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Theorem mavmul0g 18924
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.
StepHypRef Expression
1 oveq12 6304 . . 3  |-  ( ( X  =  (/)  /\  Y  =  (/) )  ->  ( X  .x.  Y )  =  ( (/)  .x.  (/) ) )
2 mavmul0.t . . . 4  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
32mavmul0 18923 . . 3  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  ( (/) 
.x.  (/) )  =  (/) )
41, 3sylan9eq 2528 . 2  |-  ( ( ( X  =  (/)  /\  Y  =  (/) )  /\  ( N  =  (/)  /\  R  e.  V ) )  -> 
( X  .x.  Y
)  =  (/) )
5 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2467 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
7 simpr 461 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  R  e.  V )
8 0fin 7759 . . . . . . . 8  |-  (/)  e.  Fin
9 eleq1 2539 . . . . . . . 8  |-  ( N  =  (/)  ->  ( N  e.  Fin  <->  (/)  e.  Fin ) )
108, 9mpbiri 233 . . . . . . 7  |-  ( N  =  (/)  ->  N  e. 
Fin )
1110adantr 465 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  N  e.  Fin )
122, 5, 6, 7, 11, 11mvmulfval 18913 . . . . 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 ) ) ) ) ) ) )
1312dmeqd 5211 . . . 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 4583 . . . . . . . . . 10  |-  (/)  e.  _V
15 eleq1 2539 . . . . . . . . . 10  |-  ( N  =  (/)  ->  ( N  e.  _V  <->  (/)  e.  _V ) )
1614, 15mpbiri 233 . . . . . . . . 9  |-  ( N  =  (/)  ->  N  e. 
_V )
17 mptexg 6141 . . . . . . . . 9  |-  ( N  e.  _V  ->  (
k  e.  N  |->  ( R  gsumg  ( l  e.  N  |->  ( ( k i l ) ( .r
`  R ) ( j `  l ) ) ) ) )  e.  _V )
1816, 17syl 16 . . . . . . . 8  |-  ( N  =  (/)  ->  ( k  e.  N  |->  ( R 
gsumg  ( l  e.  N  |->  ( ( k i l ) ( .r
`  R ) ( j `  l ) ) ) ) )  e.  _V )
1918adantr 465 . . . . . . 7  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
k  e.  N  |->  ( R  gsumg  ( l  e.  N  |->  ( ( k i l ) ( .r
`  R ) ( j `  l ) ) ) ) )  e.  _V )
2019adantr 465 . . . . . 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 )
2120ralrimivva 2888 . . . . 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 2467 . . . . . 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 ) ) ) ) ) )
2322dmmpt2ga 6867 . . . . 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 )
) )
2421, 23syl 16 . . . 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  =  (/) )
2625, 25xpeq12d 5030 . . . . . . . . . 10  |-  ( N  =  (/)  ->  ( N  X.  N )  =  ( (/)  X.  (/) ) )
27 0xp 5086 . . . . . . . . . 10  |-  ( (/)  X.  (/) )  =  (/)
2826, 27syl6eq 2524 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( N  X.  N )  =  (/) )
2928oveq2d 6311 . . . . . . . 8  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  ( N  X.  N
) )  =  ( ( Base `  R
)  ^m  (/) ) )
30 fvex 5882 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
31 map0e 7468 . . . . . . . . 9  |-  ( (
Base `  R )  e.  _V  ->  ( ( Base `  R )  ^m  (/) )  =  1o )
3230, 31mp1i 12 . . . . . . . 8  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  (/) )  =  1o )
3329, 32eqtrd 2508 . . . . . . 7  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  ( N  X.  N
) )  =  1o )
3433adantr 465 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  1o )
35 df1o2 7154 . . . . . 6  |-  1o  =  { (/) }
3634, 35syl6eq 2524 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  { (/)
} )
37 oveq2 6303 . . . . . 6  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  (/) ) )
3830, 31mp1i 12 . . . . . . 7  |-  ( R  e.  V  ->  (
( Base `  R )  ^m  (/) )  =  1o )
3938, 35syl6eq 2524 . . . . . 6  |-  ( R  e.  V  ->  (
( Base `  R )  ^m  (/) )  =  { (/)
} )
4037, 39sylan9eq 2528 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( Base `  R )  ^m  N )  =  { (/)
} )
4136, 40xpeq12d 5030 . . . 4  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( ( Base `  R
)  ^m  ( N  X.  N ) )  X.  ( ( Base `  R
)  ^m  N )
)  =  ( {
(/) }  X.  { (/) } ) )
4213, 24, 413eqtrd 2512 . . 3  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  dom  .x.  =  ( { (/) }  X.  { (/) } ) )
43 elsni 4058 . . . . 5  |-  ( X  e.  { (/) }  ->  X  =  (/) )
44 elsni 4058 . . . . 5  |-  ( Y  e.  { (/) }  ->  Y  =  (/) )
4543, 44anim12i 566 . . . 4  |-  ( ( X  e.  { (/) }  /\  Y  e.  { (/)
} )  ->  ( X  =  (/)  /\  Y  =  (/) ) )
4645con3i 135 . . 3  |-  ( -.  ( X  =  (/)  /\  Y  =  (/) )  ->  -.  ( X  e.  { (/)
}  /\  Y  e.  {
(/) } ) )
47 ndmovg 6453 . . 3  |-  ( ( dom  .x.  =  ( { (/) }  X.  { (/)
} )  /\  -.  ( X  e.  { (/) }  /\  Y  e.  { (/)
} ) )  -> 
( X  .x.  Y
)  =  (/) )
4842, 46, 47syl2anr 478 . 2  |-  ( ( -.  ( X  =  (/)  /\  Y  =  (/) )  /\  ( N  =  (/)  /\  R  e.  V
) )  ->  ( X  .x.  Y )  =  (/) )
494, 48pm2.61ian 788 1  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  ( X  .x.  Y )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   (/)c0 3790   {csn 4033   <.cop 4039    |-> cmpt 4511    X. cxp 5003   dom cdm 5005   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1oc1o 7135    ^m cmap 7432   Fincfn 7528   Basecbs 14507   .rcmulr 14573    gsumg cgsu 14713   maVecMul cmvmul 18911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-hom 14596  df-cco 14597  df-0g 14714  df-prds 14720  df-pws 14722  df-sra 17689  df-rgmod 17690  df-dsmm 18632  df-frlm 18647  df-mat 18779  df-mvmul 18912
This theorem is referenced by: (None)
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