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Theorem cramerimplem2 19476
Description: Lemma 2 for cramerimp 19478: The matrix of a system of linear equations multiplied with the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)
Hypotheses
Ref Expression
cramerimp.a  |-  A  =  ( N Mat  R )
cramerimp.b  |-  B  =  ( Base `  A
)
cramerimp.v  |-  V  =  ( ( Base `  R
)  ^m  N )
cramerimp.e  |-  E  =  ( ( ( 1r
`  A ) ( N matRepV  R ) Z ) `
 I )
cramerimp.h  |-  H  =  ( ( X ( N matRepV  R ) Y ) `
 I )
cramerimp.x  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
cramerimp.m  |-  .X.  =  ( R maMul  <. N ,  N ,  N >. )
Assertion
Ref Expression
cramerimplem2  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( X  .X.  E )  =  H )

Proof of Theorem cramerimplem2
Dummy variables  l 
i  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cramerimp.m . . 3  |-  .X.  =  ( R maMul  <. N ,  N ,  N >. )
2 eqid 2402 . . 3  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2402 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 simpl 455 . . . 4  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  R  e.  CRing )
543ad2ant1 1018 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  R  e.  CRing )
6 cramerimp.a . . . . . . 7  |-  A  =  ( N Mat  R )
7 cramerimp.b . . . . . . 7  |-  B  =  ( Base `  A
)
86, 7matrcl 19204 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
98simpld 457 . . . . 5  |-  ( X  e.  B  ->  N  e.  Fin )
109adantr 463 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  N  e.  Fin )
11103ad2ant2 1019 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  N  e.  Fin )
129anim2i 567 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( R  e.  CRing  /\  N  e.  Fin ) )
1312ancomd 449 . . . . . . . . . . . . . 14  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( N  e.  Fin  /\  R  e.  CRing ) )
146, 2matbas2 19213 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
1513, 14syl 17 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  (
Base `  A )
)
1615, 7syl6reqr 2462 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  B  =  ( ( Base `  R )  ^m  ( N  X.  N ) ) )
1716eleq2d 2472 . . . . . . . . . . 11  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( X  e.  B  <->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )
1817biimpd 207 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( X  e.  B  ->  X  e.  ( ( Base `  R )  ^m  ( N  X.  N ) ) ) )
1918ex 432 . . . . . . . . 9  |-  ( R  e.  CRing  ->  ( X  e.  B  ->  ( X  e.  B  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) ) )
2019adantr 463 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  ( X  e.  B  ->  ( X  e.  B  ->  X  e.  ( ( Base `  R )  ^m  ( N  X.  N
) ) ) ) )
2120com12 29 . . . . . . 7  |-  ( X  e.  B  ->  (
( R  e.  CRing  /\  I  e.  N )  ->  ( X  e.  B  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) ) )
2221pm2.43a 48 . . . . . 6  |-  ( X  e.  B  ->  (
( R  e.  CRing  /\  I  e.  N )  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )
2322adantr 463 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  ( ( R  e. 
