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Theorem cramerimp 18952
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 16993 . . . . . 6  |-  ( R  e.  CRing  ->  R  e.  Ring )
21adantr 465 . . . . 5  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  R  e.  Ring )
323ad2ant1 1017 . . . 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 2467 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
84, 5, 6, 7mdetf 18861 . . . . . . 7  |-  ( R  e.  CRing  ->  D : B
--> ( Base `  R
) )
98adantr 465 . . . . . 6  |-  ( ( R  e.  CRing  /\  I  e.  N )  ->  D : B --> ( Base `  R
) )
1093ad2ant1 1017 . . . . 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 18678 . . . . . . . . . . 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 1016 . . . . . . 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 3791 . . . . . . . . . . 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 1016 . . . . . . . 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 1019 . . . . . . . 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 18945 . . . . . . . 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 1017 . . . . . . 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 2467 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
305, 6, 23, 29ma1repvcl 18836 . . . . . . 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 1226 . . . . . 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 2559 . . . . 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 6020 . . . 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 1019 . . . 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 2467 . . . . 5  |-  (Unit `  R )  =  (Unit `  R )
37 cramerimp.q . . . . 5  |-  ./  =  (/r
`  R )
38 eqid 2467 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
397, 36, 37, 38dvrcan3 17122 . . . 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 1228 . . 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 1017 . . . . 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 17089 . . . . . . 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 1019 . . . . 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 16997 . . . . 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 1228 . . . 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 6297 . . 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 1176 . . . . 5  |-  ( ( ( R  e.  CRing  /\  I  e.  N )  /\  ( X  e.  B  /\  Y  e.  V ) )  -> 
( N  e.  Fin  /\  R  e.  CRing  /\  I  e.  N ) )
53523adant3 1016 . . . 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 18949 . . . 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 2517 . 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 18951 . . . 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 1225 . . 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 6297 . 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 2508 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   <.cop 4033   -->wf 5582   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Fincfn 7513   Basecbs 14483   .rcmulr 14549   1rcur 16940   Ringcrg 16983   CRingccrg 16984  Unitcui 17069  /rcdvr 17112   Mat cmat 18673   maVecMul cmvmul 18806   matRepV cmatrepV 18823   maDet cmdat 18850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-word 12502  df-concat 12504  df-s1 12505  df-substr 12506  df-splice 12507  df-reverse 12508  df-s2 12770  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-0g 14690  df-gsum 14691  df-prds 14696  df-pws 14698  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-mhm 15774  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-mulg 15858  df-subg 15990  df-ghm 16057  df-gim 16099  df-cntz 16147  df-oppg 16173  df-symg 16195  df-pmtr 16260  df-psgn 16309  df-evpm 16310  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-srg 16945  df-rng 16985  df-cring 16986  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-dvr 17113  df-rnghom 17145  df-drng 17178  df-subrg 17207  df-lmod 17294  df-lss 17359  df-sra 17598  df-rgmod 17599  df-cnfld 18189  df-zring 18254  df-zrh 18305  df-dsmm 18527  df-frlm 18542  df-mamu 18650  df-mat 18674  df-mvmul 18807  df-marrep 18824  df-marepv 18825  df-subma 18843  df-mdet 18851  df-minmar1 18901
This theorem is referenced by:  cramerlem1  18953
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