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Theorem slesolinvbi 18978
Description: The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the matrix with the right-hand side vector. (Contributed by AV, 11-Feb-2019.) (Revised by AV, 28-Feb-2019.)
Hypotheses
Ref Expression
slesolex.a  |-  A  =  ( N Mat  R )
slesolex.b  |-  B  =  ( Base `  A
)
slesolex.v  |-  V  =  ( ( Base `  R
)  ^m  N )
slesolex.x  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
slesolex.d  |-  D  =  ( N maDet  R )
slesolinv.i  |-  I  =  ( invr `  A
)
Assertion
Ref Expression
slesolinvbi  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( X  .x.  Z )  =  Y  <-> 
Z  =  ( ( I `  X ) 
.x.  Y ) ) )

Proof of Theorem slesolinvbi
StepHypRef Expression
1 simpl1 999 . . 3  |-  ( ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X )  e.  (Unit `  R )
)  /\  ( X  .x.  Z )  =  Y )  ->  ( N  =/=  (/)  /\  R  e. 
CRing ) )
2 simpl2 1000 . . 3  |-  ( ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X )  e.  (Unit `  R )
)  /\  ( X  .x.  Z )  =  Y )  ->  ( X  e.  B  /\  Y  e.  V ) )
3 simp3 998 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( D `  X
)  e.  (Unit `  R ) )
43anim1i 568 . . 3  |-  ( ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X )  e.  (Unit `  R )
)  /\  ( X  .x.  Z )  =  Y )  ->  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z
)  =  Y ) )
5 slesolex.a . . . 4  |-  A  =  ( N Mat  R )
6 slesolex.b . . . 4  |-  B  =  ( Base `  A
)
7 slesolex.v . . . 4  |-  V  =  ( ( Base `  R
)  ^m  N )
8 slesolex.x . . . 4  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
9 slesolex.d . . . 4  |-  D  =  ( N maDet  R )
10 slesolinv.i . . . 4  |-  I  =  ( invr `  A
)
115, 6, 7, 8, 9, 10slesolinv 18977 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  ->  Z  =  ( (
I `  X )  .x.  Y ) )
121, 2, 4, 11syl3anc 1228 . 2  |-  ( ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X )  e.  (Unit `  R )
)  /\  ( X  .x.  Z )  =  Y )  ->  Z  =  ( ( I `  X )  .x.  Y
) )
13 oveq2 6292 . . 3  |-  ( Z  =  ( ( I `
 X )  .x.  Y )  ->  ( X  .x.  Z )  =  ( X  .x.  (
( I `  X
)  .x.  Y )
) )
14 simpr 461 . . . . . . . . . 10  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  R  e.  CRing )
155, 6matrcl 18709 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
1615simpld 459 . . . . . . . . . . 11  |-  ( X  e.  B  ->  N  e.  Fin )
1716adantr 465 . . . . . . . . . 10  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  N  e.  Fin )
1814, 17anim12ci 567 . . . . . . . . 9  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  R  e. 
CRing ) )
19183adant3 1016 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( N  e.  Fin  /\  R  e.  CRing ) )
20 eqid 2467 . . . . . . . . 9  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
215, 20matmulr 18735 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
2219, 21syl 16 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
2322oveqd 6301 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( X ( R maMul  <. N ,  N ,  N >. ) ( I `
 X ) )  =  ( X ( .r `  A ) ( I `  X
) ) )
24 crngrng 17010 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  R  e.  Ring )
2524adantl 466 . . . . . . . . . 10  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  R  e.  Ring )
2625, 17anim12ci 567 . . . . . . . . 9  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  R  e. 
