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Theorem mamurid 19477
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 2451 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
4 mamumat1cl.r . . . . . 6  |-  ( ph  ->  R  e.  Ring )
54adantr 471 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  R  e.  Ring )
6 mamulid.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
76adantr 471 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  N  e.  Fin )
8 mamumat1cl.m . . . . . 6  |-  ( ph  ->  M  e.  Fin )
98adantr 471 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  M  e.  Fin )
10 mamurid.x . . . . . 6  |-  ( ph  ->  X  e.  ( B  ^m  ( N  X.  M ) ) )
1110adantr 471 . . . . 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 19474 . . . . . 6  |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
1615adantr 471 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  I  e.  ( B  ^m  ( M  X.  M
) ) )
17 simprl 769 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
l  e.  N )
18 simprr 771 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  m  e.  M )
191, 2, 3, 5, 7, 9, 9, 11, 16, 17, 18mamufv 19422 . . . 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 17799 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
215, 20syl 17 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  ->  R  e.  Mnd )
224ad2antrr 737 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  R  e.  Ring )
23 elmapi 7479 . . . . . . . . . 10  |-  ( X  e.  ( B  ^m  ( N  X.  M
) )  ->  X : ( N  X.  M ) --> B )
2410, 23syl 17 . . . . . . . . 9  |-  ( ph  ->  X : ( N  X.  M ) --> B )
2524ad2antrr 737 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  X : ( N  X.  M ) --> B )
26 simplrl 775 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  l  e.  N )
27 simpr 467 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  k  e.  M )
2825, 26, 27fovrnd 6428 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
l X k )  e.  B )
29 elmapi 7479 . . . . . . . . . 10  |-  ( I  e.  ( B  ^m  ( M  X.  M
) )  ->  I : ( M  X.  M ) --> B )
3015, 29syl 17 . . . . . . . . 9  |-  ( ph  ->  I : ( M  X.  M ) --> B )
3130ad2antrr 737 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  I : ( M  X.  M ) --> B )
32 simplrr 776 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  m  e.  M )
3331, 27, 32fovrnd 6428 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
k I m )  e.  B )
342, 3ringcl 17804 . . . . . . 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 1271 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
( l X k ) ( .r `  R ) ( k I m ) )  e.  B )
36 eqid 2451 . . . . . 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 6029 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( k  e.  M  |->  ( ( l X k ) ( .r
`  R ) ( k I m ) ) ) : M --> B )
38 simp2 1010 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  k  e.  M )
39323adant3 1029 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  m  e.  M )
402, 4, 12, 13, 14, 8mat1comp 19475 . . . . . . . . . 10  |-  ( ( k  e.  M  /\  m  e.  M )  ->  ( k I m )  =  if ( k  =  m ,  .1.  ,  .0.  )
)
4138, 39, 40syl2anc 671 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( k
I m )  =  if ( k  =  m ,  .1.  ,  .0.  ) )
42 ifnefalse 3860 . . . . . . . . . 10  |-  ( k  =/=  m  ->  if ( k  =  m ,  .1.  ,  .0.  )  =  .0.  )
43423ad2ant3 1032 . . . . . . . . 9  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  if (
k  =  m ,  .1.  ,  .0.  )  =  .0.  )
4441, 43eqtrd 2485 . . . . . . . 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, 13ringrz 17828 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
l X k )  e.  B )  -> 
( ( l X k ) ( .r
`  R )  .0.  )  =  .0.  )
4722, 28, 46syl2anc 671 . . . . . . . 8  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M )  ->  (
( l X k ) ( .r `  R )  .0.  )  =  .0.  )
48473adant3 1029 . . . . . . 7  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( (
l X k ) ( .r `  R
)  .0.  )  =  .0.  )
4945, 48eqtrd 2485 . . . . . 6  |-  ( ( ( ph  /\  (
l  e.  N  /\  m  e.  M )
)  /\  k  e.  M  /\  k  =/=  m
)  ->  ( (
l X k ) ( .r `  R
) ( k I m ) )  =  .0.  )
5049, 9suppsssn 6937 . . . . 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 17604 . . . 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 6303 . . . . . . 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 740 . . . . 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 1870 . . . . . . . . . 10  |-  ( i  =  m  ->  (
i  =  j  <->  m  =  j ) )
5958ifbid 3870 . . . . . . . . 9  |-  ( i  =  m  ->  if ( i  =  j ,  .1.  ,  .0.  )  =  if (
m  =  j ,  .1.  ,  .0.  )
)
60 equequ2 1871 . . . . . . . . . . 11  |-  ( j  =  m  ->  (
m  =  j  <->  m  =  m ) )
6160ifbid 3870 . . . . . . . . . 10  |-  ( j  =  m  ->  if ( m  =  j ,  .1.  ,  .0.  )  =  if ( m  =  m ,  .1.  ,  .0.  ) )
62 eqid 2451 . . . . . . . . . . 11  |-  m  =  m
6362iftruei 3855 . . . . . . . . . 10  |-  if ( m  =  m ,  .1.  ,  .0.  )  =  .1.
6461, 63syl6eq 2501 . . . . . . . . 9  |-  ( j  =  m  ->  if ( m  =  j ,  .1.  ,  .0.  )  =  .1.  )
65 fvex 5857 . . . . . . . . . 10  |-  ( 1r
`  R )  e. 
