mavmulsolcl - Metamath Proof Explorer

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Theorem mavmulsolcl 20357

Description: Every solution of the equation

for a matrix

and a vector

is a vector. (Contributed by AV, 27-Feb-2019.)

**Hypotheses**
Ref	Expression
mavmuldm.b
mavmuldm.c
mavmuldm.d
mavmuldm.t	maVecMul
mavmulsolcl.e

**Assertion**
Ref	Expression
mavmulsolcl	$/\$ $/\$ $/\$ $/\$

**Proof of Theorem mavmulsolcl**
Step	Hyp	Ref	Expression
1		2a1 28	. 2 $/\$ $/\$ $/\$ $/\$
2		simpl 473	. . . . . . . . 9 $/\$
3	2	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
4		simpl1 1064	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
5		simpl2 1065	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
6	3, 4, 5	3jca 1242	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7	6	adantl 482	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		mavmuldm.b	. . . . . . 7
9		mavmuldm.c	. . . . . . 7
10		mavmuldm.d	. . . . . . 7
11		mavmuldm.t	. . . . . . 7 maVecMul
12	8, 9, 10, 11	mavmuldm 20356	. . . . . 6 $/\$ $/\$
13	7, 12	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
14		simpl 473	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
15	14	intnand 962	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		ndmovg 6817	. . . . 5 $/\$ $/\$
17	13, 15, 16	syl2anc 693	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
18		eqeq1 2626	. . . . . 6
19		elmapi 7879	. . . . . . . . . . . . . 14
20		f0dom0 6089	. . . . . . . . . . . . . . . . . . . 20
21	20	biimprd 238	. . . . . . . . . . . . . . . . . . 19
22	21	necon3d 2815	. . . . . . . . . . . . . . . . . 18
23	22	com12 32	. . . . . . . . . . . . . . . . 17
24	23	3ad2ant3 1084	. . . . . . . . . . . . . . . 16 $/\$ $/\$
25	24	com12 32	. . . . . . . . . . . . . . 15 $/\$ $/\$
26	25	a1d 25	. . . . . . . . . . . . . 14 $/\$ $/\$
27	19, 26	syl 17	. . . . . . . . . . . . 13 $/\$ $/\$
28		mavmulsolcl.e	. . . . . . . . . . . . 13
29	27, 28	eleq2s 2719	. . . . . . . . . . . 12 $/\$ $/\$
30	29	impcom 446	. . . . . . . . . . 11 $/\$ $/\$ $/\$
31	30	impcom 446	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
32		eqneqall 2805	. . . . . . . . . 10
33	31, 32	syl5com 31	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
34	33	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
35	34	com12 32	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	eqcoms 2630	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
37	18, 36	syl6bi 243	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
38	37	com23 86	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
39	17, 38	mpcom 38	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
40	39	ex 450	. 2 $/\$ $/\$ $/\$ $/\$
41	1, 40	pm2.61i 176	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

c0 3915

cop 4183

cxp 5112

cdm 5114

wf 5884

cfv 5888 (class class class)co 6650

cmap 7857

cfn 7955

cbs 15857 maVecMul cmvmul 20346

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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-mvmul 20347

This theorem is referenced by: slesolvec 20485 cramerimplem2 20490

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