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Theorem mamurid 19111
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 2454 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamumat1cl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 463 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  R  e.  Ring )
6 mamulid.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
76adantr 463 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  N  e.  Fin )
8 mamumat1cl.m . . . . . 6  |-  ( ph  ->  M  e.  Fin )
98adantr 463 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  M  e.  Fin )
10 mamurid.x . . . . . 6  |-  ( ph  ->  X  e.  ( B  ^m  ( N  X.  M ) ) )
1110adantr 463 . . . . 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 19108 . . . . . 6  |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
1615adantr 463 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  I  e.  ( B  ^m  ( M  X.  M
) ) )
17 simprl 754 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
l  e.  N )
18 simprr 755 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  m  e.  M )
191, 2, 3, 5, 7, 9, 9, 11, 16, 17, 18mamufv 19056 . . . 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 ringmnd 17402 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
215, 20syl 16 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  R  e.  Mnd )
224ad2antrr 723 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  R  e.  Ring )
23 elmapi 7433 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( N  X.  M
) )  ->  X : ( N  X.  M ) --> B )
2410, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  X : ( N  X.  M ) --> B )
2524ad2antrr 723 . . . . . . . 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 459 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  k  e.  M )
2825, 26, 27fovrnd 6420 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
l X k )  e.  B )
29 elmapi 7433 . . . . . . . . . 10  |-  ( I  e.  ( B  ^m  ( M  X.  M
) )  ->  I : ( M  X.  M ) --> B )
3015, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  I : ( M  X.  M ) --> B )
3130ad2antrr 723 . . . . . . . 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 6420 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
k I m )  e.  B )
342, 3ringcl 17407 . . . . . . 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 1226 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
( l X k ) ( .r `  R ) ( k I m ) )  e.  B )
36 eqid 2454 . . . . . 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 6031 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) : M --> B )
38 simp2 995 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  k  e.  M )
39323adant3 1014 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  m  e.  M )
402, 4, 12, 13, 14, 8mat1comp 19109 . . . . . . . . . 10  |-  ( ( k  e.  M  /\  m  e.  M )  ->  ( k I m )  =  if ( k  =  m ,  .1.  ,  .0.  )
)
4138, 39, 40syl2anc 659 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( k
I m )  =  if ( k  =  m ,  .1.  ,  .0.  ) )
42 ifnefalse 3941 . . . . . . . . . 10  |-  ( k  =/=  m  ->  if ( k  =  m ,  .1.  ,  .0.  )  =  .0.  )
43423ad2ant3 1017 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  if (
k  =  m ,  .1.  ,  .0.  )  =  .0.  )
4441, 43eqtrd 2495 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( k
I m )  =  .0.  )
4544oveq2d 6286 . . . . . . 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, 13ringrz 17431 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B )  -> 
( ( l X k ) ( .r
`  R )  .0.  )  =  .0.  )
4722, 28, 46syl2anc 659 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
( l X k ) ( .r `  R )  .0.  )  =  .0.  )
48473adant3 1014 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( (
l X k ) ( .r `  R
)  .0.  )  =  .0.  )
4945, 48eqtrd 2495 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( (
l X k ) ( .r `  R
) ( k I m ) )  =  .0.  )
5049, 9suppsssn 6927 . . . . 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 17184 . . . 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 6278 . . . . . . . 8  |-  ( k  =  m  ->  (
l X k )  =  ( l X m ) )
53 oveq1 6277 . . . . . . . 8  |-  ( k  =  m  ->  (
k I m )  =  ( m I m ) )
5452, 53oveq12d 6288 . . . . . . 7  |-  ( k  =  m  ->  (
( l X k ) ( .r `  R ) ( k I m ) )  =  ( ( l X m ) ( .r `  R ) ( m I m ) ) )
55 ovex 6298 . . . . . . 7  |-  ( ( l X m ) ( .r `  R
) ( m I m ) )  e. 
_V
5654, 36, 55fvmpt 5931 . . . . . 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 726 . . . . 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 1803 . . . . . . . . . 10  |-  ( i  =  m  ->  (
i  =  j  <->  m  =  j ) )
5958ifbid 3951 . . . . . . . . 9  |-  ( i  =  m  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
m  =  j ,  .1.  ,  .0.  )
)
60 equequ2 1804 . . . . . . . . . . 11  |-  ( j  =  m  ->  (
m  =  j  <->  m  =  m ) )
6160ifbid 3951 . . . . . . . . . 10  |-  ( j  =  m  ->  if ( m  =  j ,  .1.  ,  .0.  )  =  if ( m  =  m ,  .1.  ,  .0.  ) )
62 eqid 2454 . . . . . . . . . . 11  |-  m  =  m
6362iftruei 3936 . . . . . . . . . 10  |-  if ( m  =  m ,  .1.  ,  .0.  )  =  .1.
6461, 63syl6eq 2511 . . . . . . . . 9  |-  ( j  =  m  ->  if ( m  =  j ,  .1.  ,  .0.  )  =  .1.  )
65 fvex 5858 . . . . . . . . . 10  |-  ( 1r
`  R )  e. 
_V
6612, 65eqeltri 2538 . . . . . . . . 9  |-  .1.  e.  _V
6759, 64, 14, 66ovmpt2 6411 . . . . . . . 8  |-  ( ( m  e.  M  /\  m  e.  M )  ->  ( m I m )  =  .1.  )
6867anidms 643 . . . . . . 7  |-  ( m  e.  M  ->  (
m I m )  =  .1.  )
6968oveq2d 6286 . . . . . 6  |-  ( m  e.  M  ->  (
( l X m ) ( .r `  R ) ( m I m ) )  =  ( ( l X m ) ( .r `  R )  .1.  ) )
7069ad2antll 726 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( l X m ) ( .r
`  R ) ( m I m ) )  =  ( ( l X m ) ( .r `  R
)  .1.  ) )
7124fovrnda 6419 . . . . . 6  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l X m )  e.  B )
722, 3, 12ringridm 17418 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
l X m )  e.  B )  -> 
( ( l X m ) ( .r
`  R )  .1.  )  =  ( l X m ) )
735, 71, 72syl2anc 659 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( l X m ) ( .r
`  R )  .1.  )  =  ( l X m ) )
7457, 70, 733eqtrd 2499 . . . 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 2499 . . 3  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l ( X F I ) m )  =  ( l X m ) )
7675ralrimivva 2875 . 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 19070 . . . . 5  |-  ( ph  ->  ( X F I )  e.  ( B  ^m  ( N  X.  M ) ) )
78 elmapi 7433 . . . . 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 5713 . . . 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 5713 . . . 4  |-  ( X : ( N  X.  M ) --> B  ->  X  Fn  ( N  X.  M ) )
8324, 82syl 16 . . 3  |-  ( ph  ->  X  Fn  ( N  X.  M ) )
84 eqfnov2 6382 . . 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 659 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106   ifcif 3929   <.cotp 4024    |-> cmpt 4497    X. cxp 4986    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272    ^m cmap 7412   Fincfn 7509   Basecbs 14716   .rcmulr 14785   0gc0g 14929    gsumg cgsu 14930   Mndcmnd 16118   1rcur 17348   Ringcrg 17393   maMul cmmul 19052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-mamu 19053
This theorem is referenced by:  matring  19112  mat1  19116
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