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Theorem ismgm 21003
Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Hypothesis
Ref Expression
ismgm.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
ismgm  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G :
( X  X.  X
) --> X ) )

Proof of Theorem ismgm
Dummy variables  g 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 5391 . . . . 5  |-  ( g  =  G  ->  (
g : ( t  X.  t ) --> t  <-> 
G : ( t  X.  t ) --> t ) )
21exbidv 1616 . . . 4  |-  ( g  =  G  ->  ( E. t  g :
( t  X.  t
) --> t  <->  E. t  G : ( t  X.  t ) --> t ) )
3 df-mgm 21002 . . . 4  |-  Magma  =  {
g  |  E. t 
g : ( t  X.  t ) --> t }
42, 3elab2g 2929 . . 3  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t  G : ( t  X.  t ) --> t ) )
5 f00 5442 . . . . . . . 8  |-  ( G : ( (/)  X.  (/) ) --> (/)  <->  ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) ) )
6 dmeq 4895 . . . . . . . . . 10  |-  ( G  =  (/)  ->  dom  G  =  dom  (/) )
7 dmeq 4895 . . . . . . . . . . 11  |-  ( dom 
G  =  dom  (/)  ->  dom  dom 
G  =  dom  dom  (/) )
8 dm0 4908 . . . . . . . . . . . . 13  |-  dom  (/)  =  (/)
98dmeqi 4896 . . . . . . . . . . . 12  |-  dom  dom  (/)  =  dom  (/)
109, 8eqtri 2316 . . . . . . . . . . 11  |-  dom  dom  (/)  =  (/)
117, 10syl6req 2345 . . . . . . . . . 10  |-  ( dom 
G  =  dom  (/)  ->  (/)  =  dom  dom 
G )
126, 11syl 15 . . . . . . . . 9  |-  ( G  =  (/)  ->  (/)  =  dom  dom 
G )
1312adantr 451 . . . . . . . 8  |-  ( ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) )  ->  (/)  =  dom  dom  G )
145, 13sylbi 187 . . . . . . 7  |-  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G )
15 xpeq12 4724 . . . . . . . . . 10  |-  ( ( t  =  (/)  /\  t  =  (/) )  ->  (
t  X.  t )  =  ( (/)  X.  (/) ) )
1615anidms 626 . . . . . . . . 9  |-  ( t  =  (/)  ->  ( t  X.  t )  =  ( (/)  X.  (/) ) )
17 feq23 5394 . . . . . . . . 9  |-  ( ( ( t  X.  t
)  =  ( (/)  X.  (/) )  /\  t  =  (/) )  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
1816, 17mpancom 650 . . . . . . . 8  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
19 eqeq1 2302 . . . . . . . 8  |-  ( t  =  (/)  ->  ( t  =  dom  dom  G  <->  (/)  =  dom  dom  G )
)
2018, 19imbi12d 311 . . . . . . 7  |-  ( t  =  (/)  ->  ( ( G : ( t  X.  t ) --> t  ->  t  =  dom  dom 
G )  <->  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G ) ) )
2114, 20mpbiri 224 . . . . . 6  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
22 fdm 5409 . . . . . . . 8  |-  ( G : ( t  X.  t ) --> t  ->  dom  G  =  ( t  X.  t ) )
23 dmeq 4895 . . . . . . . 8  |-  ( dom 
G  =  ( t  X.  t )  ->  dom  dom  G  =  dom  ( t  X.  t
) )
24 df-ne 2461 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  <->  -.  t  =  (/) )
25 dmxp 4913 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  ->  dom  ( t  X.  t )  =  t )
2624, 25sylbir 204 . . . . . . . . . . 11  |-  ( -.  t  =  (/)  ->  dom  ( t  X.  t
)  =  t )
2726eqeq1d 2304 . . . . . . . . . 10  |-  ( -.  t  =  (/)  ->  ( dom  ( t  X.  t
)  =  dom  dom  G  <-> 
t  =  dom  dom  G ) )
2827biimpcd 215 . . . . . . . . 9  |-  ( dom  ( t  X.  t
)  =  dom  dom  G  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G ) )
2928eqcoms 2299 . . . . . . . 8  |-  ( dom 
dom  G  =  dom  ( t  X.  t
)  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G )
)
3022, 23, 293syl 18 . . . . . . 7  |-  ( G : ( t  X.  t ) --> t  -> 
( -.  t  =  (/)  ->  t  =  dom  dom 
G ) )
3130com12 27 . . . . . 6  |-  ( -.  t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
3221, 31pm2.61i 156 . . . . 5  |-  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G )
3332pm4.71ri 614 . . . 4  |-  ( G : ( t  X.  t ) --> t  <->  ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
3433exbii 1572 . . 3  |-  ( E. t  G : ( t  X.  t ) --> t  <->  E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
354, 34syl6bb 252 . 2  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t
( t  =  dom  dom 
G  /\  G :
( t  X.  t
) --> t ) ) )
36 dmexg 4955 . . 3  |-  ( G  e.  A  ->  dom  G  e.  _V )
37 dmexg 4955 . . 3  |-  ( dom 
G  e.  _V  ->  dom 
dom  G  e.  _V )
38 xpeq12 4724 . . . . . . 7  |-  ( ( t  =  dom  dom  G  /\  t  =  dom  dom 
G )  ->  (
t  X.  t )  =  ( dom  dom  G  X.  dom  dom  G
) )
3938anidms 626 . . . . . 6  |-  ( t  =  dom  dom  G  ->  ( t  X.  t
)  =  ( dom 
dom  G  X.  dom  dom  G ) )
40 feq23 5394 . . . . . 6  |-  ( ( ( t  X.  t
)  =  ( dom 
dom  G  X.  dom  dom  G )  /\  t  =  dom  dom  G )  ->  ( G : ( t  X.  t ) --> t  <->  G : ( dom 
dom  G  X.  dom  dom  G ) --> dom  dom  G ) )
4139, 40mpancom 650 . . . . 5  |-  ( t  =  dom  dom  G  ->  ( G : ( t  X.  t ) --> t  <->  G : ( dom 
dom  G  X.  dom  dom  G ) --> dom  dom  G ) )
42 ismgm.1 . . . . . . . 8  |-  X  =  dom  dom  G
4342eqcomi 2300 . . . . . . 7  |-  dom  dom  G  =  X
4443, 43xpeq12i 4727 . . . . . 6  |-  ( dom 
dom  G  X.  dom  dom  G )  =  ( X  X.  X )
4544, 43feq23i 5401 . . . . 5  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) --> dom  dom  G  <->  G :
( X  X.  X
) --> X )
4641, 45syl6bb 252 . . . 4  |-  ( t  =  dom  dom  G  ->  ( G : ( t  X.  t ) --> t  <->  G : ( X  X.  X ) --> X ) )
4746ceqsexgv 2913 . . 3  |-  ( dom 
dom  G  e.  _V  ->  ( E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t )  <->  G :
( X  X.  X
) --> X ) )
4836, 37, 473syl 18 . 2  |-  ( G  e.  A  ->  ( E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t )  <-> 
G : ( X  X.  X ) --> X ) )
4935, 48bitrd 244 1  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G :
( X  X.  X
) --> X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468    X. cxp 4703   dom cdm 4705   -->wf 5267   Magmacmagm 21001
This theorem is referenced by:  clmgm  21004  opidon  21005  issmgrp  21017  mgmlion  25440  mgmrddd  25469
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-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-fun 5273  df-fn 5274  df-f 5275  df-mgm 21002
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