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Theorem cramer0 19319
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 5875 . . . . . . . . 9  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
41, 3eqtri 2486 . . . . . . . 8  |-  B  =  ( Base `  ( N Mat  R ) )
5 oveq1 6303 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( N Mat 
R )  =  (
(/) Mat  R ) )
65fveq2d 5876 . . . . . . . 8  |-  ( N  =  (/)  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  ( (/) Mat  R )
) )
74, 6syl5eq 2510 . . . . . . 7  |-  ( N  =  (/)  ->  B  =  ( Base `  ( (/) Mat  R ) ) )
87adantr 465 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  B  =  ( Base `  ( (/) Mat  R ) ) )
98eleq2d 2527 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( X  e.  B  <->  X  e.  ( Base `  ( (/) Mat  R ) ) ) )
10 mat0dimbas0 19095 . . . . . . 7  |-  ( R  e.  CRing  ->  ( Base `  ( (/) Mat  R )
)  =  { (/) } )
1110eleq2d 2527 . . . . . 6  |-  ( R  e.  CRing  ->  ( X  e.  ( Base `  ( (/) Mat  R ) )  <->  X  e.  {
(/) } ) )
1211adantl 466 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( X  e.  ( Base `  ( (/) Mat  R )
)  <->  X  e.  { (/) } ) )
139, 12bitrd 253 . . . 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 6304 . . . . . . . 8  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  (/) ) )
1716adantr 465 . . . . . . 7  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  (/) ) )
18 fvex 5882 . . . . . . . 8  |-  ( Base `  R )  e.  _V
19 map0e 7475 . . . . . . . 8  |-  ( (
Base `  R )  e.  _V  ->  ( ( Base `  R )  ^m  (/) )  =  1o )
2018, 19mp1i 12 . . . . . . 7  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( Base `  R )  ^m  (/) )  =  1o )
2115, 17, 203eqtrd 2502 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  V  =  1o )
2221eleq2d 2527 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( Y  e.  V  <->  Y  e.  1o ) )
23 el1o 7167 . . . . 5  |-  ( Y  e.  1o  <->  Y  =  (/) )
2422, 23syl6bb 261 . . . 4  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( Y  e.  V  <->  Y  =  (/) ) )
2513, 24anbi12d 710 . . 3  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( X  e.  B  /\  Y  e.  V
)  <->  ( X  e. 
{ (/) }  /\  Y  =  (/) ) ) )
26 elsni 4057 . . . 4  |-  ( X  e.  { (/) }  ->  X  =  (/) )
27 mpteq1 4537 . . . . . . . . . 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 5714 . . . . . . . . . 10  |-  ( i  e.  (/)  |->  ( ( D `
 ( ( X ( N matRepV  R ) Y ) `  i
) )  ./  ( D `  X )
) )  =  (/)
2927, 28syl6eq 2514 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( i  e.  N  |->  ( ( D `  ( ( X ( N matRepV  R
) Y ) `  i ) )  ./  ( D `  X ) ) )  =  (/) )
3029eqeq2d 2471 . . . . . . . 8  |-  ( N  =  (/)  ->  ( Z  =  ( i  e.  N  |->  ( ( D `
 ( ( X ( N matRepV  R ) Y ) `  i
) )  ./  ( D `  X )
) )  <->  Z  =  (/) ) )
3130ad2antrr 725 . . . . . . 7  |-  ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  -> 
( Z  =  ( i  e.  N  |->  ( ( D `  (
( X ( N matRepV  R ) Y ) `
 i ) ) 
./  ( D `  X ) ) )  <-> 
Z  =  (/) ) )
32 simplrl 761 . . . . . . . . . 10  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  X  =  (/) )
33 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  Z  =  (/) )
3432, 33oveq12d 6314 . . . . . . . . 9  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  ( X  .x.  Z )  =  ( (/)  .x.  (/) ) )
35 cramer.x . . . . . . . . . . 11  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
3635mavmul0 19181 . . . . . . . . . 10  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( (/) 
.x.  (/) )  =  (/) )
3736ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  ( (/) 
.x.  (/) )  =  (/) )
38 simpr 461 . . . . . . . . . . 11  |-  ( ( X  =  (/)  /\  Y  =  (/) )  ->  Y  =  (/) )
3938eqcomd 2465 . . . . . . . . . 10  |-  ( ( X  =  (/)  /\  Y  =  (/) )  ->  (/)  =  Y )
4039ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  (/)  =  Y )
4134, 37, 403eqtrd 2502 . . . . . . . 8  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  ( X  .x.  Z )  =  Y )
4241ex 434 . . . . . . 7  |-  ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  -> 
( Z  =  (/)  ->  ( X  .x.  Z
)  =  Y ) )
4331, 42sylbid 215 . . . . . 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 25 . . . . 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 434 . . . 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 654 . . 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 215 . 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 1190 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 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   {csn 4032   <.cop 4038    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   1oc1o 7141    ^m cmap 7438   Basecbs 14644   CRingccrg 17326  Unitcui 17415  /rcdvr 17458   Mat cmat 19036   maVecMul cmvmul 19169   matRepV cmatrepV 19186   maDet cmdat 19213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-hom 14736  df-cco 14737  df-0g 14859  df-prds 14865  df-pws 14867  df-sra 17945  df-rgmod 17946  df-dsmm 18890  df-frlm 18905  df-mat 19037  df-mvmul 19170
This theorem is referenced by: (None)
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