Ring ) )
27263adant3 1016 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
285matrng 18740 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
2927, 28syl 16 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  A  e.  Ring )
30 eqid 2467 . . . . . . . . . 10  |-  (Unit `  A )  =  (Unit `  A )
31 eqid 2467 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
325, 9, 6, 30, 31matunit 18975 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( X  e.  (Unit `  A
)  <->  ( D `  X )  e.  (Unit `  R ) ) )
3332ad2ant2lr 747 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( X  e.  (Unit `  A )  <->  ( D `  X )  e.  (Unit `  R
) ) )
3433biimp3ar 1329 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  X  e.  (Unit `  A
) )
35 eqid 2467 . . . . . . . 8  |-  ( .r
`  A )  =  ( .r `  A
)
36 eqid 2467 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
3730, 10, 35, 36unitrinv 17128 . . . . . . 7  |-  ( ( A  e.  Ring  /\  X  e.  (Unit `  A )
)  ->  ( X
( .r `  A
) ( I `  X ) )  =  ( 1r `  A
) )
3829, 34, 37syl2anc 661 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( X ( .r
`  A ) ( I `  X ) )  =  ( 1r
`  A ) )
3923, 38eqtrd 2508 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( X ( R maMul  <. N ,  N ,  N >. ) ( I `
 X ) )  =  ( 1r `  A ) )
4039oveq1d 6299 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( X ( R maMul  <. N ,  N ,  N >. ) ( I `
 X ) ) 
.x.  Y )  =  ( ( 1r `  A )  .x.  Y
) )
41 eqid 2467 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
42253ad2ant1 1017 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  R  e.  Ring )
43173ad2ant2 1018 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  N  e.  Fin )
447eleq2i 2545 . . . . . . . 8  |-  ( Y  e.  V  <->  Y  e.  ( ( Base `  R
)  ^m  N )
)
4544biimpi 194 . . . . . . 7  |-  ( Y  e.  V  ->  Y  e.  ( ( Base `  R
)  ^m  N )
)
4645adantl 466 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  ( (
Base `  R )  ^m  N ) )
47463ad2ant2 1018 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  Y  e.  ( ( Base `  R )  ^m  N ) )
486eleq2i 2545 . . . . . . . 8  |-  ( X  e.  B  <->  X  e.  ( Base `  A )
)
4948biimpi 194 . . . . . . 7  |-  ( X  e.  B  ->  X  e.  ( Base `  A
) )
5049adantr 465 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  X  e.  ( Base `  A ) )
51503ad2ant2 1018 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  X  e.  ( Base `  A ) )
52 eqid 2467 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  A )
5330, 10, 52rnginvcl 17126 . . . . . 6  |-  ( ( A  e.  Ring  /\  X  e.  (Unit `  A )
)  ->  ( I `  X )  e.  (
Base `  A )
)
5429, 34, 53syl2anc 661 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( I `  X
)  e.  ( Base `  A ) )
555, 41, 8, 42, 43, 47, 20, 51, 54mavmulass 18846 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( X ( R maMul  <. N ,  N ,  N >. ) ( I `
 X ) ) 
.x.  Y )  =  ( X  .x.  (
( I `  X
)  .x.  Y )
) )
565, 41, 8, 42, 43, 471mavmul 18845 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( 1r `  A )  .x.  Y
)  =  Y )
5740, 55, 563eqtr3d 2516 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( X  .x.  (
( I `  X
)  .x.  Y )
)  =  Y )
5813, 57sylan9eqr 2530 . 2  |-  ( ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X )  e.  (Unit `  R )
)  /\  Z  =  ( ( I `  X )  .x.  Y
) )  ->  ( X  .x.  Z )  =  Y )
5912, 58impbida 830 1  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( X  .x.  Z )  =  Y  <-> 
Z  =  ( ( I `  X ) 
.x.  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   <.cop 4033   <.cotp 4035   ` cfv 5588  (class class class)co 6284    ^m cmap 7420   Fincfn 7516   Basecbs 14490   .rcmulr 14556   1rcur 16955   Ringcrg 17000   CRingccrg 17001  Unitcui 17089   invrcinvr 17121   maMul cmmul 18680   Mat cmat 18704   maVecMul cmvmul 18837   maDet cmdat 18881
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 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-addf 9571  ax-mulf 9572
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-seq 12076  df-exp 12135  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-substr 12512  df-splice 12513  df-reverse 12514  df-s2 12776  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-0g 14697  df-gsum 14698  df-prds 14703  df-pws 14705  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-ghm 16070  df-gim 16112  df-cntz 16160  df-oppg 16186  df-symg 16208  df-pmtr 16273  df-psgn 16322  df-evpm 16323  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-srg 16960  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-rnghom 17165  df-drng 17198  df-subrg 17227  df-lmod 17314  df-lss 17379  df-sra 17618  df-rgmod 17619  df-assa 17760  df-cnfld 18220  df-zring 18285  df-zrh 18336  df-dsmm 18558  df-frlm 18573  df-mamu 18681  df-mat 18705  df-mvmul 18838  df-mdet 18882  df-madu 18931
This theorem is referenced by:  slesolex  18979
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