Theorem mavmul0g 20359

Description: The result of the 0-dimensional multiplication of a matrix with a vector is always the empty set. (Contributed by AV, 1-Mar-2019.)

Hypothesis

Ref	Expression
mavmul0.t	|- .x. = ( R maVecMul <. N , N >. )

Assertion

Ref	Expression
mavmul0g	|- ( ( N = (/) /\ R e. V ) -> ( X .x. Y ) = (/) )

Proof of Theorem mavmul0g

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

Step	Hyp	Ref	Expression
1		oveq12 6659	. . 3 |- ( ( X = (/) /\ Y = (/) ) -> ( X .x. Y ) = ( (/) .x. (/) ) )
2		mavmul0.t	. . . 4 |- .x. = ( R maVecMul <. N , N >. )
3	2	mavmul0 20358	. . 3 |- ( ( N = (/) /\ R e. V ) -> ( (/) .x. (/) ) = (/) )
4	1, 3	sylan9eq 2676	. 2 |- ( ( ( X = (/) /\ Y = (/) ) /\ ( N = (/) /\ R e. V ) ) -> ( X .x. Y ) = (/) )
5		eqid 2622	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
6		eqid 2622	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
7		simpr 477	. . . . . 6 |- ( ( N = (/) /\ R e. V ) -> R e. V )
8		0fin 8188	. . . . . . . 8 |- (/) e. Fin
9		eleq1 2689	. . . . . . . 8 |- ( N = (/) -> ( N e. Fin <-> (/) e. Fin ) )
10	8, 9	mpbiri 248	. . . . . . 7 |- ( N = (/) -> N e. Fin )
11	10	adantr 481	. . . . . 6 |- ( ( N = (/) /\ R e. V ) -> N e. Fin )
12	2, 5, 6, 7, 11, 11	mvmulfval 20348	. . . . 5 |- ( ( N = (/) /\ R e. V ) -> .x. = ( i e. ( ( Base ` R ) ^m ( N X. N ) ) , j e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) ) )
13	12	dmeqd 5326	. . . 4 |- ( ( N = (/) /\ R e. V ) -> dom .x. = dom ( i e. ( ( Base ` R ) ^m ( N X. N ) ) , j e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) ) )
14		0ex 4790	. . . . . . . . . 10 |- (/) e. _V
15		eleq1 2689	. . . . . . . . . 10 |- ( N = (/) -> ( N e. _V <-> (/) e. _V ) )
16	14, 15	mpbiri 248	. . . . . . . . 9 |- ( N = (/) -> N e. _V )
17		mptexg 6484	. . . . . . . . 9 |- ( N e. _V -> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) e. _V )
18	16, 17	syl 17	. . . . . . . 8 |- ( N = (/) -> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) e. _V )
19	18	adantr 481	. . . . . . 7 |- ( ( N = (/) /\ R e. V ) -> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) e. _V )
20	19	adantr 481	. . . . . 6 |- ( ( ( N = (/) /\ R e. V ) /\ ( i e. ( ( Base ` R ) ^m ( N X. N ) ) /\ j e. ( ( Base ` R ) ^m N ) ) ) -> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) e. _V )
21	20	ralrimivva 2971	. . . . 5 |- ( ( N = (/) /\ R e. V ) -> A. i e. ( ( Base ` R ) ^m ( N X. N ) ) A. j e. ( ( Base ` R ) ^m N ) ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) e. _V )
22		eqid 2622	. . . . . 6 |- ( i e. ( ( Base ` R ) ^m ( N X. N ) ) , j e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) ) = ( i e. ( ( Base ` R ) ^m ( N X. N ) ) , j e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) )
