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Theorem clmgmOLD 25537
Description: Obsolete version of mgmcl 16093 as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
clmgmOLD.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
clmgmOLD  |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )

Proof of Theorem clmgmOLD
StepHypRef Expression
1 clmgmOLD.1 . . . . 5  |-  X  =  dom  dom  G
21ismgmOLD 25536 . . . 4  |-  ( G  e.  Magma  ->  ( G  e.  Magma 
<->  G : ( X  X.  X ) --> X ) )
3 fovrn 6444 . . . . 5  |-  ( ( G : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A G B )  e.  X
)
433exp 1195 . . . 4  |-  ( G : ( X  X.  X ) --> X  -> 
( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) )
52, 4syl6bi 228 . . 3  |-  ( G  e.  Magma  ->  ( G  e.  Magma  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) ) )
65pm2.43i 47 . 2  |-  ( G  e.  Magma  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( A G B )  e.  X ) ) )
763imp 1190 1  |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    e. wcel 1819    X. cxp 5006   dom cdm 5008   -->wf 5590  (class class class)co 6296   Magmacmagm 25534
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-8 1821  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  ax-un 6591
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-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-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-mgmOLD 25535
This theorem is referenced by:  exidcl  30586
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