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Theorem cramerimp 18625
Description: One direction of Cramer's rule (according to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations."): The ith component of the solution vector of a system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the righthand side vector of the system of linear equations divided by the determinant of the matrix 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.d  |-  D  =  ( N maDet  R )
cramerimp.q  |-  ./  =  (/r
`  R )
Assertion
Ref Expression
cramerimp  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( Z `  I
)  =  ( ( D `  H ) 
./  ( D `  X ) ) )

Proof of Theorem cramerimp
StepHypRef Expression
1 crngrng 16779 . . . . . 6  |-  ( R  e.  CRing  ->  R  e.  Ring )
21adantr 465 . . . . 5  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  R  e.  Ring )
323ad2ant1 1009 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  ->  R  e.  Ring )
4 cramerimp.d . . . . . . . 8  |-  D  =  ( N maDet  R )
5 cramerimp.a . . . . . . . 8  |-  A  =  ( N Mat  R )
6 cramerimp.b . . . . . . . 8  |-  B  =  ( Base `  A
)
7 eqid 2454 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
84, 5, 6, 7mdetf 18534 . . . . . . 7  |-  ( R  e.  CRing  ->  D : B
--> ( Base `  R
) )
98adantr 465 . . . . . 6  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  D : B --> ( Base `  R
) )
1093ad2ant1 1009 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  ->  D : B --> ( Base `  R ) )
11 cramerimp.e . . . . . 6  |-  E  =  ( ( ( 1r
`  A ) ( N matRepV  R ) Z ) `
 I )
125, 6matrcl 18438 . . . . . . . . . . 11  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
1312simpld 459 . . . . . . . . . 10  |-  ( X  e.  B  ->  N  e.  Fin )
1413adantr 465 . . . . . . . . 9  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  N  e.  Fin )
152, 14anim12i 566 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( R  e.  Ring  /\  N  e.  Fin )
)
16153adant3 1008 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( R  e.  Ring  /\  N  e.  Fin )
)
17 ne0i 3752 . . . . . . . . . . 11  |-  ( I  e.  N  ->  N  =/=  (/) )
181, 17anim12ci 567 . . . . . . . . . 10  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  ( N  =/=  (/)  /\  R  e. 
Ring ) )
1918anim1i 568 . . . . . . . . 9  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V ) ) )
20193adant3 1008 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V ) ) )
21 simpl 457 . . . . . . . . 9  |-  ( ( ( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) )  ->  ( X  .x.  Z )  =  Y )
22213ad2ant3 1011 . . . . . . . 8  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( X  .x.  Z
)  =  Y )
23 cramerimp.v . . . . . . . . 9  |-  V  =  ( ( Base `  R
)  ^m  N )
24 cramerimp.x . . . . . . . . 9  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
255, 6, 23, 24slesolvec 18618 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( X  .x.  Z )  =  Y  ->  Z  e.  V ) )
2620, 22, 25sylc 60 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  ->  Z  e.  V )
27 simpr 461 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  I  e.  N )
28273ad2ant1 1009 . . . . . . 7  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  ->  I  e.  N )
29 eqid 2454 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
305, 6, 23, 29ma1repvcl 18509 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( Z  e.  V  /\  I  e.  N
) )  ->  (
( ( 1r `  A ) ( N matRepV  R ) Z ) `
 I )  e.  B )
3116, 26, 28, 30syl12anc 1217 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( ( ( 1r
`  A ) ( N matRepV  R ) Z ) `
 I )  e.  B )
3211, 31syl5eqel 2546 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  ->  E  e.  B )
3310, 32ffvelrnd 5954 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( D `  E
)  e.  ( Base `  R ) )
34 simpr 461 . . . . 5  |-  ( ( ( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) )  ->  ( D `  X )  e.  (Unit `  R )
)
35343ad2ant3 1011 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( D `  X
)  e.  (Unit `  R ) )
36 eqid 2454 . . . . 5  |-  (Unit `  R )  =  (Unit `  R )
37 cramerimp.q . . . . 5  |-  ./  =  (/r
`  R )
38 eqid 2454 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
397, 36, 37, 38dvrcan3 16908 . . . 4  |-  ( ( R  e.  Ring  /\  ( D `  E )  e.  ( Base `  R
)  /\  ( D `  X )  e.  (Unit `  R ) )  -> 
( ( ( D `
 E ) ( .r `  R ) ( D `  X
) )  ./  ( D `  X )
)  =  ( D `
 E ) )
403, 33, 35, 39syl3anc 1219 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( ( ( D `
 E ) ( .r `  R ) ( D `  X
) )  ./  ( D `  X )
)  =  ( D `
 E ) )
41 simpl 457 . . . . . 6  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  R  e.  CRing )
42413ad2ant1 1009 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  ->  R  e.  CRing )
