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Theorem submcl 15502
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 15495 . . . . . . 7  |-  ( S  e.  (SubMnd `  M
)  ->  M  e.  Mnd )
2 eqid 2443 . . . . . . . 8  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2443 . . . . . . . 8  |-  ( 0g
`  M )  =  ( 0g `  M
)
4 submcl.p . . . . . . . 8  |-  .+  =  ( +g  `  M )
52, 3, 4issubm 15496 . . . . . . 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 1002 . . . 4  |-  ( S  e.  (SubMnd `  M
)  ->  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S
)
9 proplem2 14648 . . . 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 1183 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 965    = wceq 1369    e. wcel 1756   A.wral 2736    C_ wss 3349   ` cfv 5439  (class class class)co 6112   Basecbs 14195   +g cplusg 14259   0gc0g 14399   Mndcmnd 15430  SubMndcsubmnd 15484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fv 5447  df-ov 6115  df-submnd 15486
This theorem is referenced by:  resmhm  15508  mhmima  15512  gsumwsubmcl  15537  submmulgcl  15682  symggen  15997  lsmsubm  16173  gsumzadd  16430  gsumzaddOLD  16432  gsumzoppg  16462  gsumzoppgOLD  16463
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