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Theorem cramer0 19727
Description: Special case of Cramer's rule for 0-dimensional matrices/vectors. (Contributed by AV, 28-Feb-2019.)
Hypotheses
Ref Expression
cramer.a  |-  A  =  ( N Mat  R )
cramer.b  |-  B  =  ( Base `  A
)
cramer.v  |-  V  =  ( ( Base `  R
)  ^m  N )
cramer.d  |-  D  =  ( N maDet  R )
cramer.x  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
cramer.q  |-  ./  =  (/r
`  R )
Assertion
Ref Expression
cramer0  |-  ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V
)  /\  ( D `  X )  e.  (Unit `  R ) )  -> 
( Z  =  ( i  e.  N  |->  ( ( D `  (
( X ( N matRepV  R ) Y ) `
 i ) ) 
./  ( D `  X ) ) )  ->  ( X  .x.  Z )  =  Y ) )
Distinct variable groups:    B, i    D, i    i, N    R, i    i, V    i, X    i, Y    i, Z    .x. , i    ./ , i
Allowed substitution hint:    A( i)

Proof of Theorem cramer0
StepHypRef Expression
1 cramer.b . . . . . . . . 9  |-  B  =  ( Base `  A
)
2 cramer.a . . . . . . . . . 10  |-  A  =  ( N Mat  R )
32fveq2i 5873 . . . . . . . . 9  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
41, 3eqtri 2475 . . . . . . . 8  |-  B  =  ( Base `  ( N Mat  R ) )
5 oveq1 6302 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( N Mat 
R )  =  (
(/) Mat  R ) )
65fveq2d 5874 . . . . . . . 8  |-  ( N  =  (/)  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  ( (/) Mat  R )
) )
74, 6syl5eq 2499 . . . . . . 7  |-  ( N  =  (/)  ->  B  =  ( Base `  ( (/) Mat  R ) ) )
87adantr 467 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  B  =  ( Base `  ( (/) Mat  R ) ) )
98eleq2d 2516 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( X  e.  B  <->  X  e.  ( Base `  ( (/) Mat  R ) ) ) )
10 mat0dimbas0 19503 . . . . . . 7  |-  ( R  e.  CRing  ->  ( Base `  ( (/) Mat  R )
)  =  { (/) } )
1110eleq2d 2516 . . . . . 6  |-  ( R  e.  CRing  ->  ( X  e.  ( Base `  ( (/) Mat  R ) )  <->  X  e.  {
(/) } ) )
1211adantl 468 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( X  e.  ( Base `  ( (/) Mat  R )
)  <->  X  e.  { (/) } ) )
139, 12bitrd 257 . . . 4  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( X  e.  B  <->  X  e.  {
(/) } ) )
14 cramer.v . . . . . . . 8  |-  V  =  ( ( Base `  R
)  ^m  N )
1514a1i 11 . . . . . . 7  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  V  =  ( ( Base `  R )  ^m  N
) )
16 oveq2 6303 . . . . . . . 8  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  (/) ) )
1716adantr 467 . . . . . . 7  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  (/) ) )
18 fvex 5880 . . . . . . . 8  |-  ( Base `  R )  e.  _V
19 map0e 7514 . . . . . . . 8  |-  ( (
Base `  R )  e.  _V  ->  ( ( Base `  R )  ^m  (/) )  =  1o )
2018, 19mp1i 13 . . . . . . 7  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( Base `  R )  ^m  (/) )  =  1o )
2115, 17, 203eqtrd 2491 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  V  =  1o )
2221eleq2d 2516 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( Y  e.  V  <->  Y  e.  1o ) )
23 el1o 7206 . . . . 5  |-  ( Y  e.  1o  <->  Y  =  (/) )
2422, 23syl6bb 265 . . . 4  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( Y  e.  V  <->  Y  =  (/) ) )
2513, 24anbi12d 718 . . 3  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( X  e.  B  /\  Y  e.  V
)  <->  ( X  e. 
