Theorem cramer0 20496

Description: Special case of Cramer's rule for 0-dimensional matrices/vectors. (Contributed by AV, 28-Feb-2019.)

Hypotheses

Ref	Expression
cramer.a	|- A = ( N Mat R )
cramer.b	|- B = ( Base ` A )
cramer.v	|- V = ( ( Base ` R ) ^m N )
cramer.d	|- D = ( N maDet R )
cramer.x	|- .x. = ( R maVecMul <. N , N >. )
cramer.q	|- ./ = (/r ` R )

Assertion

Ref	Expression
cramer0	|- ( ( ( N = (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) /\ ( D ` X ) e. (Unit ` R ) ) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) -> ( X .x. Z ) = Y ) )

Distinct variable groups:   B, i   D, i   i, N   R, i   i, V   i, X   i, Y   i, Z   .x. , i   ./ , i

Allowed substitution hint:   A( i)

Proof of Theorem cramer0

Step	Hyp	Ref	Expression
1		cramer.b	. . . . . . . . 9 |- B = ( Base ` A )
2		cramer.a	. . . . . . . . . 10 |- A = ( N Mat R )
3	2	fveq2i 6194	. . . . . . . . 9 |- ( Base ` A ) = ( Base ` ( N Mat R ) )
4	1, 3	eqtri 2644	. . . . . . . 8 |- B = ( Base ` ( N Mat R ) )
5		oveq1 6657	. . . . . . . . 9 |- ( N = (/) -> ( N Mat R ) = ( (/) Mat R ) )
6	5	fveq2d 6195	. . . . . . . 8 |- ( N = (/) -> ( Base ` ( N Mat R ) ) = ( Base ` ( (/) Mat R ) ) )
7	4, 6	syl5eq 2668	. . . . . . 7 |- ( N = (/) -> B = ( Base ` ( (/) Mat R ) ) )
8	7	adantr 481	. . . . . 6 |- ( ( N = (/) /\ R e. CRing ) -> B = ( Base ` ( (/) Mat R ) ) )
9	8	eleq2d 2687	. . . . 5 |- ( ( N = (/) /\ R e. CRing ) -> ( X e. B <-> X e. ( Base ` ( (/) Mat R ) ) ) )
10		mat0dimbas0 20272	. . . . . . 7 |- ( R e. CRing -> ( Base ` ( (/) Mat R ) ) = { (/) } )
11	10	eleq2d 2687	. . . . . 6 |- ( R e. CRing -> ( X e. ( Base ` ( (/) Mat R ) ) <-> X e. { (/) } ) )
12	11	adantl 482	. . . . 5 |- ( ( N = (/) /\ R e. CRing ) -> ( X e. ( Base ` ( (/) Mat R ) ) <-> X e. { (/) } ) )
13	9, 12	bitrd 268	. . . 4 |- ( ( N = (/) /\ R e. CRing ) -> ( X e. B <-> X e. { (/) } ) )
14		cramer.v	. . . . . . . 8 |- V = ( ( Base ` R ) ^m N )
15	14	a1i 11	. . . . . . 7 |- ( ( N = (/) /\ R e. CRing ) -> V = ( ( Base ` R ) ^m N ) )
16		oveq2 6658	. . . . . . . 8 |- ( N = (/) -> ( ( Base ` R ) ^m N ) = ( ( Base ` R ) ^m (/) ) )
17	16	adantr 481	. . . . . . 7 |- ( ( N = (/) /\ R e. CRing ) -> ( ( Base ` R ) ^m N ) = ( ( Base ` R ) ^m (/) ) )
18		fvex 6201	. . . . . . . 8 |- ( Base ` R ) e. _V
19		map0e 7895	. . . . . . . 8 |- ( ( Base ` R ) e. _V -> ( ( Base ` R ) ^m (/) ) = 1o )
20	18, 19	mp1i 13	. . . . . . 7 |- ( ( N = (/) /\ R e. CRing ) -> ( ( Base ` R ) ^m (/) ) = 1o )
21	15, 17, 20	3eqtrd 2660	. . . . . 6 |- ( ( N = (/) /\ R e. CRing ) -> V = 1o )
22	21	eleq2d 2687	. . . . 5 |- ( ( N = (/) /\ R e. CRing ) -> ( Y e. V <-> Y e. 1o ) )
23		el1o 7579	. . . . 5 |- ( Y e. 1o <-> Y = (/) )
24	22, 23	syl6bb 276	. . . 4 |- ( ( N = (/) /\ R e. CRing ) -> ( Y e. V <-> Y = (/) ) )
25	13, 24	anbi12d 747	. . 3 |- ( ( N = (/) /\ R e. CRing ) -> ( ( X e. B /\ Y e. V ) <-> ( X e. { (/) } /\ Y = (/) ) ) )
26		elsni 4194	. . . 4 |- ( X e. { (/) } -> X = (/) )
27		mpteq1 4737	. . . . . . . . . 10 |- ( N = (/) -> ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) = ( i e. (/) |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) )
