Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnghmresel Structured version   Unicode version

Theorem rnghmresel 39308
Description: An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020.)
Hypothesis
Ref Expression
rnghmresel.h  |-  ( ph  ->  H  =  ( RngHomo  |`  ( B  X.  B ) ) )
Assertion
Ref Expression
rnghmresel  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B )  /\  F  e.  ( X H Y ) )  ->  F  e.  ( X RngHomo  Y )
)

Proof of Theorem rnghmresel
StepHypRef Expression
1 rnghmresel.h . . . . . 6  |-  ( ph  ->  H  =  ( RngHomo  |`  ( B  X.  B ) ) )
21adantr 467 . . . . 5  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  H  =  ( RngHomo  |`  ( B  X.  B ) ) )
32oveqd 6320 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X H Y )  =  ( X ( RngHomo  |`  ( B  X.  B ) ) Y ) )
4 ovres 6448 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X ( RngHomo  |`  ( B  X.  B ) ) Y )  =  ( X RngHomo  Y ) )
54adantl 468 . . . 4  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X ( RngHomo  |`  ( B  X.  B ) ) Y )  =  ( X RngHomo  Y ) )
63, 5eqtrd 2464 . . 3  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( X H Y )  =  ( X RngHomo  Y ) )
76eleq2d 2493 . 2  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B ) )  -> 
( F  e.  ( X H Y )  <-> 
F  e.  ( X RngHomo  Y ) ) )
87biimp3a 1365 1  |-  ( (
ph  /\  ( X  e.  B  /\  Y  e.  B )  /\  F  e.  ( X H Y ) )  ->  F  e.  ( X RngHomo  Y )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    X. cxp 4849    |` cres 4853  (class class class)co 6303   RngHomo crngh 39227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-xp 4857  df-res 4863  df-iota 5563  df-fv 5607  df-ov 6306
This theorem is referenced by:  rnghmsubcsetclem2  39320
  Copyright terms: Public domain W3C validator