Theorem slesolex 18613

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

Step	Hyp	Ref	Expression
1		slesolex.a	. . . . 5 |- A = ( N Mat R )
2		slesolex.x	. . . . 5 |- .x. = ( R maVecMul <. N , N >. )
3		eqid 2451	. . . . 5 |- ( Base ` R ) = ( Base ` R )
4		eqid 2451	. . . . 5 |- ( .r ` R ) = ( .r ` R )
5		crngrng 16770	. . . . . . 7 |- ( R e. CRing -> R e. Ring )
6	5	adantl 466	. . . . . 6 |- ( ( N =/= (/) /\ R e. CRing ) -> R e. Ring )
7	6	3ad2ant1 1009	. . . . 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 )
9	1, 8	matrcl 18430	. . . . . . . 8 |- ( X e. B -> ( N e. Fin /\ R e. _V ) )
10	9	simpld 459	. . . . . . 7 |- ( X e. B -> N e. Fin )
11	10	adantr 465	. . . . . 6 |- ( ( X e. B /\ Y e. V ) -> N e. Fin )
12	11	3ad2ant2 1010	. . . . 5 |- ( ( ( N =/= (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) /\ ( D ` X ) e. (Unit ` R ) ) -> N e. Fin )
13	6, 11	anim12ci 567	. . . . . . . 8 |- ( ( ( N =/= (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) ) -> ( N e. Fin /\ R e. Ring ) )
14	13	3adant3 1008	. . . . . . 7 |- ( ( ( N =/= (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) /\ ( D ` X ) e. (Unit ` R ) ) -> ( N e. Fin /\ R e. Ring ) )
15	1	matrng 18449	. . . . . . 7 |- ( ( N e. Fin /\ R e. Ring ) -> A e. Ring )
16	14, 15	syl 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 2451	. . . . . . . . . 10 |- (Unit ` A ) = (Unit ` A )
19		eqid 2451	. . . . . . . . . 10 |- (Unit ` R ) = (Unit ` R )
20	1, 17, 8, 18, 19	matunit 18609	. . . . . . . . 9 |- ( ( R e. CRing /\ X e. B ) -> ( X e. (Unit ` A ) <-> ( D ` X ) e. (Unit ` R ) ) )
21	20	bicomd 201	. . . . . . . 8 |- ( ( R e. CRing /\ X e. B ) -> ( ( D ` X ) e. (Unit ` R ) <-> X e. (Unit ` A ) ) )
22	21	ad2ant2lr 747	. . . . . . 7 |- ( ( ( N =/= (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) ) -> ( ( D ` X ) e. (Unit ` R ) <-> X e. (Unit ` A ) ) )
23	22	biimp3a 1319	. . . . . 6 |- ( ( ( N =/= (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) /\ ( D ` X ) e. (Unit ` R ) ) -> X e. (Unit ` A ) )
24		eqid 2451	. . . . . . 7 |- ( invr ` A ) = ( invr ` A )
25		eqid 2451	. . . . . . 7 |- ( Base ` A ) = ( Base ` A )
26	18, 24, 25	rnginvcl 16883	. . . . . 6 |- ( ( A e. Ring /\ X e. (Unit ` A ) ) -> ( ( invr ` A ) ` X ) e. ( Base ` A ) )
27	16, 23, 26	syl2anc 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 )
29	28	eleq2i 2529	. . . . . . . 8 |- ( Y e. V <-> Y e. ( ( Base ` R ) ^m N ) )
30	29	biimpi 194	. . . . . . 7 |- ( Y e. V -> Y e. ( ( Base ` R ) ^m N ) )
31	30	adantl 466	. . . . . 6 |- ( ( X e. B /\ Y e. V ) -> Y e. ( ( Base ` R ) ^m N ) )
32	31	3ad2ant2 1010	. . . . 5 |- ( ( ( N =/= (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) /\ ( D ` X ) e. (Unit ` R ) ) -> Y e. ( ( Base ` R ) ^m N ) )
33	1, 2, 3, 4, 7, 12, 27, 32	mavmulcl 18478	. . . 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 ) )
34	33, 28	syl6eleqr 2550	. . 3 |- ( ( ( N =/= (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) /\ ( D ` X ) e. (Unit ` R ) ) -> ( ( ( invr ` A ) ` X ) .x. Y ) e. V )
35	1, 8, 28, 2, 17, 24	slesolinvbi 18612	. . . . . 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 ) ) )
36	35	adantr 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 ) ) )
37	36	biimprd 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 ) )
38	37	impancom 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 ) )
39	34, 38	rspcimedv 3174	. 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 ) )
40	39	pm2.43i 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 )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   =/= wne 2644   E.wrex 2796   _Vcvv 3071   (/)c0 3738   <.cop 3984   ` cfv 5519  (class class class)co 6193   ^m cmap 7317   Fincfn 7413   Basecbs 14285   .rcmulr 14350   Ringcrg 16760   CRingccrg 16761  Unitcui 16846   invrcinvr 16878   Mat cmat 18398   maVecMul cmvmul 18471   maDet cmdat 18515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-addf 9465  ax-mulf 9466

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-ot 3987  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-substr 12344  df-splice 12345  df-reverse 12346  df-s2 12586  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-0g 14491  df-gsum 14492  df-prds 14497  df-pws 14499  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mulg 15659  df-subg 15789  df-ghm 15856  df-gim 15898  df-cntz 15946  df-oppg 15972  df-symg 15994  df-pmtr 16059  df-psgn 16108  df-evpm 16109  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-srg 16722  df-rng 16762  df-cring 16763  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-dvr 16890  df-rnghom 16921  df-drng 16949  df-subrg 16978  df-lmod 17065  df-lss 17129  df-sra 17368  df-rgmod 17369  df-assa 17499  df-cnfld 17937  df-zring 18002  df-zrh 18053  df-dsmm 18275  df-frlm 18290  df-mamu 18399  df-mat 18400  df-mvmul 18472  df-mdet 18516  df-madu 18565

This theorem is referenced by:  cramerlem3  18620