MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  slesolinvbi Structured version   Unicode version

Theorem slesolinvbi 19641
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 1008 . . 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 1009 . . 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 1007 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( D `  X
)  e.  (Unit `  R ) )
43anim1i 570 . . 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 19640 . . 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 1264 . 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 6250 . . 3  |-  ( Z  =  ( ( I `
 X )  .x.  Y )  ->  ( X  .x.  Z )  =  ( X  .x.  (
( I `  X
)  .x.  Y )
) )
14 simpr 462 . . . . . . . . . 10  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  R  e.  CRing )
155, 6matrcl 19372 . . . . . . . . . . . 12  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
1615simpld 460 . . . . . . . . . . 11  |-  ( X  e.  B  ->  N  e.  Fin )
1716adantr 466 . . . . . . . . . 10  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  N  e.  Fin )
1814, 17anim12ci 569 . . . . . . . . 9  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  R  e. 
CRing ) )
19183adant3 1025 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( N  e.  Fin  /\  R  e.  CRing ) )
20 eqid 2422 . . . . . . . . 9  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
215, 20matmulr 19398 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
2219, 21syl 17 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
2322oveqd 6259 . . . . . 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 crngring 17727 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  R  e.  Ring )
2524adantl 467 . . . . . . . . . 10  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  R  e.  Ring )
2625, 17anim12ci 569 . . . . . . . . 9  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  R  e. 
Ring ) )
27263adant3 1025 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
285matring 19403 . . . . . . . 8  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
2927, 28syl 17 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  A  e.  Ring )
30 eqid 2422 . . . . . . . . . 10  |-  (Unit `  A )  =  (Unit `  A )
31 eqid 2422 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
325, 9, 6, 30, 31matunit 19638 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( X  e.  (Unit `  A
)  <->  ( D `  X )  e.  (Unit `  R ) ) )
3332ad2ant2lr 752 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( X  e.  (Unit `  A )  <->  ( D `  X )  e.  (Unit `  R
) ) )
3433biimp3ar 1365 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  X  e.  (Unit `  A
) )
35 eqid 2422 . . . . . . . 8  |-  ( .r
`  A )  =  ( .r `  A
)
36 eqid 2422 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
3730, 10, 35, 36unitrinv 17842 . . . . . . 7  |-  ( ( A  e.  Ring  /\  X  e.  (Unit `  A )
)  ->  ( X
( .r `  A
) ( I `  X ) )  =  ( 1r `  A
) )
3829, 34, 37syl2anc 665 . . . . . 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 2456 . . . . 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 6257 . . . 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 2422 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
42253ad2ant1 1026 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  R  e.  Ring )
43173ad2ant2 1027 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  N  e.  Fin )
447eleq2i 2492 . . . . . . . 8  |-  ( Y  e.  V  <->  Y  e.  ( ( Base `  R
)  ^m  N )
)
4544biimpi 197 . . . . . . 7  |-  ( Y  e.  V  ->  Y  e.  ( ( Base `  R
)  ^m  N )
)
4645adantl 467 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  ( (
Base `  R )  ^m  N ) )
47463ad2ant2 1027 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  Y  e.  ( ( Base `  R )  ^m  N ) )
486eleq2i 2492 . . . . . . . 8  |-  ( X  e.  B  <->  X  e.  ( Base `  A )
)
4948biimpi 197 . . . . . . 7  |-  ( X  e.  B  ->  X  e.  ( Base `  A
) )
5049adantr 466 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  X  e.  ( Base `  A ) )
51503ad2ant2 1027 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  X  e.  ( Base `  A ) )
52 eqid 2422 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  A )
5330, 10, 52ringinvcl 17840 . . . . . 6  |-  ( ( A  e.  Ring  /\  X  e.  (Unit `  A )
)  ->  ( I `  X )  e.  (
Base `  A )
)
5429, 34, 53syl2anc 665 . . . . 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 19509 . . . 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 19508 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( 1r `  A )  .x.  Y
)  =  Y )
5740, 55, 563eqtr3d 2464 . . 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 2478 . 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 840 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2593   _Vcvv 3016   (/)c0 3697   <.cop 3940   <.cotp 3942   ` cfv 5537  (class class class)co 6242    ^m cmap 7420   Fincfn 7517   Basecbs 15057   .rcmulr 15127   1rcur 17671   Ringcrg 17716   CRingccrg 17717  Unitcui 17803   invrcinvr 17835   maMul cmmul 19343   Mat cmat 19367   maVecMul cmvmul 19500   maDet cmdat 19544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-inf2 8092  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-xor 1401  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rmo 2716  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-ot 3943  df-uni 4156  df-int 4192  df-iun 4237  df-iin 4238  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-se 4749  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-isom 5546  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-of 6482  df-om 6644  df-1st 6744  df-2nd 6745  df-supp 6863  df-tpos 6921  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7830  df-sup 7902  df-oi 7971  df-card 8318  df-cda 8542  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-div 10214  df-nn 10554  df-2 10612  df-3 10613  df-4 10614  df-5 10615  df-6 10616  df-7 10617  df-8 10618  df-9 10619  df-10 10620  df-n0 10814  df-z 10882  df-dec 10996  df-uz 11104  df-rp 11247  df-fz 11729  df-fzo 11860  df-seq 12157  df-exp 12216  df-hash 12459  df-word 12605  df-lsw 12606  df-concat 12607  df-s1 12608  df-substr 12609  df-splice 12610  df-reverse 12611  df-s2 12883  df-struct 15059  df-ndx 15060  df-slot 15061  df-base 15062  df-sets 15063  df-ress 15064  df-plusg 15139  df-mulr 15140  df-starv 15141  df-sca 15142  df-vsca 15143  df-ip 15144  df-tset 15145  df-ple 15146  df-ds 15148  df-unif 15149  df-hom 15150  df-cco 15151  df-0g 15276  df-gsum 15277  df-prds 15282  df-pws 15284  df-mre 15428  df-mrc 15429  df-acs 15431  df-mgm 16424  df-sgrp 16463  df-mnd 16473  df-mhm 16518  df-submnd 16519  df-grp 16609  df-minusg 16610  df-sbg 16611  df-mulg 16612  df-subg 16750  df-ghm 16817  df-gim 16859  df-cntz 16907  df-oppg 16933  df-symg 16955  df-pmtr 17019  df-psgn 17068  df-evpm 17069  df-cmn 17368  df-abl 17369  df-mgp 17660  df-ur 17672  df-srg 17676  df-ring 17718  df-cring 17719  df-oppr 17787  df-dvdsr 17805  df-unit 17806  df-invr 17836  df-dvr 17847  df-rnghom 17879  df-drng 17913  df-subrg 17942  df-lmod 18029  df-lss 18092  df-sra 18331  df-rgmod 18332  df-assa 18472  df-cnfld 18907  df-zring 18975  df-zrh 19010  df-dsmm 19230  df-frlm 19245  df-mamu 19344  df-mat 19368  df-mvmul 19501  df-mdet 19545  df-madu 19594
This theorem is referenced by:  slesolex  19642
  Copyright terms: Public domain W3C validator