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Theorem mavmul0g 19225
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 6279 . . 3  |-  ( ( X  =  (/)  /\  Y  =  (/) )  ->  ( X  .x.  Y )  =  ( (/)  .x.  (/) ) )
2 mavmul0.t . . . 4  |-  .x.  =  ( R maVecMul  <. N ,  N >. )
32mavmul0 19224 . . 3  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  ( (/) 
.x.  (/) )  =  (/) )
41, 3sylan9eq 2515 . 2  |-  ( ( ( X  =  (/)  /\  Y  =  (/) )  /\  ( N  =  (/)  /\  R  e.  V ) )  -> 
( X  .x.  Y
)  =  (/) )
5 eqid 2454 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2454 . . . . . 6  |-  ( .r
`  R )  =  ( .r `  R
)
7 simpr 459 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  R  e.  V )
8 0fin 7740 . . . . . . . 8  |-  (/)  e.  Fin
9 eleq1 2526 . . . . . . . 8  |-  ( N  =  (/)  ->  ( N  e.  Fin  <->  (/)  e.  Fin ) )
108, 9mpbiri 233 . . . . . . 7  |-  ( N  =  (/)  ->  N  e. 
Fin )
1110adantr 463 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  N  e.  Fin )
122, 5, 6, 7, 11, 11mvmulfval 19214 . . . . 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 5194 . . . 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 4569 . . . . . . . . . 10  |-  (/)  e.  _V
15 eleq1 2526 . . . . . . . . . 10  |-  ( N  =  (/)  ->  ( N  e.  _V  <->  (/)  e.  _V ) )
1614, 15mpbiri 233 . . . . . . . . 9  |-  ( N  =  (/)  ->  N  e. 
_V )
17 mptexg 6117 . . . . . . . . 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 463 . . . . . . 7  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
k  e.  N  |->  ( R  gsumg  ( l  e.  N  |->  ( ( k i l ) ( .r
`  R ) ( j `  l ) ) ) ) )  e.  _V )
2019adantr 463 . . . . . 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 2875 . . . . 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 2454 . . . . . 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 6845 . . . . 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 5013 . . . . . . . . . 10  |-  ( N  =  (/)  ->  ( N  X.  N )  =  ( (/)  X.  (/) ) )
27 0xp 5069 . . . . . . . . . 10  |-  ( (/)  X.  (/) )  =  (/)
2826, 27syl6eq 2511 . . . . . . . . 9  |-  ( N  =  (/)  ->  ( N  X.  N )  =  (/) )
2928oveq2d 6286 . . . . . . . 8  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  ( N  X.  N
) )  =  ( ( Base `  R
)  ^m  (/) ) )
30 fvex 5858 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
31 map0e 7449 . . . . . . . . 9  |-  ( (
Base `  R )  e.  _V  ->  ( ( Base `  R )  ^m  (/) )  =  1o )
3230, 31mp1i 12 . . . . . . . 8  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  (/) )  =  1o )
3329, 32eqtrd 2495 . . . . . . 7  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  ( N  X.  N
) )  =  1o )
3433adantr 463 . . . . . 6  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  1o )
35 df1o2 7134 . . . . . 6  |-  1o  =  { (/) }
3634, 35syl6eq 2511 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( Base `  R )  ^m  ( N  X.  N
) )  =  { (/)
} )
37 oveq2 6278 . . . . . 6  |-  ( N  =  (/)  ->  ( (
Base `  R )  ^m  N )  =  ( ( Base `  R
)  ^m  (/) ) )
3830, 31mp1i 12 . . . . . . 7  |-  ( R  e.  V  ->  (
( Base `  R )  ^m  (/) )  =  1o )
3938, 35syl6eq 2511 . . . . . 6  |-  ( R  e.  V  ->  (
( Base `  R )  ^m  (/) )  =  { (/)
} )
4037, 39sylan9eq 2515 . . . . 5  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( Base `  R )  ^m  N )  =  { (/)
} )
4136, 40xpeq12d 5013 . . . 4  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  (
( ( Base `  R
)  ^m  ( N  X.  N ) )  X.  ( ( Base `  R
)  ^m  N )
)  =  ( {
(/) }  X.  { (/) } ) )
4213, 24, 413eqtrd 2499 . . 3  |-  ( ( N  =  (/)  /\  R  e.  V )  ->  dom  .x.  =  ( { (/) }  X.  { (/) } ) )
43 elsni 4041 . . . . 5  |-  ( X  e.  { (/) }  ->  X  =  (/) )
44 elsni 4041 . . . . 5  |-  ( Y  e.  { (/) }  ->  Y  =  (/) )
4543, 44anim12i 564 . . . 4  |-  ( ( X  e.  { (/) }  /\  Y  e.  { (/)
} )  ->  ( X  =  (/)  /\  Y  =  (/) ) )
4645con3i 135 . . 3  |-  ( -.  ( X  =  (/)  /\  Y  =  (/) )  ->  -.  ( X  e.  { (/)
}  /\  Y  e.  {
(/) } ) )
47 ndmovg 6431 . . 3  |-  ( ( dom  .x.  =  ( { (/) }  X.  { (/)
} )  /\  -.  ( X  e.  { (/) }  /\  Y  e.  { (/)
} ) )  -> 
( X  .x.  Y
)  =  (/) )
4842, 46, 47syl2anr 476 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   (/)c0 3783   {csn 4016   <.cop 4022    |-> cmpt 4497    X. cxp 4986   dom cdm 4988   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1oc1o 7115    ^m cmap 7412   Fincfn 7509   Basecbs 14719   .rcmulr 14788    gsumg cgsu 14933   maVecMul cmvmul 19212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-hom 14811  df-cco 14812  df-0g 14934  df-prds 14940  df-pws 14942  df-sra 18016  df-rgmod 18017  df-dsmm 18939  df-frlm 18954  df-mat 19080  df-mvmul 19213
This theorem is referenced by: (None)
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