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Theorem mamurid 18704
Description: The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (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 )
mamurid.f  |-  F  =  ( R maMul  <. N ,  M ,  M >. )
mamurid.x  |-  ( ph  ->  X  e.  ( B  ^m  ( N  X.  M ) ) )
Assertion
Ref Expression
mamurid  |-  ( ph  ->  ( X F I )  =  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 mamurid
Dummy variables  k 
l  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamurid.f . . . . 5  |-  F  =  ( R maMul  <. N ,  M ,  M >. )
2 mamumat1cl.b . . . . 5  |-  B  =  ( Base `  R
)
3 eqid 2460 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamumat1cl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  R  e.  Ring )
6 mamulid.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
76adantr 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  N  e.  Fin )
8 mamumat1cl.m . . . . . 6  |-  ( ph  ->  M  e.  Fin )
98adantr 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  M  e.  Fin )
10 mamurid.x . . . . . 6  |-  ( ph  ->  X  e.  ( B  ^m  ( N  X.  M ) ) )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  X  e.  ( B  ^m  ( N  X.  M
) ) )
12 mamumat1cl.o . . . . . . 7  |-  .1.  =  ( 1r `  R )
13 mamumat1cl.z . . . . . . 7  |-  .0.  =  ( 0g `  R )
14 mamumat1cl.i . . . . . . 7  |-  I  =  ( i  e.  M ,  j  e.  M  |->  if ( i  =  j ,  .1.  ,  .0.  ) )
152, 4, 12, 13, 14, 8mamumat1cl 18701 . . . . . 6  |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
1615adantr 465 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  I  e.  ( B  ^m  ( M  X.  M
) ) )
17 simprl 755 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
l  e.  N )
18 simprr 756 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  m  e.  M )
191, 2, 3, 5, 7, 9, 9, 11, 16, 17, 18mamufv 18649 . . . 4  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l ( X F I ) m )  =  ( R 
gsumg  ( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) ) )
20 rngmnd 16988 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
215, 20syl 16 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  R  e.  Mnd )
224ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  R  e.  Ring )
23 elmapi 7430 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( N  X.  M
) )  ->  X : ( N  X.  M ) --> B )
2410, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  X : ( N  X.  M ) --> B )
2524ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  X : ( N  X.  M ) --> B )
26 simplrl 759 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  l  e.  N )
27 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  k  e.  M )
2825, 26, 27fovrnd 6422 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
l X k )  e.  B )
29 elmapi 7430 . . . . . . . . . 10  |-  ( I  e.  ( B  ^m  ( M  X.  M
) )  ->  I : ( M  X.  M ) --> B )
3015, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  I : ( M  X.  M ) --> B )
3130ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  I : ( M  X.  M ) --> B )
32 simplrr 760 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  m  e.  M )
3331, 27, 32fovrnd 6422 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
k I m )  e.  B )
342, 3rngcl 16992 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B  /\  (
k I m )  e.  B )  -> 
( ( l X k ) ( .r
`  R ) ( k I m ) )  e.  B )
3522, 28, 33, 34syl3anc 1223 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
( l X k ) ( .r `  R ) ( k I m ) )  e.  B )
36 eqid 2460 . . . . . 6  |-  ( k  e.  M  |->  ( ( l X k ) ( .r `  R
) ( k I m ) ) )  =  ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) )
3735, 36fmptd 6036 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) : M --> B )
38 simp2 992 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  k  e.  M )
39323adant3 1011 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  m  e.  M )
402, 4, 12, 13, 14, 8mat1comp 18702 . . . . . . . . . 10  |-  ( ( k  e.  M  /\  m  e.  M )  ->  ( k I m )  =  if ( k  =  m ,  .1.  ,  .0.  )
)
4138, 39, 40syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( k
I m )  =  if ( k  =  m ,  .1.  ,  .0.  ) )
42 ifnefalse 3944 . . . . . . . . . 10  |-  ( k  =/=  m  ->  if ( k  =  m ,  .1.  ,  .0.  )  =  .0.  )
43423ad2ant3 1014 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  if (
k  =  m ,  .1.  ,  .0.  )  =  .0.  )
4441, 43eqtrd 2501 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( k
I m )  =  .0.  )
4544oveq2d 6291 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( (
l X k ) ( .r `  R
) ( k I m ) )  =  ( ( l X k ) ( .r
`  R )  .0.  ) )
462, 3, 13rngrz 17016 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B )  -> 
( ( l X k ) ( .r
`  R )  .0.  )  =  .0.  )
4722, 28, 46syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
( l X k ) ( .r `  R )  .0.  )  =  .0.  )
48473adant3 1011 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( (
l X k ) ( .r `  R
)  .0.  )  =  .0.  )
4945, 48eqtrd 2501 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( (
l X k ) ( .r `  R
) ( k I m ) )  =  .0.  )
5049, 9suppsssn 6925 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) ) supp  .0.  )  C_  { m }
)
512, 13, 21, 9, 18, 37, 50gsumpt 16772 . . . 4  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( R  gsumg  ( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) )  =  ( ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) ) `  m ) )
52 oveq2 6283 . . . . . . . 8  |-  ( k  =  m  ->  (
l X k )  =  ( l X m ) )
53 oveq1 6282 . . . . . . . 8  |-  ( k  =  m  ->  (
k I m )  =  ( m I m ) )
5452, 53oveq12d 6293 . . . . . . 7  |-  ( k  =  m  ->  (
( l X k ) ( .r `  R ) ( k I m ) )  =  ( ( l X m ) ( .r `  R ) ( m I m ) ) )
55 ovex 6300 . . . . . . 7  |-  ( ( l X m ) ( .r `  R
) ( m I m ) )  e. 
