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Theorem clmgm 21004
Description: Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.)
Hypothesis
Ref Expression
clmgm.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
clmgm  |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )

Proof of Theorem clmgm
StepHypRef Expression
1 clmgm.1 . . . . 5  |-  X  =  dom  dom  G
21ismgm 21003 . . . 4  |-  ( G  e.  Magma  ->  ( G  e.  Magma 
<->  G : ( X  X.  X ) --> X ) )
3 fovrn 6006 . . . . 5  |-  ( ( G : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A G B )  e.  X
)
433exp 1150 . . . 4  |-  ( G : ( X  X.  X ) --> X  -> 
( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) )
52, 4syl6bi 219 . . 3  |-  ( G  e.  Magma  ->  ( G  e.  Magma  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) ) )
65pm2.43i 43 . 2  |-  ( G  e.  Magma  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) )
763imp 1145 1  |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    X. cxp 4703   dom cdm 4705   -->wf 5267  (class class class)co 5874   Magmacmagm 21001
This theorem is referenced by:  reacomsmgrp2  25447  reacomsmgrp3  25448  clfsebs  25450  resgcom  25454  fprodadd  25455  isppm  25457  seqzp2  25458  exidcl  26669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-mgm 21002
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