_V
6612, 65eqeltri 2525 . . . . . . . . 9  |-  .1.  e.  _V
6759, 64, 14, 66ovmpt2 6419 . . . . . . . 8  |-  ( ( m  e.  M  /\  m  e.  M )  ->  ( m I m )  =  .1.  )
6867anidms 655 . . . . . . 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 740 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( l X m ) ( .r
`  R ) ( m I m ) )  =  ( ( l X m ) ( .r `  R
)  .1.  ) )
7124fovrnda 6427 . . . . . 6  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l X m )  e.  B )
722, 3, 12ringridm 17815 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
l X m )  e.  B )  -> 
( ( l X m ) ( .r
`  R )  .1.  )  =  ( l X m ) )
735, 71, 72syl2anc 671 . . . . 5  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( ( l X m ) ( .r
`  R )  .1.  )  =  ( l X m ) )
7457, 70, 733eqtrd 2489 . . . 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 2489 . . 3  |-  ( (
ph  /\  ( l  e.  N  /\  m  e.  M ) )  -> 
( l ( X F I ) m )  =  ( l X m ) )
7675ralrimivva 2794 . 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 19436 . . . . 5  |-  ( ph  ->  ( X F I )  e.  ( B  ^m  ( N  X.  M ) ) )
78 elmapi 7479 . . . . 5  |-  ( ( X F I )  e.  ( B  ^m  ( N  X.  M
) )  ->  ( X F I ) : ( N  X.  M
) --> B )
7977, 78syl 17 . . . 4  |-  ( ph  ->  ( X F I ) : ( N  X.  M ) --> B )
80 ffn 5710 . . . 4  |-  ( ( X F I ) : ( N  X.  M ) --> B  -> 
( X F I )  Fn  ( N  X.  M ) )
8179, 80syl 17 . . 3  |-  ( ph  ->  ( X F I )  Fn  ( N  X.  M ) )
82 ffn 5710 . . . 4  |-  ( X : ( N  X.  M ) --> B  ->  X  Fn  ( N  X.  M ) )
8324, 82syl 17 . . 3  |-  ( ph  ->  X  Fn  ( N  X.  M ) )
84 eqfnov2 6390 . . 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 671 . 2  |-  ( ph  ->  ( ( X F I )  =  X  <->  A. l  e.  N  A. m  e.  M  ( l ( X F I ) m )  =  ( l X m ) ) )
8676, 85mpbird 240 1  |-  ( ph  ->  ( X F I )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 986    = wceq 1447    e. wcel 1890    =/= wne 2621   A.wral 2736   _Vcvv 3012   ifcif 3848   <.cotp 3943    |-> cmpt 4432    X. cxp 4809    Fn wfn 5555   -->wf 5556   ` cfv 5560  (class class class)co 6275    |-> cmpt2 6277    ^m cmap 7458   Fincfn 7555   Basecbs 15131   .rcmulr 15201   0gc0g 15348    gsumg cgsu 15349   Mndcmnd 16545   1rcur 17745   Ringcrg 17790   maMul cmmul 19418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-rep 4486  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570  ax-inf2 8132  ax-cnex 9581  ax-resscn 9582  ax-1cn 9583  ax-icn 9584  ax-addcl 9585  ax-addrcl 9586  ax-mulcl 9587  ax-mulrcl 9588  ax-mulcom 9589  ax-addass 9590  ax-mulass 9591  ax-distr 9592  ax-i2m1 9593  ax-1ne0 9594  ax-1rid 9595  ax-rnegex 9596  ax-rrecex 9597  ax-cnre 9598  ax-pre-lttri 9599  ax-pre-lttrn 9600  ax-pre-ltadd 9601  ax-pre-mulgt0 9602
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-pss 3387  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-tp 3940  df-op 3942  df-ot 3944  df-uni 4168  df-int 4204  df-iun 4249  df-iin 4250  df-br 4374  df-opab 4433  df-mpt 4434  df-tr 4469  df-eprel 4722  df-id 4726  df-po 4732  df-so 4733  df-fr 4770  df-se 4771  df-we 4772  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-pred 5358  df-ord 5404  df-on 5405  df-lim 5406  df-suc 5407  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-isom 5569  df-riota 6237  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6680  df-1st 6780  df-2nd 6781  df-supp 6902  df-wrecs 7014  df-recs 7076  df-rdg 7114  df-1o 7168  df-oadd 7172  df-er 7349  df-map 7460  df-en 7556  df-dom 7557  df-sdom 7558  df-fin 7559  df-fsupp 7870  df-oi 8011  df-card 8359  df-pnf 9663  df-mnf 9664  df-xr 9665  df-ltxr 9666  df-le 9667  df-sub 9848  df-neg 9849  df-nn 10598  df-2 10656  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11775  df-fzo 11908  df-seq 12207  df-hash 12509  df-ndx 15134  df-slot 15135  df-base 15136  df-sets 15137  df-ress 15138  df-plusg 15213  df-0g 15350  df-gsum 15351  df-mre 15502  df-mrc 15503  df-acs 15505  df-mgm 16498  df-sgrp 16537  df-mnd 16547  df-submnd 16593  df-grp 16683  df-minusg 16684  df-mulg 16686  df-cntz 16981  df-cmn 17442  df-abl 17443  df-mgp 17734  df-ur 17746  df-ring 17792  df-mamu 19419
This theorem is referenced by:  matring  19478  mat1  19482
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