CRing  /\  I  e.  N
)  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )
2423impcom 428 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  ->  X  e.  ( ( Base `  R )  ^m  ( N  X.  N
) ) )
25243adant3 1017 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
26 crngring 17527 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
2726adantr 463 . . . . . . 7  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  R  e.  Ring )
2827, 10anim12i 564 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( R  e.  Ring  /\  N  e.  Fin )
)
29283adant3 1017 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( R  e.  Ring  /\  N  e.  Fin ) )
30 ne0i 3743 . . . . . . . . 9  |-  ( I  e.  N  ->  N  =/=  (/) )
3130adantl 464 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  N  =/=  (/) )
32313ad2ant1 1018 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  N  =/=  (/) )
3311, 11, 323jca 1177 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) ) )
34 cramerimp.v . . . . . . . . . . 11  |-  V  =  ( ( Base `  R
)  ^m  N )
3534eleq2i 2480 . . . . . . . . . 10  |-  ( Y  e.  V  <->  Y  e.  ( ( Base `  R
)  ^m  N )
)
3635biimpi 194 . . . . . . . . 9  |-  ( Y  e.  V  ->  Y  e.  ( ( Base `  R
)  ^m  N )
)
3736adantl 464 . . . . . . . 8  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  ( (
Base `  R )  ^m  N ) )
384, 37anim12i 564 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( R  e.  CRing  /\  Y  e.  ( (
Base `  R )  ^m  N ) ) )
39383adant3 1017 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( R  e.  CRing  /\  Y  e.  ( ( Base `  R
)  ^m  N )
) )
40 simp3 999 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( X  .x.  Z )  =  Y )
41 eqid 2402 . . . . . . . 8  |-  ( (
Base `  R )  ^m  ( N  X.  N
) )  =  ( ( Base `  R
)  ^m  ( N  X.  N ) )
42 cramerimp.x . . . . . . . 8  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
43 eqid 2402 . . . . . . . 8  |-  ( (
Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  N )
442, 41, 34, 42, 43mavmulsolcl 19343 . . . . . . 7  |-  ( ( ( N  e.  Fin  /\  N  e.  Fin  /\  N  =/=  (/) )  /\  ( R  e.  CRing  /\  Y  e.  ( ( Base `  R
)  ^m  N )
) )  ->  (
( X  .x.  Z
)  =  Y  ->  Z  e.  V )
)
4544imp 427 . . . . . 6  |-  ( ( ( ( N  e. 
Fin  /\  N  e.  Fin  /\  N  =/=  (/) )  /\  ( R  e.  CRing  /\  Y  e.  ( ( Base `  R
)  ^m  N )
) )  /\  ( X  .x.  Z )  =  Y )  ->  Z  e.  V )
4633, 39, 40, 45syl21anc 1229 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  Z  e.  V )
47 simpr 459 . . . . . 6  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  I  e.  N )
48473ad2ant1 1018 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  I  e.  N )
49 cramerimp.e . . . . . 6  |-  E  =  ( ( ( 1r
`  A ) ( N matRepV  R ) Z ) `
 I )
50 eqid 2402 . . . . . . 7  |-  ( 1r
`  A )  =  ( 1r `  A
)
516, 7, 34, 50ma1repvcl 19362 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( Z  e.  V  /\  I  e.  N
) )  ->  (
( ( 1r `  A ) ( N matRepV  R ) Z ) `
 I )  e.  B )
5249, 51syl5eqel 2494 . . . . 5  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( Z  e.  V  /\  I  e.  N
) )  ->  E  e.  B )
5329, 46, 48, 52syl12anc 1228 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  E  e.  B )
5416eqcomd 2410 . . . . . 6  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  B )
5554ad2ant2r 745 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  B )
56553adant3 1017 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  B )
5753, 56eleqtrrd 2493 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  E  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
581, 2, 3, 5, 11, 11, 11, 25, 57mamuval 19178 . 2  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( X  .X.  E )  =  ( i  e.  N ,  j  e.  N  |->  ( R  gsumg  ( l  e.  N  |->  ( ( i X l ) ( .r
`  R ) ( l E j ) ) ) ) ) )
59273ad2ant1 1018 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  R  e.  Ring )
60593ad2ant1 1018 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  I  e.  N
)  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  /\  i  e.  N  /\  j  e.  N )  ->  R  e.  Ring )
61 simpl 455 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  X  e.  B )
62613ad2ant2 1019 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  X  e.  B )
6362, 46, 483jca 1177 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( X  e.  B  /\  Z  e.  V  /\  I  e.  N )
)
64633ad2ant1 1018 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  I  e.  N
)  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  /\  i  e.  N  /\  j  e.  N )  ->  ( X  e.  B  /\  Z  e.  V  /\  I  e.  N )
)
65 simp2 998 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  I  e.  N
)  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  /\  i  e.  N  /\  j  e.  N )  ->  i  e.  N )
66 simp3 999 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  I  e.  N
)  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  /\  i  e.  N  /\  j  e.  N )  ->  j  e.  N )
67403ad2ant1 1018 . . . 4  |-  ( ( ( ( R  e. 
