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Theorem cramerimplem2 18625
Description: Lemma 2 for cramerimp 18627: 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 2454 . . 3  |-  ( Base `  R )  =  (
Base `  R )
3 eqid 2454 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 simpl 457 . . . 4  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  R  e.  CRing )
543ad2ant1 1009 . . 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 18440 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
98simpld 459 . . . . 5  |-  ( X  e.  B  ->  N  e.  Fin )
109adantr 465 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  N  e.  Fin )
11103ad2ant2 1010 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  N  e.  Fin )
129anim2i 569 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( R  e.  CRing  /\  N  e.  Fin ) )
1312ancomd 451 . . . . . . . . . . . . . 14  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( N  e.  Fin  /\  R  e.  CRing ) )
146, 2matbas2 18450 . . . . . . . . . . . . . 14  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  ( Base `  A
) )
1513, 14syl 16 . . . . . . . . . . . . 13  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  (
Base `  A )
)
1615, 7syl6reqr 2514 . . . . . . . . . . . 12  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  B  =  ( ( Base `  R )  ^m  ( N  X.  N ) ) )
1716eleq2d 2524 . . . . . . . . . . 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 434 . . . . . . . . 9  |-  ( R  e.  CRing  ->  ( X  e.  B  ->  ( X  e.  B  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) ) )
2019adantr 465 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  ( X  e.  B  ->  ( X  e.  B  ->  X  e.  ( ( Base `  R )  ^m  ( N  X.  N
) ) ) ) )
2120com12 31 . . . . . . 7  |-  ( X  e.  B  ->  (
( R  e.  CRing  /\  I  e.  N )  ->  ( X  e.  B  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) ) )
2221pm2.43a 49 . . . . . 6  |-  ( X  e.  B  ->  (
( R  e.  CRing  /\  I  e.  N )  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )
2322adantr 465 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  ( ( R  e. 
CRing  /\  I  e.  N
)  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) ) )
2423impcom 430 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  ->  X  e.  ( ( Base `  R )  ^m  ( N  X.  N
) ) )
25243adant3 1008 . . 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 crngrng 16781 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
2726adantr 465 . . . . . . 7  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  R  e.  Ring )
2827, 10anim12i 566 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( R  e.  Ring  /\  N  e.  Fin )
)
29283adant3 1008 . . . . 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 3754 . . . . . . . . 9  |-  ( I  e.  N  ->  N  =/=  (/) )
3130adantl 466 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  N  =/=  (/) )
32313ad2ant1 1009 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  N  =/=  (/) )
3311, 11, 323jca 1168 . . . . . 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 2532 . . . . . . . . . 10  |-  ( Y  e.  V  <->  Y  e.  ( ( Base `  R
)  ^m  N )
)
3635biimpi 194 . . . . . . . . 9  |-  ( Y  e.  V  ->  Y  e.  ( ( Base `  R
)  ^m  N )
)
3736adantl 466 . . . . . . . 8  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  ( (
Base `  R )  ^m  N ) )
384, 37anim12i 566 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( R  e.  CRing  /\  Y  e.  ( (
Base `  R )  ^m  N ) ) )
39383adant3 1008 . . . . . 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 990 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  ( X  .x.  Z )  =  Y )
41 eqid 2454 . . . . . . . 8  |-  ( (
Base `  R )  ^m  ( N  X.  N
) )  =  ( ( Base `  R
)  ^m  ( N  X.  N ) )
42 cramerimp.x . . . . . . . 8  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
43 eqid 2454 . . . . . . . 8  |-  ( (
Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  N )
442, 41, 34, 42, 43mavmulsolcl 18492 . . . . . . 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 429 . . . . . 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 1218 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  Z  e.  V )
47 simpr 461 . . . . . 6  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  I  e.  N )
48473ad2ant1 1009 . . . . 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 2454 . . . . . . 7  |-  ( 1r
`  A )  =  ( 1r `  A
)
516, 7, 34, 50ma1repvcl 18511 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( Z  e.  V  /\  I  e.  N
) )  ->  (
( ( 1r `  A ) ( N matRepV  R ) Z ) `
 I )  e.  B )
5249, 51syl5eqel 2546 . . . . 5  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( Z  e.  V  /\  I  e.  N
) )  ->  E  e.  B )
5329, 46, 48, 52syl12anc 1217 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  E  e.  B )
5416eqcomd 2462 . . . . . 6  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  B )
5554ad2ant2r 746 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( ( Base `  R
)  ^m  ( N  X.  N ) )  =  B )
56553adant3 1008 . . . 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 2545 . . 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 18412 . 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 1009 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  R  e.  Ring )
60593ad2ant1 1009 . . . 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 457 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  X  e.  B )
62613ad2ant2 1010 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  X  e.  B )
6362, 46, 483jca 1168 . . . . 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 1009 . . . 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 989 . . . 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 990 . . . 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 1009 . . . 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 2454 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
696, 7, 34, 50, 68, 49, 42mulmarep1gsum2 18515 . . . 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 1231 . . 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 6262 . 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 461 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  V )
74733ad2ant2 1010 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  Y  e.  V )
75 eqid 2454 . . . . 5  |-  ( N matRepV  R )  =  ( N matRepV  R )
766, 7, 75, 34marepvval 18508 . . . 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 1219 . . 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 2508 . 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 2499 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   (/)c0 3748   ifcif 3902   <.cop 3994   <.cotp 3996    |-> cmpt 4461    X. cxp 4949   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205    ^m cmap 7327   Fincfn 7423   Basecbs 14295   .rcmulr 14361   0gc0g 14500    gsumg cgsu 14501   1rcur 16728   Ringcrg 16771   CRingccrg 16772   maMul cmmul 18407   Mat cmat 18408   maVecMul cmvmul 18481   matRepV cmatrepV 18498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-ot 3997  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-fzo 11669  df-seq 11927  df-hash 12224  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-sca 14376  df-vsca 14377  df-ip 14378  df-tset 14379  df-ple 14380  df-ds 14382  df-hom 14384  df-cco 14385  df-0g 14502  df-gsum 14503  df-prds 14508  df-pws 14510  df-mre 14646  df-mrc 14647  df-acs 14649  df-mnd 15537  df-mhm 15586  df-submnd 15587  df-grp 15667  df-minusg 15668  df-sbg 15669  df-mulg 15670  df-subg 15800  df-ghm 15867  df-cntz 15957  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-rng 16773  df-cring 16774  df-subrg 16989  df-lmod 17076  df-lss 17140  df-sra 17379  df-rgmod 17380  df-dsmm 18285  df-frlm 18300  df-mamu 18409  df-mat 18410  df-mvmul 18482  df-marepv 18500
This theorem is referenced by:  cramerimplem3  18626
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