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Theorem slesolinv 18484
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, 10-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
slesolinv  |-  ( ( ( 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 slesolinv
StepHypRef Expression
1 slesolex.a . . 3  |-  A  =  ( N Mat  R )
2 eqid 2441 . . 3  |-  ( Base `  R )  =  (
Base `  R )
3 slesolex.x . . 3  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
4 crngrng 16653 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
54adantl 466 . . . 4  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  R  e.  Ring )
653ad2ant1 1009 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  ->  R  e.  Ring )
7 slesolex.b . . . . . . 7  |-  B  =  ( Base `  A
)
81, 7matrcl 18310 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
98simpld 459 . . . . 5  |-  ( X  e.  B  ->  N  e.  Fin )
109adantr 465 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  N  e.  Fin )
11103ad2ant2 1010 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  ->  N  e.  Fin )
124anim2i 569 . . . . . . 7  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  ( N  =/=  (/)  /\  R  e. 
Ring ) )
1312anim1i 568 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( N  =/=  (/)  /\  R  e. 
Ring )  /\  ( X  e.  B  /\  Y  e.  V )
) )
14133adant3 1008 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V ) ) )
15 simpr 461 . . . . . 6  |-  ( ( ( D `  X
)  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y )  ->  ( X  .x.  Z )  =  Y )
16153ad2ant3 1011 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( X  .x.  Z
)  =  Y )
17 slesolex.v . . . . . 6  |-  V  =  ( ( Base `  R
)  ^m  N )
181, 7, 17, 3slesolvec 18483 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  Ring )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( X  .x.  Z )  =  Y  ->  Z  e.  V ) )
1914, 16, 18sylc 60 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  ->  Z  e.  V )
2019, 17syl6eleq 2531 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  ->  Z  e.  ( ( Base `  R )  ^m  N ) )
21 eqid 2441 . . 3  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
225, 10anim12ci 567 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  R  e. 
Ring ) )
23223adant3 1008 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
241matrng 18328 . . . . 5  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
2523, 24syl 16 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  ->  A  e.  Ring )
26 slesolex.d . . . . . . . . . 10  |-  D  =  ( N maDet  R )
27 eqid 2441 . . . . . . . . . 10  |-  (Unit `  A )  =  (Unit `  A )
28 eqid 2441 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
291, 26, 7, 27, 28matunit 18482 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( X  e.  (Unit `  A
)  <->  ( D `  X )  e.  (Unit `  R ) ) )
3029bicomd 201 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( D `  X
)  e.  (Unit `  R )  <->  X  e.  (Unit `  A ) ) )
3130ad2ant2lr 747 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( D `  X )  e.  (Unit `  R )  <->  X  e.  (Unit `  A
) ) )
3231biimpd 207 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( D `  X )  e.  (Unit `  R )  ->  X  e.  (Unit `  A ) ) )
3332adantrd 468 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( (
( D `  X
)  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y )  ->  X  e.  (Unit `  A )
) )
34333impia 1184 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  ->  X  e.  (Unit `  A
) )
35 slesolinv.i . . . . 5  |-  I  =  ( invr `  A
)
36 eqid 2441 . . . . 5  |-  ( Base `  A )  =  (
Base `  A )
3727, 35, 36rnginvcl 16766 . . . 4  |-  ( ( A  e.  Ring  /\  X  e.  (Unit `  A )
)  ->  ( I `  X )  e.  (
Base `  A )
)
3825, 34, 37syl2anc 661 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( I `  X
)  e.  ( Base `  A ) )
397eleq2i 2505 . . . . . 6  |-  ( X  e.  B  <->  X  e.  ( Base `  A )
)
4039biimpi 194 . . . . 5  |-  ( X  e.  B  ->  X  e.  ( Base `  A
) )
4140adantr 465 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  X  e.  ( Base `  A ) )
42413ad2ant2 1010 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  ->  X  e.  ( Base `  A ) )
431, 2, 3, 6, 11, 20, 21, 38, 42mavmulass 18358 . 2  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( ( I `
 X ) ( R maMul  <. N ,  N ,  N >. ) X ) 
.x.  Z )  =  ( ( I `  X )  .x.  ( X  .x.  Z ) ) )
44 simpr 461 . . . . . . . . 9  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  R  e.  CRing )
4544, 10anim12ci 567 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  R  e. 
