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Theorem mavmul0g 18379
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 6115 . . 3  |-  ( ( X  =  (/)  /\  Y  =  (/) )  ->  ( X  .x.  Y )  =  ( (/)  .x.  (/) ) )
2 mavmul0.t . . . 4  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
32mavmul0 18378 . . 3  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  ( (/) 
.x.  (/) )  =  (/) )
41, 3sylan9eq 2495 . 2  |-  ( ( ( X  =  (/)  /\  Y  =  (/) )  /\  ( N  =  (/)  /\  R  e.  V ) )  -> 
( X  .x.  Y
)  =  (/) )
5 eqid 2443 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2443 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
7 simpr 461 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  R  e.  V )
8 0fin 7555 . . . . . . . 8  |-  (/)  e.  Fin
9 eleq1 2503 . . . . . . . 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 18368 . . . . 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 5057 . . . 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 4437 . . . . . . . . . 10  |-  (/)  e.  _V
15 eleq1 2503 . . . . . . . . . 10  |-  ( N  =  (/)  ->  ( N  e.  _V  <->  (/)  e.  _V ) )
1614, 15mpbiri 233 . . . . . . . . 9  |-  ( N  =  (/)  ->  N  e. 
_V )
17 mptexg 5962 . . . . . . . . 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 2823 . . . . 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 2443 . . . . . 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 6660 . . . . 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 4880 . . . . . . . . . 10  |-  ( N  =  (/)  ->  ( N  X.  N )  =  ( (/)  X.  (/) ) )
27 0xp 4932 . . . . . . . . . 10  |-  ( (/)  X.  (/) )  =  (/)
2826, 27syl6eq 2491 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( N  X.  N )  =  (/) )
2928oveq2d 6122 . . . . . . . 8  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  ( N  X.  N
) )  =  ( ( Base `  R
)  ^m  (/) ) )
30 fvex 5716 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
31 map0e 7265 . . . . . . . . 9  |-  ( (
Base `  R )  e.  _V  ->  ( ( Base `  R )  ^m  (/) )  =  1o )
3230, 31mp1i 12 . . . . . . . 8  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  (/) )  =  1o )
3329, 32eqtrd 2475 . . . . . . 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 6947 . . . . . 6  |-  1o  =  { (/) }
3634, 35syl6eq 2491 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  { (/)
} )
37 oveq2 6114 . . . . . 6  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  (/) ) )
3830, 31mp1i 12 . . . . . . 7  |-  ( R  e.  V  ->  (
( Base `  R )  ^m  (/) )  =  1o )
3938, 35syl6eq 2491 . . . . . 6  |-  ( R  e.  V  ->  (
( Base `  R )  ^m  (/) )  =  { (/)
} )
4037, 39sylan9eq 2495 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( Base `  R )  ^m  N )  =  { (/)
} )
4136, 40xpeq12d 4880 . . . 4  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( ( Base `  R
)  ^m  ( N  X.  N ) )  X.  ( ( Base `  R
)  ^m  N )
)  =  ( {
(/) }  X.  { (/) } ) )
4213, 24, 413eqtrd 2479 . . 3  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  dom  .x.  =  ( { (/) }  X.  { (/) } ) )
43 elsni 3917 . . . . 5  |-  ( X  e.  { (/) }  ->  X  =  (/) )
44 elsni 3917 . . . . 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 6261 . . 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 1369    e. wcel 1756   A.wral 2730   _Vcvv 2987   (/)c0 3652   {csn 3892   <.cop 3898    e. cmpt 4365    X. cxp 4853   dom cdm 4855   ` cfv 5433  (class class class)co 6106    e. cmpt2 6108   1oc1o 6928    ^m cmap 7229   Fincfn 7325   Basecbs 14189   .rcmulr 14254    gsumg cgsu 14394   maVecMul cmvmul 18366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-ot 3901  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-supp 6706  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fsupp 7636  df-sup 7706  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-fz 11453  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-sca 14269  df-vsca 14270  df-ip 14271  df-tset 14272  df-ple 14273  df-ds 14275  df-hom 14277  df-cco 14278  df-0g 14395  df-prds 14401  df-pws 14403  df-sra 17268  df-rgmod 17269  df-dsmm 18172  df-frlm 18187  df-mat 18297  df-mvmul 18367
This theorem is referenced by: (None)
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