Theorem cramerimplem2 20490

Description: Lemma 2 for cramerimp 20492: The matrix of a system of linear equations multiplied with the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)

Hypotheses

Ref	Expression
cramerimp.a	|- A = ( N Mat R )
cramerimp.b	|- B = ( Base ` A )
cramerimp.v	|- V = ( ( Base ` R ) ^m N )
cramerimp.e	|- E = ( ( ( 1r ` A ) ( N matRepV R ) Z ) ` I )
cramerimp.h	|- H = ( ( X ( N matRepV R ) Y ) ` I )
cramerimp.x	|- .x. = ( R maVecMul <. N , N >. )
cramerimp.m	|- .X. = ( R maMul <. N , N , N >. )

Assertion

Ref	Expression
cramerimplem2	|- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( X .X. E ) = H )

Proof of Theorem cramerimplem2

Dummy variables l i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cramerimp.m	. . 3 |- .X. = ( R maMul <. N , N , N >. )
2		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
3		eqid 2622	. . 3 |- ( .r ` R ) = ( .r ` R )
4		simpl 473	. . . 4 |- ( ( R e. CRing /\ I e. N ) -> R e. CRing )
5	4	3ad2ant1 1082	. . 3 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> R e. CRing )
6		cramerimp.a	. . . . . . 7 |- A = ( N Mat R )
7		cramerimp.b	. . . . . . 7 |- B = ( Base ` A )
8	6, 7	matrcl 20218	. . . . . 6 |- ( X e. B -> ( N e. Fin /\ R e. _V ) )
9	8	simpld 475	. . . . 5 |- ( X e. B -> N e. Fin )
10	9	adantr 481	. . . 4 |- ( ( X e. B /\ Y e. V ) -> N e. Fin )
11	10	3ad2ant2 1083	. . 3 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> N e. Fin )
12	9	anim2i 593	. . . . . . . . . . . . . . 15 |- ( ( R e. CRing /\ X e. B ) -> ( R e. CRing /\ N e. Fin ) )
13	12	ancomd 467	. . . . . . . . . . . . . 14 |- ( ( R e. CRing /\ X e. B ) -> ( N e. Fin /\ R e. CRing ) )
14	6, 2	matbas2 20227	. . . . . . . . . . . . . 14 |- ( ( N e. Fin /\ R e. CRing ) -> ( ( Base ` R ) ^m ( N X. N ) ) = ( Base ` A ) )
15	13, 14	syl 17	. . . . . . . . . . . . 13 |- ( ( R e. CRing /\ X e. B ) -> ( ( Base ` R ) ^m ( N X. N ) ) = ( Base ` A ) )
16	15, 7	syl6reqr 2675	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ X e. B ) -> B = ( ( Base ` R ) ^m ( N X. N ) ) )
17	16	eleq2d 2687	. . . . . . . . . . 11 |- ( ( R e. CRing /\ X e. B ) -> ( X e. B <-> X e. ( ( Base ` R ) ^m ( N X. N ) ) ) )
18	17	biimpd 219	. . . . . . . . . 10 |- ( ( R e. CRing /\ X e. B ) -> ( X e. B -> X e. ( ( Base ` R ) ^m ( N X. N ) ) ) )
19	18	ex 450	. . . . . . . . 9 |- ( R e. CRing -> ( X e. B -> ( X e. B -> X e. ( ( Base ` R ) ^m ( N X. N ) ) ) ) )
20	19	adantr 481	. . . . . . . 8 |- ( ( R e. CRing /\ I e. N ) -> ( X e. B -> ( X e. B -> X e. ( ( Base ` R ) ^m ( N X. N ) ) ) ) )
21	20	com12 32	. . . . . . 7 |- ( X e. B -> ( ( R e. CRing /\ I e. N ) -> ( X e. B -> X e. ( ( Base ` R ) ^m ( N X. N ) ) ) ) )
22	21	pm2.43a 54	. . . . . 6 |- ( X e. B -> ( ( R e. CRing /\ I e. N ) -> X e. ( ( Base ` R ) ^m ( N X. N ) ) ) )
23	22	adantr 481	. . . . 5 |- ( ( X e. B /\ Y e. V ) -> ( ( R e. CRing /\ I e. N ) -> X e. ( ( Base ` R ) ^m ( N X. N ) ) ) )
24	23	impcom 446	. . . 4 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) ) -> X e. ( ( Base ` R ) ^m ( N X. N ) ) )
25	24	3adant3 1081	. . 3 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> X e. ( ( Base ` R ) ^m ( N X. N ) ) )
26		crngring 18558	. . . . . . . 8 |- ( R e. CRing -> R e. Ring )
27	26	adantr 481	. . . . . . 7 |- ( ( R e. CRing /\ I e. N ) -> R e. Ring )
28	27, 10	anim12i 590	. . . . . 6 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) ) -> ( R e. Ring /\ N e. Fin ) )
