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

Theorem mat1rhmval 19108
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 4220 . . . 4  |-  ( x  =  X  ->  <. O ,  x >.  =  <. O ,  X >. )
43sneqd 4044 . . 3  |-  ( x  =  X  ->  { <. O ,  x >. }  =  { <. O ,  X >. } )
54adantl 466 . 2  |-  ( ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  /\  x  =  X
)  ->  { <. O ,  x >. }  =  { <. O ,  X >. } )
6 simp3 998 . 2  |-  ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  X  e.  K )
7 snex 4697 . . 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 5961 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   {csn 4032   <.cop 4038    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   Basecbs 14644   Ringcrg 17325   Mat cmat 19036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602
This theorem is referenced by:  mat1rhmelval  19109  mat1rhmcl  19110  mat1mhm  19113
  Copyright terms: Public domain W3C validator