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

Theorem submcl 15856
Description: Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
submcl.p  |-  .+  =  ( +g  `  M )
Assertion
Ref Expression
submcl  |-  ( ( S  e.  (SubMnd `  M )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .+  Y )  e.  S )

Proof of Theorem submcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submrcl 15849 . . . . . . 7  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )
2 eqid 2467 . . . . . . . 8  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2467 . . . . . . . 8  |-  ( 0g
`  M )  =  ( 0g `  M
)
4 submcl.p . . . . . . . 8  |-  .+  =  ( +g  `  M )
52, 3, 4issubm 15850 . . . . . . 7  |-  ( M  e.  Mnd  ->  ( S  e.  (SubMnd `  M
)  <->  ( S  C_  ( Base `  M )  /\  ( 0g `  M
)  e.  S  /\  A. x  e.  S  A. y  e.  S  (
x  .+  y )  e.  S ) ) )
61, 5syl 16 . . . . . 6  |-  ( S  e.  (SubMnd `  M
)  ->  ( S  e.  (SubMnd `  M )  <->  ( S  C_  ( Base `  M )  /\  ( 0g `  M )  e.  S  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S
) ) )
76ibi 241 . . . . 5  |-  ( S  e.  (SubMnd `  M
)  ->  ( S  C_  ( Base `  M
)  /\  ( 0g `  M )  e.  S  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S ) )
87simp3d 1010 . . . 4  |-  ( S  e.  (SubMnd `  M
)  ->  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S
)
9 proplem2 14961 . . . 4  |-  ( ( ( X  e.  S  /\  Y  e.  S
)  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S
)  ->  ( X  .+  Y )  e.  S
)
108, 9sylan2 474 . . 3  |-  ( ( ( X  e.  S  /\  Y  e.  S
)  /\  S  e.  (SubMnd `  M ) )  ->  ( X  .+  Y )  e.  S
)
1110ancoms 453 . 2  |-  ( ( S  e.  (SubMnd `  M )  /\  ( X  e.  S  /\  Y  e.  S )
)  ->  ( X  .+  Y )  e.  S
)
12113impb 1192 1  |-  ( ( S  e.  (SubMnd `  M )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .+  Y )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Mndcmnd 15793  SubMndcsubmnd 15838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-submnd 15840
This theorem is referenced by:  resmhm  15862  mhmima  15866  gsumwsubmcl  15878  submmulgcl  16048  symggen  16368  lsmsubm  16546  gsumzadd  16808  gsumzaddOLD  16810  gsumzoppg  16840  gsumzoppgOLD  16841
  Copyright terms: Public domain W3C validator