29	28	3adant3 1081	. . . . 5 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( R e. Ring /\ N e. Fin ) )
30		ne0i 3921	. . . . . . . . 9 |- ( I e. N -> N =/= (/) )
31	30	adantl 482	. . . . . . . 8 |- ( ( R e. CRing /\ I e. N ) -> N =/= (/) )
32	31	3ad2ant1 1082	. . . . . . 7 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> N =/= (/) )
33	11, 11, 32	3jca 1242	. . . . . 6 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( N e. Fin /\ N e. Fin /\ N =/= (/) ) )
34		cramerimp.v	. . . . . . . . . . 11 |- V = ( ( Base ` R ) ^m N )
35	34	eleq2i 2693	. . . . . . . . . 10 |- ( Y e. V <-> Y e. ( ( Base ` R ) ^m N ) )
36	35	biimpi 206	. . . . . . . . 9 |- ( Y e. V -> Y e. ( ( Base ` R ) ^m N ) )
37	36	adantl 482	. . . . . . . 8 |- ( ( X e. B /\ Y e. V ) -> Y e. ( ( Base ` R ) ^m N ) )
38	4, 37	anim12i 590	. . . . . . 7 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) ) -> ( R e. CRing /\ Y e. ( ( Base ` R ) ^m N ) ) )
39	38	3adant3 1081	. . . . . 6 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( R e. CRing /\ Y e. ( ( Base ` R ) ^m N ) ) )
40		simp3 1063	. . . . . 6 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( X .x. Z ) = Y )
41		eqid 2622	. . . . . . . 8 |- ( ( Base ` R ) ^m ( N X. N ) ) = ( ( Base ` R ) ^m ( N X. N ) )
42		cramerimp.x	. . . . . . . 8 |- .x. = ( R maVecMul <. N , N >. )
43		eqid 2622	. . . . . . . 8 |- ( ( Base ` R ) ^m N ) = ( ( Base ` R ) ^m N )
44	2, 41, 34, 42, 43	mavmulsolcl 20357	. . . . . . 7 |- ( ( ( N e. Fin /\ N e. Fin /\ N =/= (/) ) /\ ( R e. CRing /\ Y e. ( ( Base ` R ) ^m N ) ) ) -> ( ( X .x. Z ) = Y -> Z e. V ) )
45	44	imp 445	. . . . . 6 |- ( ( ( ( N e. Fin /\ N e. Fin /\ N =/= (/) ) /\ ( R e. CRing /\ Y e. ( ( Base ` R ) ^m N ) ) ) /\ ( X .x. Z ) = Y ) -> Z e. V )
46	33, 39, 40, 45	syl21anc 1325	. . . . 5 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> Z e. V )
47		simpr 477	. . . . . 6 |- ( ( R e. CRing /\ I e. N ) -> I e. N )
48	47	3ad2ant1 1082	. . . . 5 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> I e. N )
49		cramerimp.e	. . . . . 6 |- E = ( ( ( 1r ` A ) ( N matRepV R ) Z ) ` I )
50		eqid 2622	. . . . . . 7 |- ( 1r ` A ) = ( 1r ` A )
51	6, 7, 34, 50	ma1repvcl 20376	. . . . . 6 |- ( ( ( R e. Ring /\ N e. Fin ) /\ ( Z e. V /\ I e. N ) ) -> ( ( ( 1r ` A ) ( N matRepV R ) Z ) ` I ) e. B )
52	49, 51	syl5eqel 2705	. . . . 5 |- ( ( ( R e. Ring /\ N e. Fin ) /\ ( Z e. V /\ I e. N ) ) -> E e. B )
53	29, 46, 48, 52	syl12anc 1324	. . . 4 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> E e. B )
54	16	eqcomd 2628	. . . . . 6 |- ( ( R e. CRing /\ X e. B ) -> ( ( Base ` R ) ^m ( N X. N ) ) = B )
55	54	ad2ant2r 783	. . . . 5 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) ) -> ( ( Base ` R ) ^m ( N X. N ) ) = B )
56	55	3adant3 1081	. . . 4 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( ( Base ` R ) ^m ( N X. N ) ) = B )
57	53, 56	eleqtrrd 2704	. . 3 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> E e. ( ( Base ` R ) ^m ( N X. N ) ) )
58	1, 2, 3, 5, 11, 11, 11, 25, 57	mamuval 20192	. 2 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( X .X. E ) = ( i e. N , j e. N |-> ( R gsumg ( l e. N |-> ( ( i X l ) ( .r ` R ) ( l E j ) ) ) ) ) )
59	27	3ad2ant1 1082	. . . . 5 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> R e. Ring )
60	59	3ad2ant1 1082	. . . 4 |- ( ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) /\ i e. N /\ j e. N ) -> R e. Ring )
61		simpl 473	. . . . . . 7 |- ( ( X e. B /\ Y e. V ) -> X e. B )
