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Theorem cramer0 18508
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 5706 . . . . . . . . 9  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
41, 3eqtri 2463 . . . . . . . 8  |-  B  =  ( Base `  ( N Mat  R ) )
5 oveq1 6110 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( N Mat 
R )  =  (
(/) Mat  R ) )
65fveq2d 5707 . . . . . . . 8  |-  ( N  =  (/)  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  ( (/) Mat  R )
) )
74, 6syl5eq 2487 . . . . . . 7  |-  ( N  =  (/)  ->  B  =  ( Base `  ( (/) Mat  R ) ) )
87adantr 465 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  B  =  ( Base `  ( (/) Mat  R ) ) )
98eleq2d 2510 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( X  e.  B  <->  X  e.  ( Base `  ( (/) Mat  R ) ) ) )
10 mat0dimbas0 18337 . . . . . . 7  |-  ( R  e.  CRing  ->  ( Base `  ( (/) Mat  R )
)  =  { (/) } )
1110eleq2d 2510 . . . . . 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 6111 . . . . . . . 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 5713 . . . . . . . 8  |-  ( Base `  R )  e.  _V
19 map0e 7262 . . . . . . . 8  |-  ( (
Base `  R )  e.  _V  ->  ( ( Base `  R )  ^m  (/) )  =  1o )
2018, 19mp1i 12 . . . . . . 7  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  (
( Base `  R )  ^m  (/) )  =  1o )
2115, 17, 203eqtrd 2479 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  V  =  1o )
2221eleq2d 2510 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  CRing )  ->  ( Y  e.  V  <->  Y  e.  1o ) )
23 el1o 6951 . . . . 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 3914 . . . 4  |-  ( X  e.  { (/) }  ->  X  =  (/) )
27 mpteq1 4384 . . . . . . . . . 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 5550 . . . . . . . . . 10  |-  ( i  e.  (/)  |->  ( ( D `
 ( ( X ( N matRepV  R ) Y ) `  i
) )  ./  ( D `  X )
) )  =  (/)
2927, 28syl6eq 2491 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( i  e.  N  |->  ( ( D `  ( ( X ( N matRepV  R
) Y ) `  i ) )  ./  ( D `  X ) ) )  =  (/) )
3029eqeq2d 2454 . . . . . . . 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 759 . . . . . . . . . 10  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  X  =  (/) )
33 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  Z  =  (/) )
3432, 33oveq12d 6121 . . . . . . . . 9  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  ( X  .x.  Z )  =  ( (/)  .x.  (/) ) )
35 cramer.x . . . . . . . . . . 11  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
3635mavmul0 18375 . . . . . . . . . 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 2448 . . . . . . . . . 10  |-  ( ( X  =  (/)  /\  Y  =  (/) )  ->  (/)  =  Y )
4039ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( N  =  (/)  /\  R  e.  CRing )  /\  ( X  =  (/)  /\  Y  =  (/) ) )  /\  Z  =  (/) )  ->  (/)  =  Y )
4134, 37, 403eqtrd 2479 . . . . . . . 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 1181 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2984   (/)c0 3649   {csn 3889   <.cop 3895    e. cmpt 4362   ` cfv 5430  (class class class)co 6103   1oc1o 6925    ^m cmap 7226   Basecbs 14186   CRingccrg 16658  Unitcui 16743  /rcdvr 16786   Mat cmat 18292   maVecMul cmvmul 18363   matRepV cmatrepV 18380   maDet cmdat 18407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-ot 3898  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-hom 14274  df-cco 14275  df-0g 14392  df-prds 14398  df-pws 14400  df-sra 17265  df-rgmod 17266  df-dsmm 18169  df-frlm 18184  df-mat 18294  df-mvmul 18364
This theorem is referenced by: (None)
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