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Theorem slesolex 19699
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 2423 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
4 eqid 2423 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 crngring 17784 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
65adantl 468 . . . . . 6  |-  ( ( N  =/=  (/)  /\  R  e.  CRing )  ->  R  e.  Ring )
763ad2ant1 1027 . . . . 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 19429 . . . . . . . 8  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
109simpld 461 . . . . . . 7  |-  ( X  e.  B  ->  N  e.  Fin )
1110adantr 467 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  N  e.  Fin )
12113ad2ant2 1028 . . . . 5  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  N  e.  Fin )
136, 11anim12ci 570 . . . . . . . 8  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( N  e.  Fin  /\  R  e. 
Ring ) )
14133adant3 1026 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  -> 
( N  e.  Fin  /\  R  e.  Ring )
)
151matring 19460 . . . . . . 7  |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
1614, 15syl 17 . . . . . 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 2423 . . . . . . . . . 10  |-  (Unit `  A )  =  (Unit `  A )
19 eqid 2423 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
201, 17, 8, 18, 19matunit 19695 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( X  e.  (Unit `  A
)  <->  ( D `  X )  e.  (Unit `  R ) ) )
2120bicomd 205 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( D `  X
)  e.  (Unit `  R )  <->  X  e.  (Unit `  A ) ) )
2221ad2ant2lr 753 . . . . . . 7  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )
)  ->  ( ( D `  X )  e.  (Unit `  R )  <->  X  e.  (Unit `  A
) ) )
2322biimp3a 1365 . . . . . 6  |-  ( ( ( N  =/=  (/)  /\  R  e.  CRing )  /\  ( X  e.  B  /\  Y  e.  V )  /\  ( D `  X
)  e.  (Unit `  R ) )  ->  X  e.  (Unit `  A
) )
24 eqid 2423 . . . . . . 7  |-  ( invr `  A )  =  (
invr `  A )
25 eqid 2423 . . . . . . 7  |-  ( Base `  A )  =  (
Base `  A )
2618, 24, 25ringinvcl 17897 . . . . . 6  |-  ( ( A  e.  Ring  /\  X  e.  (Unit `  A )
)  ->  ( ( invr `  A ) `  X )  e.  (
Base `  A )
)
2716, 23, 26syl2anc 666 . . . . 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 2501 . . . . . . . 8  |-  ( Y  e.  V  <->  Y  e.  ( ( Base `  R
)  ^m  N )
)
3029biimpi 198 . . . . . . 7  |-  ( Y  e.  V  ->  Y  e.  ( ( Base `  R
)  ^m  N )
)
3130adantl 468 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  V )  ->  Y  e.  ( (
Base `  R )  ^m  N ) )
32313ad2ant2 1028 . . . . 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 19564 . . . 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 2522 . . 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 19698 . . . . . 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 467 . . . . 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 227 . . . 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 442 . . 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 3185 . 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 50 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   E.wrex 2777   _Vcvv 3082   (/)c0 3762   <.cop 4003   ` cfv 5599  (class class class)co 6303    ^m cmap 7478   Fincfn 7575   Basecbs 15114   .rcmulr 15184   Ringcrg 17773   CRingccrg 17774  Unitcui 17860   invrcinvr 17892   Mat cmat 19424   maVecMul cmvmul 19557   maDet cmdat 19601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-xor 1402  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-ot 4006  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-tpos 6979  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-sup 7960  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-rp 11305  df-fz 11787  df-fzo 11918  df-seq 12215  df-exp 12274  df-hash 12517  df-word 12662  df-lsw 12663  df-concat 12664  df-s1 12665  df-substr 12666  df-splice 12667  df-reverse 12668  df-s2 12940  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-0g 15333  df-gsum 15334  df-prds 15339  df-pws 15341  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-subg 16807  df-ghm 16874  df-gim 16916  df-cntz 16964  df-oppg 16990  df-symg 17012  df-pmtr 17076  df-psgn 17125  df-evpm 17126  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-srg 17733  df-ring 17775  df-cring 17776  df-oppr 17844  df-dvdsr 17862  df-unit 17863  df-invr 17893  df-dvr 17904  df-rnghom 17936  df-drng 17970  df-subrg 17999  df-lmod 18086  df-lss 18149  df-sra 18388  df-rgmod 18389  df-assa 18529  df-cnfld 18964  df-zring 19032  df-zrh 19067  df-dsmm 19287  df-frlm 19302  df-mamu 19401  df-mat 19425  df-mvmul 19558  df-mdet 19602  df-madu 19651
This theorem is referenced by:  cramerlem3  19706
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