{ (/) }  /\  Y  =  (/) ) ) )
26 elsni 3995 . . . 4  |-  ( X  e.  { (/) }  ->  X  =  (/) )
27 mpteq1 4486 . . . . . . . . . 10  |-  ( N  =  (/)  ->  ( i  e.  N  |->  ( ( D `  ( ( X ( N matRepV  R
) Y ) `  i ) )  ./  ( D `  X ) ) )  =  ( i  e.  (/)  |->  ( ( D `  ( ( X ( N matRepV  R
) Y ) `  i ) )  ./  ( D `  X ) ) ) )
28 mpt0 5710 . . . . . . . . . 10  |-  ( i  e.  (/)  |->  ( ( D `
 ( ( X ( N matRepV  R ) Y ) `  i
) )  ./  ( D `  X )
) )  =  (/)
2927, 28syl6eq 2503 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( i  e.  N  |->  ( ( D `  ( ( X ( N matRepV  R
) Y ) `  i ) )  ./  ( D `  X ) ) )  =  (/) )
3029eqeq2d 2463 . . . . . . . 8  |-  ( N  =  (/)  ->  ( Z  =  ( i  e.  N  |->  ( ( D `
 ( ( X ( N matRepV  R ) Y ) `  i
) )  ./  ( D `  X )
) )  <->  Z  =  (/) ) )
3130ad2antrr 733 . . . . . . 7  |-  ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  -> 
( Z  =  ( i  e.  N  |->  ( ( D `  (
( X ( N matRepV  R ) Y ) `
 i ) ) 
./  ( D `  X ) ) )  <-> 
Z  =  (/) ) )
32 simplrl 771 . . . . . . . . . 10  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  X  =  (/) )
33 simpr 463 . . . . . . . . . 10  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  Z  =  (/) )
3432, 33oveq12d 6313 . . . . . . . . 9  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  ( X  .x.  Z )  =  ( (/)  .x.  (/) ) )
35 cramer.x . . . . . . . . . . 11  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
3635mavmul0 19589 . . . . . . . . . 10  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( (/) 
.x.  (/) )  =  (/) )
3736ad2antrr 733 . . . . . . . . 9  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  ( (/) 
.x.  (/) )  =  (/) )
38 simpr 463 . . . . . . . . . . 11  |-  ( ( X  =  (/)  /\  Y  =  (/) )  ->  Y  =  (/) )
3938eqcomd 2459 . . . . . . . . . 10  |-  ( ( X  =  (/)  /\  Y  =  (/) )  ->  (/)  =  Y )
4039ad2antlr 734 . . . . . . . . 9  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  (/)  =  Y )
4134, 37, 403eqtrd 2491 . . . . . . . 8  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  ( X  .x.  Z )  =  Y )
4241ex 436 . . . . . . 7  |-  ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  -> 
( Z  =  (/)  ->  ( X  .x.  Z
)  =  Y ) )
4331, 42sylbid 219 . . . . . 6  |-  ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  -> 
( Z  =  ( i  e.  N  |->  ( ( D `  (
( X ( N matRepV  R ) Y ) `
 i ) ) 
./  ( D `  X ) ) )  ->  ( X  .x.  Z )  =  Y ) )
4443a1d 26 . . . . 5  |-  ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  -> 
( ( D `  X )  e.  (Unit `  R )  ->  ( Z  =  ( i  e.  N  |->  ( ( D `  ( ( X ( N matRepV  R
) Y ) `  i ) )  ./  ( D `  X ) ) )  ->  ( X  .x.  Z )  =  Y ) ) )
4544ex 436 . . . 4  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( X  =  (/)  /\  Y  =  (/) )  -> 
( ( D `  X )  e.  (Unit `  R )  ->  ( Z  =  ( i  e.  N  |->  ( ( D `  ( ( X ( N matRepV  R
) Y ) `  i ) )  ./  ( D `  X ) ) )  ->  ( X  .x.  Z )  =  Y ) ) ) )
4626, 45sylani 660 . . 3  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( X  e.  { (/)
}  /\  Y  =  (/) )  ->  ( ( D `  X )  e.  (Unit `  R )  ->  ( Z  =  ( i  e.  N  |->  ( ( D `  (
( X ( N matRepV  R ) Y ) `
 i ) ) 
./  ( D `  X ) ) )  ->  ( X  .x.  Z )  =  Y ) ) ) )
4725, 46sylbid 219 . 2  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( X  e.  B  /\  Y  e.  V
)  ->  ( ( D `  X )  e.  (Unit `  R )  ->  ( Z  =  ( i  e.  N  |->  ( ( D `  (
( X ( N matRepV  R ) Y ) `
 i ) ) 
./  ( D `  X ) ) )  ->  ( X  .x.  Z )  =  Y ) ) ) )
48473imp 1203 1  |-  ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V
)  /\  ( D `  X )  e.  (Unit `  R ) )  -> 
( Z  =  ( i  e.  N  |->  ( ( D `  (
( X ( N matRepV  R ) Y ) `
 i ) ) 
./  ( D `  X ) ) )  ->  ( X  .x.  Z )  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   _Vcvv 3047   (/)c0 3733   {csn 3970   <.cop 3976    |-> cmpt 4464   ` cfv 5585  (class class class)co 6295   1oc1o 7180    ^m cmap 7477   Basecbs 15133   CRingccrg 17793  Unitcui 17879  /rcdvr 17922   Mat cmat 19444   maVecMul cmvmul 19577   matRepV cmatrepV 19594   maDet cmdat 19621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-ot 3979  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-sup 7961  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-hom 15226  df-cco 15227  df-0g 15352  df-prds 15358  df-pws 15360  df-sra 18407  df-rgmod 18408  df-dsmm 19307  df-frlm 19322  df-mat 19445  df-mvmul 19578
This theorem is referenced by: (None)
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