I l )  =  .1.  )
7473oveq1d 6293 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( ( l I l ) ( .r
`  R ) ( l X k ) )  =  (  .1.  ( .r `  R
) ( l X k ) ) )
7530fovrnda 6427 . . . . . 6  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l X k )  e.  B )
762, 3, 10ringlidm 17542 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B )  -> 
(  .1.  ( .r
`  R ) ( l X k ) )  =  ( l X k ) )
775, 75, 76syl2anc 659 . . . . 5  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
(  .1.  ( .r
`  R ) ( l X k ) )  =  ( l X k ) )
7861, 74, 773eqtrd 2447 . . . 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 2447 . . 3  |-  ( (
ph  /\  ( l  e.  M  /\  k  e.  N ) )  -> 
( l ( I F X ) k )  =  ( l X k ) )
8079ralrimivva 2825 . 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 19195 . . . . 5  |-  ( ph  ->  ( I F X )  e.  ( B  ^m  ( M  X.  N ) ) )
82 elmapi 7478 . . . . 5  |-  ( ( I F X )  e.  ( B  ^m  ( M  X.  N
) )  ->  (
I F X ) : ( M  X.  N ) --> B )
8381, 82syl 17 . . . 4  |-  ( ph  ->  ( I F X ) : ( M  X.  N ) --> B )
84 ffn 5714 . . . 4  |-  ( ( I F X ) : ( M  X.  N ) --> B  -> 
( I F X )  Fn  ( M  X.  N ) )
8583, 84syl 17 . . 3  |-  ( ph  ->  ( I F X )  Fn  ( M  X.  N ) )
86 ffn 5714 . . . 4  |-  ( X : ( M  X.  N ) --> B  ->  X  Fn  ( M  X.  N ) )
8730, 86syl 17 . . 3  |-  ( ph  ->  X  Fn  ( M  X.  N ) )
88 eqfnov2 6390 . . 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 659 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   _Vcvv 3059   ifcif 3885   <.cotp 3980    |-> cmpt 4453    X. cxp 4821    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280    ^m cmap 7457   Fincfn 7554   Basecbs 14841   .rcmulr 14910   0gc0g 15054    gsumg cgsu 15055   Mndcmnd 16243   1rcur 17473   Ringcrg 17518   maMul cmmul 19177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-gsum 15057  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-mamu 19178
This theorem is referenced by:  matring  19237  mat1  19241
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