28		mpt0 6021	. . . . . . . . . 10 |- ( i e. (/) |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) = (/)
29	27, 28	syl6eq 2672	. . . . . . . . 9 |- ( N = (/) -> ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) = (/) )
30	29	eqeq2d 2632	. . . . . . . 8 |- ( N = (/) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) <-> Z = (/) ) )
31	30	ad2antrr 762	. . . . . . 7 |- ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) <-> Z = (/) ) )
32		simplrl 800	. . . . . . . . . 10 |- ( ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) /\ Z = (/) ) -> X = (/) )
33		simpr 477	. . . . . . . . . 10 |- ( ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) /\ Z = (/) ) -> Z = (/) )
34	32, 33	oveq12d 6668	. . . . . . . . 9 |- ( ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) /\ Z = (/) ) -> ( X .x. Z ) = ( (/) .x. (/) ) )
35		cramer.x	. . . . . . . . . . 11 |- .x. = ( R maVecMul <. N , N >. )
36	35	mavmul0 20358	. . . . . . . . . 10 |- ( ( N = (/) /\ R e. CRing ) -> ( (/) .x. (/) ) = (/) )
37	36	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) /\ Z = (/) ) -> ( (/) .x. (/) ) = (/) )
38		simpr 477	. . . . . . . . . . 11 |- ( ( X = (/) /\ Y = (/) ) -> Y = (/) )
39	38	eqcomd 2628	. . . . . . . . . 10 |- ( ( X = (/) /\ Y = (/) ) -> (/) = Y )
40	39	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) /\ Z = (/) ) -> (/) = Y )
41	34, 37, 40	3eqtrd 2660	. . . . . . . 8 |- ( ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) /\ Z = (/) ) -> ( X .x. Z ) = Y )
42	41	ex 450	. . . . . . 7 |- ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) -> ( Z = (/) -> ( X .x. Z ) = Y ) )
43	31, 42	sylbid 230	. . . . . 6 |- ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) -> ( X .x. Z ) = Y ) )
44	43	a1d 25	. . . . 5 |- ( ( ( N = (/) /\ R e. CRing ) /\ ( X = (/) /\ Y = (/) ) ) -> ( ( D ` X ) e. (Unit ` R ) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) -> ( X .x. Z ) = Y ) ) )
45	44	ex 450	. . . 4 |- ( ( N = (/) /\ R e. CRing ) -> ( ( X = (/) /\ Y = (/) ) -> ( ( D ` X ) e. (Unit ` R ) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) -> ( X .x. Z ) = Y ) ) ) )
46	26, 45	sylani 686	. . 3 |- ( ( N = (/) /\ R e. CRing ) -> ( ( X e. { (/) } /\ Y = (/) ) -> ( ( D ` X ) e. (Unit ` R ) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) -> ( X .x. Z ) = Y ) ) ) )
47	25, 46	sylbid 230	. 2 |- ( ( N = (/) /\ R e. CRing ) -> ( ( X e. B /\ Y e. V ) -> ( ( D ` X ) e. (Unit ` R ) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) -> ( X .x. Z ) = Y ) ) ) )
48	47	3imp 1256	1 |- ( ( ( N = (/) /\ R e. CRing ) /\ ( X e. B /\ Y e. V ) /\ ( D ` X ) e. (Unit ` R ) ) -> ( Z = ( i e. N |-> ( ( D ` ( ( X ( N matRepV R ) Y ) ` i ) ) ./ ( D ` X ) ) ) -> ( X .x. Z ) = Y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   Basecbs 15857   CRingccrg 18548  Unitcui 18639  /_rcdvr 18682   Mat cmat 20213   maVecMul cmvmul 20346   matRepV cmatrepV 20363   maDet cmdat 20390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mat 20214  df-mvmul 20347

This theorem is referenced by: (None)