62	61	3ad2ant2 1083	. . . . . 6 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> X e. B )
63	62, 46, 48	3jca 1242	. . . . 5 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( X e. B /\ Z e. V /\ I e. N ) )
64	63	3ad2ant1 1082	. . . 4 |- ( ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) /\ i e. N /\ j e. N ) -> ( X e. B /\ Z e. V /\ I e. N ) )
65		simp2 1062	. . . 4 |- ( ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) /\ i e. N /\ j e. N ) -> i e. N )
66		simp3 1063	. . . 4 |- ( ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) /\ i e. N /\ j e. N ) -> j e. N )
67	40	3ad2ant1 1082	. . . 4 |- ( ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) /\ i e. N /\ j e. N ) -> ( X .x. Z ) = Y )
68		eqid 2622	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
69	6, 7, 34, 50, 68, 49, 42	mulmarep1gsum2 20380	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Z e. V /\ I e. N ) /\ ( i e. N /\ j e. N /\ ( X .x. Z ) = Y ) ) -> ( R gsumg ( l e. N |-> ( ( i X l ) ( .r ` R ) ( l E j ) ) ) ) = if ( j = I , ( Y ` i ) , ( i X j ) ) )
70	60, 64, 65, 66, 67, 69	syl113anc 1338	. . 3 |- ( ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) /\ i e. N /\ j e. N ) -> ( R gsumg ( l e. N |-> ( ( i X l ) ( .r ` R ) ( l E j ) ) ) ) = if ( j = I , ( Y ` i ) , ( i X j ) ) )
71	70	mpt2eq3dva 6719	. 2 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( i e. N , j e. N |-> ( R gsumg ( l e. N |-> ( ( i X l ) ( .r ` R ) ( l E j ) ) ) ) ) = ( i e. N , j e. N |-> if ( j = I , ( Y ` i ) , ( i X j ) ) ) )
72		cramerimp.h	. . 3 |- H = ( ( X ( N matRepV R ) Y ) ` I )
73		simpr 477	. . . . 5 |- ( ( X e. B /\ Y e. V ) -> Y e. V )
74	73	3ad2ant2 1083	. . . 4 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> Y e. V )
75		eqid 2622	. . . . 5 |- ( N matRepV R ) = ( N matRepV R )
76	6, 7, 75, 34	marepvval 20373	. . . 4 |- ( ( X e. B /\ Y e. V /\ I e. N ) -> ( ( X ( N matRepV R ) Y ) ` I ) = ( i e. N , j e. N |-> if ( j = I , ( Y ` i ) , ( i X j ) ) ) )
77	62, 74, 48, 76	syl3anc 1326	. . 3 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( ( X ( N matRepV R ) Y ) ` I ) = ( i e. N , j e. N |-> if ( j = I , ( Y ` i ) , ( i X j ) ) ) )
78	72, 77	syl5req 2669	. 2 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( i e. N , j e. N |-> if ( j = I , ( Y ` i ) , ( i X j ) ) ) = H )
79	58, 71, 78	3eqtrd 2660	1 |- ( ( ( R e. CRing /\ I e. N ) /\ ( X e. B /\ Y e. V ) /\ ( X .x. Z ) = Y ) -> ( X .X. E ) = H )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   ifcif 4086   <.cop 4183   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Fincfn 7955   Basecbs 15857   .rcmulr 15942   0gc0g 16100   gsum_g cgsu 16101   1rcur 18501   Ringcrg 18547   CRingccrg 18548   maMul cmmul 20189   Mat cmat 20213   maVecMul cmvmul 20346   matRepV cmatrepV 20363

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-inf2 8538  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-fal 1489  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-rmo 2920  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-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  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-oi 8415  df-card 8765  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-fzo 12466  df-seq 12802  df-hash 13118  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-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-mamu 20190  df-mat 20214  df-mvmul 20347  df-marepv 20365

This theorem is referenced by:  cramerimplem3  20491