437, 36unitcl 16875 . . . . . . 7  |-  ( ( D `  X )  e.  (Unit `  R
)  ->  ( D `  X )  e.  (
Base `  R )
)
4443adantl 466 . . . . . 6  |-  ( ( ( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) )  ->  ( D `  X )  e.  ( Base `  R
) )
45443ad2ant3 1011 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( D `  X
)  e.  ( Base `  R ) )
467, 38crngcom 16783 . . . . 5  |-  ( ( R  e.  CRing  /\  ( D `  E )  e.  ( Base `  R
)  /\  ( D `  X )  e.  (
Base `  R )
)  ->  ( ( D `  E )
( .r `  R
) ( D `  X ) )  =  ( ( D `  X ) ( .r
`  R ) ( D `  E ) ) )
4742, 33, 45, 46syl3anc 1219 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( ( D `  E ) ( .r
`  R ) ( D `  X ) )  =  ( ( D `  X ) ( .r `  R
) ( D `  E ) ) )
4847oveq1d 6216 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( ( ( D `
 E ) ( .r `  R ) ( D `  X
) )  ./  ( D `  X )
)  =  ( ( ( D `  X
) ( .r `  R ) ( D `
 E ) ) 
./  ( D `  X ) ) )
4914adantl 466 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  ->  N  e.  Fin )
5041adantr 465 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  ->  R  e.  CRing )
5127adantr 465 . . . . . 6  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  ->  I  e.  N )
5249, 50, 513jca 1168 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( N  e.  Fin  /\  R  e.  CRing  /\  I  e.  N ) )
53523adant3 1008 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( N  e.  Fin  /\  R  e.  CRing  /\  I  e.  N ) )
545, 6, 23, 11, 4cramerimplem1 18622 . . . 4  |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  I  e.  N )  /\  Z  e.  V )  ->  ( D `  E )  =  ( Z `  I ) )
5553, 26, 54syl2anc 661 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( D `  E
)  =  ( Z `
 I ) )
5640, 48, 553eqtr3rd 2504 . 2  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( Z `  I
)  =  ( ( ( D `  X
) ( .r `  R ) ( D `
 E ) ) 
./  ( D `  X ) ) )
57 cramerimp.h . . . . 5  |-  H  =  ( ( X ( N matRepV  R ) Y ) `
 I )
585, 6, 23, 11, 57, 24, 4, 38cramerimplem3 18624 . . . 4  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( X  .x.  Z )  =  Y )  ->  (
( D `  X
) ( .r `  R ) ( D `
 E ) )  =  ( D `  H ) )
59583adant3r 1216 . . 3  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( ( D `  X ) ( .r
`  R ) ( D `  E ) )  =  ( D `
 H ) )
6059oveq1d 6216 . 2  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( ( ( D `
 X ) ( .r `  R ) ( D `  E
) )  ./  ( D `  X )
)  =  ( ( D `  H ) 
./  ( D `  X ) ) )
6156, 60eqtrd 2495 1  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V )  /\  (
( X  .x.  Z
)  =  Y  /\  ( D `  X )  e.  (Unit `  R
) ) )  -> 
( Z `  I
)  =  ( ( D `  H ) 
./  ( D `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   (/)c0 3746   <.cop 3992   -->wf 5523   ` cfv 5527  (class class class)co 6201    ^m cmap 7325   Fincfn 7421   Basecbs 14293   .rcmulr 14359   1rcur 16726   Ringcrg 16769   CRingccrg 16770  Unitcui 16855  /rcdvr 16898   Mat cmat 18406   maVecMul cmvmul 18479   matRepV cmatrepV 18496   maDet cmdat 18523
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 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-addf 9473  ax-mulf 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  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 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-ot 3995  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-tpos 6856  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-sup 7803  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-rp 11104  df-fz 11556  df-fzo 11667  df-seq 11925  df-exp 11984  df-hash 12222  df-word 12348  df-concat 12350  df-s1 12351  df-substr 12352  df-splice 12353  df-reverse 12354  df-s2 12594  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-starv 14373  df-sca 14374  df-vsca 14375  df-ip 14376  df-tset 14377  df-ple 14378  df-ds 14380  df-unif 14381  df-hom 14382  df-cco 14383  df-0g 14500  df-gsum 14501  df-prds 14506  df-pws 14508  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-mulg 15668  df-subg 15798  df-ghm 15865  df-gim 15907  df-cntz 15955  df-oppg 15981  df-symg 16003  df-pmtr 16068  df-psgn 16117  df-evpm 16118  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-srg 16731  df-rng 16771  df-cring 16772  df-oppr 16839  df-dvdsr 16857  df-unit 16858  df-invr 16888  df-dvr 16899  df-rnghom 16930  df-drng 16958  df-subrg 16987  df-lmod 17074  df-lss 17138  df-sra 17377  df-rgmod 17378  df-cnfld 17945  df-zring 18010  df-zrh 18061  df-dsmm 18283  df-frlm 18298  df-mamu 18407  df-mat 18408  df-mvmul 18480  df-marrep 18497  df-marepv 18498  df-subma 18516  df-mdet 18524  df-minmar1 18574
This theorem is referenced by:  cramerlem1  18626
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