CRing ) )
46453adant3 1008 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( N  e.  Fin  /\  R  e.  CRing ) )
471, 21matmulr 18311 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  CRing )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
4846, 47syl 16 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( R maMul  <. N ,  N ,  N >. )  =  ( .r `  A ) )
4948oveqd 6106 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( I `  X ) ( R maMul  <. N ,  N ,  N >. ) X )  =  ( ( I `
 X ) ( .r `  A ) X ) )
50 eqid 2441 . . . . . . 7  |-  ( .r
`  A )  =  ( .r `  A
)
51 eqid 2441 . . . . . . 7  |-  ( 1r
`  A )  =  ( 1r `  A
)
5227, 35, 50, 51unitlinv 16767 . . . . . 6  |-  ( ( A  e.  Ring  /\  X  e.  (Unit `  A )
)  ->  ( (
I `  X )
( .r `  A
) X )  =  ( 1r `  A
) )
5325, 34, 52syl2anc 661 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( I `  X ) ( .r
`  A ) X )  =  ( 1r
`  A ) )
5449, 53eqtrd 2473 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( I `  X ) ( R maMul  <. N ,  N ,  N >. ) X )  =  ( 1r `  A ) )
5554oveq1d 6104 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( ( I `
 X ) ( R maMul  <. N ,  N ,  N >. ) X ) 
.x.  Z )  =  ( ( 1r `  A )  .x.  Z
) )
561, 2, 3, 6, 11, 201mavmul 18357 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( 1r `  A )  .x.  Z
)  =  Z )
5755, 56eqtrd 2473 . 2  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( ( I `
 X ) ( R maMul  <. N ,  N ,  N >. ) X ) 
.x.  Z )  =  Z )
58 oveq2 6097 . . . 4  |-  ( ( X  .x.  Z )  =  Y  ->  (
( I `  X
)  .x.  ( X  .x.  Z ) )  =  ( ( I `  X )  .x.  Y
) )
5958adantl 466 . . 3  |-  ( ( ( D `  X
)  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y )  ->  (
( I `  X
)  .x.  ( X  .x.  Z ) )  =  ( ( I `  X )  .x.  Y
) )
60593ad2ant3 1011 . 2  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( ( D `  X )  e.  (Unit `  R )  /\  ( X  .x.  Z )  =  Y ) )  -> 
( ( I `  X )  .x.  ( X  .x.  Z ) )  =  ( ( I `
 X )  .x.  Y ) )
6143, 57, 603eqtr3d 2481 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 965    = wceq 1369    e. wcel 1756    =/= wne 2604   _Vcvv 2970   (/)c0 3635   <.cop 3881   <.cotp 3883   ` cfv 5416  (class class class)co 6089    ^m cmap 7212   Fincfn 7308   Basecbs 14172   .rcmulr 14237   1rcur 16601   Ringcrg 16643   CRingccrg 16644  Unitcui 16729   invrcinvr 16761   maMul cmmul 18277   Mat cmat 18278   maVecMul cmvmul 18349   maDet cmdat 18393
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-ot 3884  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-tpos 6743  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-sup 7689  df-oi 7722  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-seq 11805  df-exp 11864  df-hash 12102  df-word 12227  df-concat 12229  df-s1 12230  df-substr 12231  df-splice 12232  df-reverse 12233  df-s2 12473  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-0g 14378  df-gsum 14379  df-prds 14384  df-pws 14386  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-mhm 15462  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546  df-subg 15676  df-ghm 15743  df-gim 15785  df-cntz 15833  df-oppg 15859  df-symg 15881  df-pmtr 15946  df-psgn 15995  df-evpm 15996  df-cmn 16277  df-abl 16278  df-mgp 16590  df-ur 16602  df-rng 16645  df-cring 16646  df-oppr 16713  df-dvdsr 16731  df-unit 16732  df-invr 16762  df-dvr 16773  df-rnghom 16804  df-drng 16832  df-subrg 16861  df-lmod 16948  df-lss 17012  df-sra 17251  df-rgmod 17252  df-assa 17382  df-cnfld 17817  df-zring 17882  df-zrh 17933  df-dsmm 18155  df-frlm 18170  df-mamu 18279  df-mat 18280  df-mvmul 18350  df-mdet 18394  df-madu 18438
This theorem is referenced by:  slesolinvbi  18485
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