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Theorem matbas0 19096
Description: There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
Assertion
Ref Expression
matbas0  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )

Proof of Theorem matbas0
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mat 19094 . . . 4  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
21mpt2ndm0 6453 . . 3  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
32fveq2d 5809 . 2  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  (/) ) )
4 base0 14774 . 2  |-  (/)  =  (
Base `  (/) )
53, 4syl6eqr 2461 1  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   (/)c0 3737   <.cop 3977   <.cotp 3979    X. cxp 4940   ` cfv 5525  (class class class)co 6234   Fincfn 7474   ndxcnx 14730   sSet csts 14731   Basecbs 14733   .rcmulr 14802   freeLMod cfrlm 18967   maMul cmmul 19069   Mat cmat 19093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-slot 14737  df-base 14738  df-mat 19094
This theorem is referenced by:  nfimdetndef  19275  mdetfval1  19276
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