MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mat1rhmval Structured version   Visualization version   Unicode version

Theorem mat1rhmval 19497
Description: The value of the ring homomorphism  F. (Contributed by AV, 22-Dec-2019.)
Hypotheses
Ref Expression
mat1rhmval.k  |-  K  =  ( Base `  R
)
mat1rhmval.a  |-  A  =  ( { E } Mat  R )
mat1rhmval.b  |-  B  =  ( Base `  A
)
mat1rhmval.o  |-  O  = 
<. E ,  E >.
mat1rhmval.f  |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )
Assertion
Ref Expression
mat1rhmval  |-  ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  ( F `  X )  =  { <. O ,  X >. } )
Distinct variable groups:    x, K    x, O    x, E    x, R    x, V    x, X
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem mat1rhmval
StepHypRef Expression
1 mat1rhmval.f . . 3  |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )
21a1i 11 . 2  |-  ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  F  =  ( x  e.  K  |->  { <. O ,  x >. } ) )
3 opeq2 4166 . . . 4  |-  ( x  =  X  ->  <. O ,  x >.  =  <. O ,  X >. )
43sneqd 3979 . . 3  |-  ( x  =  X  ->  { <. O ,  x >. }  =  { <. O ,  X >. } )
54adantl 468 . 2  |-  ( ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  /\  x  =  X
)  ->  { <. O ,  x >. }  =  { <. O ,  X >. } )
6 simp3 1009 . 2  |-  ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  X  e.  K )
7 snex 4640 . . 3  |-  { <. O ,  X >. }  e.  _V
87a1i 11 . 2  |-  ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  { <. O ,  X >. }  e.  _V )
92, 5, 6, 8fvmptd 5952 1  |-  ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  ( F `  X )  =  { <. O ,  X >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 984    = wceq 1443    e. wcel 1886   _Vcvv 3044   {csn 3967   <.cop 3973    |-> cmpt 4460   ` cfv 5581  (class class class)co 6288   Basecbs 15114   Ringcrg 17773   Mat cmat 19425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589
This theorem is referenced by:  mat1rhmelval  19498  mat1rhmcl  19499  mat1mhm  19502
  Copyright terms: Public domain W3C validator