23	22	dmmpt2ga 7242	. . . . 5 |- ( A. i e. ( ( Base ` R ) ^m ( N X. N ) ) A. j e. ( ( Base ` R ) ^m N ) ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) e. _V -> dom ( i e. ( ( Base ` R ) ^m ( N X. N ) ) , j e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) ) = ( ( ( Base ` R ) ^m ( N X. N ) ) X. ( ( Base ` R ) ^m N ) ) )
24	21, 23	syl 17	. . . 4 |- ( ( N = (/) /\ R e. V ) -> dom ( i e. ( ( Base ` R ) ^m ( N X. N ) ) , j e. ( ( Base ` R ) ^m N ) |-> ( k e. N |-> ( R gsumg ( l e. N |-> ( ( k i l ) ( .r ` R ) ( j ` l ) ) ) ) ) ) = ( ( ( Base ` R ) ^m ( N X. N ) ) X. ( ( Base ` R ) ^m N ) ) )
25		id 22	. . . . . . . . . . 11 |- ( N = (/) -> N = (/) )
26	25, 25	xpeq12d 5140	. . . . . . . . . 10 |- ( N = (/) -> ( N X. N ) = ( (/) X. (/) ) )
27		0xp 5199	. . . . . . . . . 10 |- ( (/) X. (/) ) = (/)
28	26, 27	syl6eq 2672	. . . . . . . . 9 |- ( N = (/) -> ( N X. N ) = (/) )
29	28	oveq2d 6666	. . . . . . . 8 |- ( N = (/) -> ( ( Base ` R ) ^m ( N X. N ) ) = ( ( Base ` R ) ^m (/) ) )
30		fvex 6201	. . . . . . . . 9 |- ( Base ` R ) e. _V
31		map0e 7895	. . . . . . . . 9 |- ( ( Base ` R ) e. _V -> ( ( Base ` R ) ^m (/) ) = 1o )
32	30, 31	mp1i 13	. . . . . . . 8 |- ( N = (/) -> ( ( Base ` R ) ^m (/) ) = 1o )
33	29, 32	eqtrd 2656	. . . . . . 7 |- ( N = (/) -> ( ( Base ` R ) ^m ( N X. N ) ) = 1o )
34	33	adantr 481	. . . . . 6 |- ( ( N = (/) /\ R e. V ) -> ( ( Base ` R ) ^m ( N X. N ) ) = 1o )
35		df1o2 7572	. . . . . 6 |- 1o = { (/) }
36	34, 35	syl6eq 2672	. . . . 5 |- ( ( N = (/) /\ R e. V ) -> ( ( Base ` R ) ^m ( N X. N ) ) = { (/) } )
37		oveq2 6658	. . . . . 6 |- ( N = (/) -> ( ( Base ` R ) ^m N ) = ( ( Base ` R ) ^m (/) ) )
38	30, 31	mp1i 13	. . . . . . 7 |- ( R e. V -> ( ( Base ` R ) ^m (/) ) = 1o )
39	38, 35	syl6eq 2672	. . . . . 6 |- ( R e. V -> ( ( Base ` R ) ^m (/) ) = { (/) } )
40	37, 39	sylan9eq 2676	. . . . 5 |- ( ( N = (/) /\ R e. V ) -> ( ( Base ` R ) ^m N ) = { (/) } )
41	36, 40	xpeq12d 5140	. . . 4 |- ( ( N = (/) /\ R e. V ) -> ( ( ( Base ` R ) ^m ( N X. N ) ) X. ( ( Base ` R ) ^m N ) ) = ( { (/) } X. { (/) } ) )
42	13, 24, 41	3eqtrd 2660	. . 3 |- ( ( N = (/) /\ R e. V ) -> dom .x. = ( { (/) } X. { (/) } ) )
43		elsni 4194	. . . . 5 |- ( X e. { (/) } -> X = (/) )
44		elsni 4194	. . . . 5 |- ( Y e. { (/) } -> Y = (/) )
45	43, 44	anim12i 590	. . . 4 |- ( ( X e. { (/) } /\ Y e. { (/) } ) -> ( X = (/) /\ Y = (/) ) )
46	45	con3i 150	. . 3 |- ( -. ( X = (/) /\ Y = (/) ) -> -. ( X e. { (/) } /\ Y e. { (/) } ) )
47		ndmovg 6817	. . 3 |- ( ( dom .x. = ( { (/) } X. { (/) } ) /\ -. ( X e. { (/) } /\ Y e. { (/) } ) ) -> ( X .x. Y ) = (/) )
48	42, 46, 47	syl2anr 495	. 2 |- ( ( -. ( X = (/) /\ Y = (/) ) /\ ( N = (/) /\ R e. V ) ) -> ( X .x. Y ) = (/) )
49	4, 48	pm2.61ian 831	1 |- ( ( N = (/) /\ R e. V ) -> ( X .x. Y ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   ^m cmap 7857   Fincfn 7955   Basecbs 15857   .rcmulr 15942   gsum_g cgsu 16101   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  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)