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Theorem slesolinvbi 18492
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 991 . . 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 992 . . 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 990 . . . 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 18491 . . 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 1218 . 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 6104 . . 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 18317 . . . . . . . . . . . 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 1008 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( N  e.  Fin  /\  R  e.  CRing ) )
20 eqid 2443 . . . . . . . . 9  |-  ( R maMul  <. N ,  N ,  N >. )  =  ( R maMul  <. N ,  N ,  N >. )
215, 20matmulr 18318 . . . . . . . 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 6113 . . . . . 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 16660 . . . . . . . . . . 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 1008 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
285matrng 18335 . . . . . . . 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 2443 . . . . . . . . . 10  |-  (Unit `  A )  =  (Unit `  A )
31 eqid 2443 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
325, 9, 6, 30, 31matunit 18489 . . . . . . . . 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 1319 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  X  e.  (Unit `  A
) )
35 eqid 2443 . . . . . . . 8  |-  ( .r
`  A )  =  ( .r `  A
)
36 eqid 2443 . . . . . . . 8  |-  ( 1r
`  A )  =  ( 1r `  A
)
3730, 10, 35, 36unitrinv 16775 . . . . . . 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 2475 . . . . 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 6111 . . . 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 2443 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
42253ad2ant1 1009 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  R  e.  Ring )
43173ad2ant2 1010 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  N  e.  Fin )
447eleq2i 2507 . . . . . . . 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 1010 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  Y  e.  ( ( Base `  R )  ^m  N ) )
486eleq2i 2507 . . . . . . . 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 1010 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  X  e.  ( Base `  A ) )
52 eqid 2443 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  A )
5330, 10, 52rnginvcl 16773 . . . . . 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 18365 . . . 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 18364 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( 1r `  A )  .x.  Y
)  =  Y )
5740, 55, 563eqtr3d 2483 . . 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 2497 . 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 828 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 2611   _Vcvv 2977   (/)c0 3642   <.cop 3888   <.cotp 3890   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   Fincfn 7315   Basecbs 14179   .rcmulr 14244   1rcur 16608   Ringcrg 16650   CRingccrg 16651  Unitcui 16736   invrcinvr 16768   maMul cmmul 18284   Mat cmat 18285   maVecMul cmvmul 18356   maDet cmdat 18400
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-addf 9366  ax-mulf 9367
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-ot 3891  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-word 12234  df-concat 12236  df-s1 12237  df-substr 12238  df-splice 12239  df-reverse 12240  df-s2 12480  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-0g 14385  df-gsum 14386  df-prds 14391  df-pws 14393  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-mhm 15469  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-ghm 15750  df-gim 15792  df-cntz 15840  df-oppg 15866  df-symg 15888  df-pmtr 15953  df-psgn 16002  df-evpm 16003  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-rng 16652  df-cring 16653  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-dvr 16780  df-rnghom 16811  df-drng 16839  df-subrg 16868  df-lmod 16955  df-lss 17019  df-sra 17258  df-rgmod 17259  df-assa 17389  df-cnfld 17824  df-zring 17889  df-zrh 17940  df-dsmm 18162  df-frlm 18177  df-mamu 18286  df-mat 18287  df-mvmul 18357  df-mdet 18401  df-madu 18445
This theorem is referenced by:  slesolex  18493
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