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Theorem matbas0pc 18285
Description: There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
Assertion
Ref Expression
matbas0pc  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )

Proof of Theorem matbas0pc
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mat 18281 . . . . 5  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
21reldmmpt2 6200 . . . 4  |-  Rel  dom Mat
32ovprc 6117 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
43fveq2d 5694 . 2  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  (/) ) )
5 base0 14212 . 2  |-  (/)  =  (
Base `  (/) )
64, 5syl6eqr 2492 1  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971   (/)c0 3636   <.cop 3882   <.cotp 3884    X. cxp 4837   ` cfv 5417  (class class class)co 6090   Fincfn 7309   ndxcnx 14170   sSet csts 14171   Basecbs 14173   .rcmulr 14238   freeLMod cfrlm 18170   maMul cmmul 18278   Mat cmat 18279
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-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-slot 14177  df-base 14178  df-mat 18281
This theorem is referenced by:  marrepfval  18370  marepvfval  18375  submafval  18389  minmar1fval  18451
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