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Theorem matbas0pc 19203
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 19202 . . . . 5  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
21reldmmpt2 6394 . . . 4  |-  Rel  dom Mat
32ovprc 6308 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
43fveq2d 5853 . 2  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  ( Base `  (/) ) )
5 base0 14882 . 2  |-  (/)  =  (
Base `  (/) )
64, 5syl6eqr 2461 1  |-  ( -.  ( N  e.  _V  /\  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 3059   (/)c0 3738   <.cop 3978   <.cotp 3980    X. cxp 4821   ` cfv 5569  (class class class)co 6278   Fincfn 7554   ndxcnx 14838   sSet csts 14839   Basecbs 14841   .rcmulr 14910   freeLMod cfrlm 19075   maMul cmmul 19177   Mat cmat 19201
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 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
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 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-slot 14845  df-base 14846  df-mat 19202
This theorem is referenced by:  marrepfval  19354  marepvfval  19359  submafval  19373  minmar1fval  19440
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