CRing  /\  I  e.  N
)  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  /\  i  e.  N  /\  j  e.  N )  ->  ( X  .x.  Z )  =  Y )
68 eqid 2402 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
696, 7, 34, 50, 68, 49, 42mulmarep1gsum2 19366 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Z  e.  V  /\  I  e.  N )  /\  ( i  e.  N  /\  j  e.  N  /\  ( X  .x.  Z
)  =  Y ) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( i X l ) ( .r `  R ) ( l E j ) ) ) )  =  if ( j  =  I ,  ( Y `  i ) ,  ( i X j ) ) )
7060, 64, 65, 66, 67, 69syl113anc 1242 . . 3  |-  ( ( ( ( R  e. 
CRing  /\  I  e.  N
)  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  /\  i  e.  N  /\  j  e.  N )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( i X l ) ( .r
`  R ) ( l E j ) ) ) )  =  if ( j  =  I ,  ( Y `
 i ) ,  ( i X j ) ) )
7170mpt2eq3dva 6341 . 2  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  (
i  e.  N , 
j  e.  N  |->  ( R  gsumg  ( l  e.  N  |->  ( ( i X l ) ( .r
`  R ) ( l E j ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  I ,  ( Y `  i ) ,  ( i X j ) ) ) )
72 cramerimp.h . . 3  |-  H  =  ( ( X ( N matRepV  R ) Y ) `
 I )
73 simpr 459 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  V )
74733ad2ant2 1019 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  Y  e.  V )
75 eqid 2402 . . . . 5  |-  ( N matRepV  R )  =  ( N matRepV  R )
766, 7, 75, 34marepvval 19359 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  V  /\  I  e.  N )  ->  ( ( X ( N matRepV  R ) Y ) `
 I )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  I ,  ( Y `
 i ) ,  ( i X j ) ) ) )
7762, 74, 48, 76syl3anc 1230 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  (
( X ( N matRepV  R ) Y ) `
 I )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  I ,  ( Y `
 i ) ,  ( i X j ) ) ) )
7872, 77syl5req 2456 . 2  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  (
i  e.  N , 
j  e.  N  |->  if ( j  =  I ,  ( Y `  i ) ,  ( i X j ) ) )  =  H )
7958, 71, 783eqtrd 2447 1  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( X  .X.  E )  =  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3058   (/)c0 3737   ifcif 3884   <.cop 3977   <.cotp 3979    |-> cmpt 4452    X. cxp 4820   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279    ^m cmap 7456   Fincfn 7553   Basecbs 14839   .rcmulr 14908   0gc0g 15052    gsumg cgsu 15053   1rcur 17471   Ringcrg 17516   CRingccrg 17517   maMul cmmul 19175   Mat cmat 19199   maVecMul cmvmul 19332   matRepV cmatrepV 19349
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 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  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 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-ot 3980  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-fz 11725  df-fzo 11853  df-seq 12150  df-hash 12451  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-hom 14931  df-cco 14932  df-0g 15054  df-gsum 15055  df-prds 15060  df-pws 15062  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-submnd 16289  df-grp 16379  df-minusg 16380  df-sbg 16381  df-mulg 16382  df-subg 16520  df-ghm 16587  df-cntz 16677  df-cmn 17122  df-abl 17123  df-mgp 17460  df-ur 17472  df-ring 17518  df-cring 17519  df-subrg 17745  df-lmod 17832  df-lss 17897  df-sra 18136  df-rgmod 18137  df-dsmm 19059  df-frlm 19074  df-mamu 19176  df-mat 19200  df-mvmul 19333  df-marepv 19351
This theorem is referenced by:  cramerimplem3  19477
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