_V
5654, 36, 55fvmpt 5941 . . . . . 6  |-  ( m  e.  M  ->  (
( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) `  m
)  =  ( ( l X m ) ( .r `  R
) ( m I m ) ) )
5756ad2antll 728 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) ) `  m )  =  ( ( l X m ) ( .r `  R ) ( m I m ) ) )
58 equequ1 1742 . . . . . . . . . 10  |-  ( i  =  m  ->  (
i  =  j  <->  m  =  j ) )
5958ifbid 3954 . . . . . . . . 9  |-  ( i  =  m  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
m  =  j ,  .1.  ,  .0.  )
)
60 equequ2 1743 . . . . . . . . . . 11  |-  ( j  =  m  ->  (
m  =  j  <->  m  =  m ) )
6160ifbid 3954 . . . . . . . . . 10  |-  ( j  =  m  ->  if ( m  =  j ,  .1.  ,  .0.  )  =  if ( m  =  m ,  .1.  ,  .0.  ) )
62 eqid 2460 . . . . . . . . . . 11  |-  m  =  m
6362iftruei 3939 . . . . . . . . . 10  |-  if ( m  =  m ,  .1.  ,  .0.  )  =  .1.
6461, 63syl6eq 2517 . . . . . . . . 9  |-  ( j  =  m  ->  if ( m  =  j ,  .1.  ,  .0.  )  =  .1.  )
65 fvex 5867 . . . . . . . . . 10  |-  ( 1r
`  R )  e. 
_V
6612, 65eqeltri 2544 . . . . . . . . 9  |-  .1.  e.  _V
6759, 64, 14, 66ovmpt2 6413 . . . . . . . 8  |-  ( ( m  e.  M  /\  m  e.  M )  ->  ( m I m )  =  .1.  )
6867anidms 645 . . . . . . 7  |-  ( m  e.  M  ->  (
m I m )  =  .1.  )
6968oveq2d 6291 . . . . . 6  |-  ( m  e.  M  ->  (
( l X m ) ( .r `  R ) ( m I m ) )  =  ( ( l X m ) ( .r `  R )  .1.  ) )
7069ad2antll 728 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( l X m ) ( .r
`  R ) ( m I m ) )  =  ( ( l X m ) ( .r `  R
)  .1.  ) )
7124fovrnda 6421 . . . . . 6  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l X m )  e.  B )
722, 3, 12rngridm 17003 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
l X m )  e.  B )  -> 
( ( l X m ) ( .r
`  R )  .1.  )  =  ( l X m ) )
735, 71, 72syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( l X m ) ( .r
`  R )  .1.  )  =  ( l X m ) )
7457, 70, 733eqtrd 2505 . . . 4  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( k  e.  M  |->  ( ( l X k ) ( .r `  R ) ( k I m ) ) ) `  m )  =  ( l X m ) )
7519, 51, 743eqtrd 2505 . . 3  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l ( X F I ) m )  =  ( l X m ) )
7675ralrimivva 2878 . 2  |-  ( ph  ->  A. l  e.  N  A. m  e.  M  ( l ( X F I ) m )  =  ( l X m ) )
772, 4, 1, 6, 8, 8, 10, 15mamucl 18663 . . . . 5  |-  ( ph  ->  ( X F I )  e.  ( B  ^m  ( N  X.  M ) ) )
78 elmapi 7430 . . . . 5  |-  ( ( X F I )  e.  ( B  ^m  ( N  X.  M
) )  ->  ( X F I ) : ( N  X.  M
) --> B )
7977, 78syl 16 . . . 4  |-  ( ph  ->  ( X F I ) : ( N  X.  M ) --> B )
80 ffn 5722 . . . 4  |-  ( ( X F I ) : ( N  X.  M ) --> B  -> 
( X F I )  Fn  ( N  X.  M ) )
8179, 80syl 16 . . 3  |-  ( ph  ->  ( X F I )  Fn  ( N  X.  M ) )
82 ffn 5722 . . . 4  |-  ( X : ( N  X.  M ) --> B  ->  X  Fn  ( N  X.  M ) )
8324, 82syl 16 . . 3  |-  ( ph  ->  X  Fn  ( N  X.  M ) )
84 eqfnov2 6384 . . 3  |-  ( ( ( X F I )  Fn  ( N  X.  M )  /\  X  Fn  ( N  X.  M ) )  -> 
( ( X F I )  =  X  <->  A. l  e.  N  A. m  e.  M  ( l ( X F I ) m )  =  ( l X m ) ) )
8581, 83, 84syl2anc 661 . 2  |-  ( ph  ->  ( ( X F I )  =  X  <->  A. l  e.  N  A. m  e.  M  ( l ( X F I ) m )  =  ( l X m ) ) )
8676, 85mpbird 232 1  |-  ( ph  ->  ( X F I )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   _Vcvv 3106   ifcif 3932   <.cotp 4028    |-> cmpt 4498    X. cxp 4990    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277    ^m cmap 7410   Fincfn 7506   Basecbs 14479   .rcmulr 14545   0gc0g 14684    gsumg cgsu 14685   Mndcmnd 15715   1rcur 16936   Ringcrg 16979   maMul cmmul 18645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-gsum 14687  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-mamu 18646
This theorem is referenced by:  matrng  18705  mat1  18709
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