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Theorem slesolex 18951
Description: Every system of linear equations represented by a matrix with a unit as determinant has a solution. (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 )
Assertion
Ref Expression
slesolex  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  E. z  e.  V  ( X  .x.  z )  =  Y )
Distinct variable groups:    z, A    z, B    z, D    z, N    z, R    z, V    z, X    z, Y    z,  .x.

Proof of Theorem slesolex
StepHypRef Expression
1 slesolex.a . . . . 5  |-  A  =  ( N Mat  R )
2 slesolex.x . . . . 5  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
3 eqid 2467 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2467 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 crngrng 16996 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
65adantl 466 . . . . . 6  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  R  e.  Ring )
763ad2ant1 1017 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  R  e.  Ring )
8 slesolex.b . . . . . . . . 9  |-  B  =  ( Base `  A
)
91, 8matrcl 18681 . . . . . . . 8  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
109simpld 459 . . . . . . 7  |-  ( X  e.  B  ->  N  e.  Fin )
1110adantr 465 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  N  e.  Fin )
12113ad2ant2 1018 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  N  e.  Fin )
136, 11anim12ci 567 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  R  e. 
Ring ) )
14133adant3 1016 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
151matrng 18712 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
1614, 15syl 16 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  A  e.  Ring )
17 slesolex.d . . . . . . . . . 10  |-  D  =  ( N maDet  R )
18 eqid 2467 . . . . . . . . . 10  |-  (Unit `  A )  =  (Unit `  A )
19 eqid 2467 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
201, 17, 8, 18, 19matunit 18947 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( X  e.  (Unit `  A
)  <->  ( D `  X )  e.  (Unit `  R ) ) )
2120bicomd 201 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( D `  X
)  e.  (Unit `  R )  <->  X  e.  (Unit `  A ) ) )
2221ad2ant2lr 747 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( D `  X )  e.  (Unit `  R )  <->  X  e.  (Unit `  A
) ) )
2322biimp3a 1328 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  X  e.  (Unit `  A
) )
24 eqid 2467 . . . . . . 7  |-  ( invr `  A )  =  (
invr `  A )
25 eqid 2467 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  A )
2618, 24, 25rnginvcl 17109 . . . . . 6  |-  ( ( A  e.  Ring  /\  X  e.  (Unit `  A )
)  ->  ( ( invr `  A ) `  X )  e.  (
Base `  A )
)
2716, 23, 26syl2anc 661 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( invr `  A
) `  X )  e.  ( Base `  A
) )
28 slesolex.v . . . . . . . . 9  |-  V  =  ( ( Base `  R
)  ^m  N )
2928eleq2i 2545 . . . . . . . 8  |-  ( Y  e.  V  <->  Y  e.  ( ( Base `  R
)  ^m  N )
)
3029biimpi 194 . . . . . . 7  |-  ( Y  e.  V  ->  Y  e.  ( ( Base `  R
)  ^m  N )
)
3130adantl 466 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  ( (
Base `  R )  ^m  N ) )
32313ad2ant2 1018 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  Y  e.  ( ( Base `  R )  ^m  N ) )
331, 2, 3, 4, 7, 12, 27, 32mavmulcl 18816 . . . 4  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( ( invr `  A ) `  X
)  .x.  Y )  e.  ( ( Base `  R
)  ^m  N )
)
3433, 28syl6eleqr 2566 . . 3  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( ( invr `  A ) `  X
)  .x.  Y )  e.  V )
351, 8, 28, 2, 17, 24slesolinvbi 18950 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( X  .x.  z )  =  Y  <-> 
z  =  ( ( ( invr `  A
) `  X )  .x.  Y ) ) )
3635adantr 465 . . . . 5  |-  ( ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X )  e.  (Unit `  R )
)  /\  ( ( N  =/=  (/)  /\  R  e. 
CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) ) )  ->  ( ( X 
.x.  z )  =  Y  <->  z  =  ( ( ( invr `  A
) `  X )  .x.  Y ) ) )
3736biimprd 223 . . . 4  |-  ( ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X )  e.  (Unit `  R )
)  /\  ( ( N  =/=  (/)  /\  R  e. 
CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) ) )  ->  ( z  =  ( ( ( invr `  A ) `  X
)  .x.  Y )  ->  ( X  .x.  z
)  =  Y ) )
3837impancom 440 . . 3  |-  ( ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X )  e.  (Unit `  R )
)  /\  z  =  ( ( ( invr `  A ) `  X
)  .x.  Y )
)  ->  ( (
( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( X  .x.  z
)  =  Y ) )
3934, 38rspcimedv 3216 . 2  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( ( ( N  =/=  (/)  /\  R  e. 
CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  E. z  e.  V  ( X  .x.  z )  =  Y ) )
4039pm2.43i 47 1  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  E. z  e.  V  ( X  .x.  z )  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   _Vcvv 3113   (/)c0 3785   <.cop 4033   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Fincfn 7513   Basecbs 14486   .rcmulr 14552   Ringcrg 16986   CRingccrg 16987  Unitcui 17072   invrcinvr 17104   Mat cmat 18676   maVecMul cmvmul 18809   maDet cmdat 18853
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 12072  df-exp 12131  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-substr 12508  df-splice 12509  df-reverse 12510  df-s2 12772  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-0g 14693  df-gsum 14694  df-prds 14699  df-pws 14701  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-mulg 15861  df-subg 15993  df-ghm 16060  df-gim 16102  df-cntz 16150  df-oppg 16176  df-symg 16198  df-pmtr 16263  df-psgn 16312  df-evpm 16313  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-srg 16948  df-rng 16988  df-cring 16989  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-dvr 17116  df-rnghom 17148  df-drng 17181  df-subrg 17210  df-lmod 17297  df-lss 17362  df-sra 17601  df-rgmod 17602  df-assa 17732  df-cnfld 18192  df-zring 18257  df-zrh 18308  df-dsmm 18530  df-frlm 18545  df-mamu 18653  df-mat 18677  df-mvmul 18810  df-mdet 18854  df-madu 18903
This theorem is referenced by:  cramerlem3  18958
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