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Theorem matbas0 18674
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 18672 . . . 4  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
21mpt2ndm0 6493 . . 3  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
32fveq2d 5863 . 2  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  (/) ) )
4 base0 14520 . 2  |-  (/)  =  (
Base `  (/) )
53, 4syl6eqr 2521 1  |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108   (/)c0 3780   <.cop 4028   <.cotp 4030    X. cxp 4992   ` cfv 5581  (class class class)co 6277   Fincfn 7508   ndxcnx 14478   sSet csts 14479   Basecbs 14481   .rcmulr 14547   freeLMod cfrlm 18539   maMul cmmul 18647   Mat cmat 18671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-slot 14485  df-base 14486  df-mat 18672
This theorem is referenced by:  nfimdetndef  18853  mdetfval1  18854
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