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Theorem ismgmOLD 26033
Description: Obsolete version of ismgm 16476 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
ismgmOLD.1  |-  X  =  dom  dom  G
Assertion
Ref Expression
ismgmOLD  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G :
( X  X.  X
) --> X ) )

Proof of Theorem ismgmOLD
Dummy variables  g 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 5724 . . . . 5  |-  ( g  =  G  ->  (
g : ( t  X.  t ) --> t  <-> 
G : ( t  X.  t ) --> t ) )
21exbidv 1758 . . . 4  |-  ( g  =  G  ->  ( E. t  g :
( t  X.  t
) --> t  <->  E. t  G : ( t  X.  t ) --> t ) )
3 df-mgmOLD 26032 . . . 4  |-  Magma  =  {
g  |  E. t 
g : ( t  X.  t ) --> t }
42, 3elab2g 3220 . . 3  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t  G : ( t  X.  t ) --> t ) )
5 f00 5778 . . . . . . . 8  |-  ( G : ( (/)  X.  (/) ) --> (/)  <->  ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) ) )
6 dmeq 5050 . . . . . . . . . 10  |-  ( G  =  (/)  ->  dom  G  =  dom  (/) )
7 dmeq 5050 . . . . . . . . . . 11  |-  ( dom 
G  =  dom  (/)  ->  dom  dom 
G  =  dom  dom  (/) )
8 dm0 5063 . . . . . . . . . . . . 13  |-  dom  (/)  =  (/)
98dmeqi 5051 . . . . . . . . . . . 12  |-  dom  dom  (/)  =  dom  (/)
109, 8eqtri 2451 . . . . . . . . . . 11  |-  dom  dom  (/)  =  (/)
117, 10syl6req 2480 . . . . . . . . . 10  |-  ( dom 
G  =  dom  (/)  ->  (/)  =  dom  dom 
G )
126, 11syl 17 . . . . . . . . 9  |-  ( G  =  (/)  ->  (/)  =  dom  dom 
G )
1312adantr 466 . . . . . . . 8  |-  ( ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) )  ->  (/)  =  dom  dom  G )
145, 13sylbi 198 . . . . . . 7  |-  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G )
15 xpeq12 4868 . . . . . . . . . 10  |-  ( ( t  =  (/)  /\  t  =  (/) )  ->  (
t  X.  t )  =  ( (/)  X.  (/) ) )
1615anidms 649 . . . . . . . . 9  |-  ( t  =  (/)  ->  ( t  X.  t )  =  ( (/)  X.  (/) ) )
17 feq23 5727 . . . . . . . . 9  |-  ( ( ( t  X.  t
)  =  ( (/)  X.  (/) )  /\  t  =  (/) )  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
1816, 17mpancom 673 . . . . . . . 8  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
19 eqeq1 2426 . . . . . . . 8  |-  ( t  =  (/)  ->  ( t  =  dom  dom  G  <->  (/)  =  dom  dom  G )
)
2018, 19imbi12d 321 . . . . . . 7  |-  ( t  =  (/)  ->  ( ( G : ( t  X.  t ) --> t  ->  t  =  dom  dom 
G )  <->  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G ) ) )
2114, 20mpbiri 236 . . . . . 6  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
22 fdm 5746 . . . . . . . 8  |-  ( G : ( t  X.  t ) --> t  ->  dom  G  =  ( t  X.  t ) )
23 dmeq 5050 . . . . . . . 8  |-  ( dom 
G  =  ( t  X.  t )  ->  dom  dom  G  =  dom  ( t  X.  t
) )
24 df-ne 2620 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  <->  -.  t  =  (/) )
25 dmxp 5068 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  ->  dom  ( t  X.  t )  =  t )
2624, 25sylbir 216 . . . . . . . . . . 11  |-  ( -.  t  =  (/)  ->  dom  ( t  X.  t
)  =  t )
2726eqeq1d 2424 . . . . . . . . . 10  |-  ( -.  t  =  (/)  ->  ( dom  ( t  X.  t
)  =  dom  dom  G  <-> 
t  =  dom  dom  G ) )
2827biimpcd 227 . . . . . . . . 9  |-  ( dom  ( t  X.  t
)  =  dom  dom  G  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G ) )
2928eqcoms 2434 . . . . . . . 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 32 . . . . . 6  |-  ( -.  t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
3221, 31pm2.61i 167 . . . . 5  |-  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G )
3332pm4.71ri 637 . . . 4  |-  ( G : ( t  X.  t ) --> t  <->  ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
3433exbii 1712 . . 3  |-  ( E. t  G : ( t  X.  t ) --> t  <->  E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
354, 34syl6bb 264 . 2  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t
( t  =  dom  dom 
G  /\  G :
( t  X.  t
) --> t ) ) )
36 dmexg 6734 . . 3  |-  ( G  e.  A  ->  dom  G  e.  _V )
37 dmexg 6734 . . 3  |-  ( dom 
G  e.  _V  ->  dom 
dom  G  e.  _V )
38 xpeq12 4868 . . . . . . 7  |-  ( ( t  =  dom  dom  G  /\  t  =  dom  dom 
G )  ->  (
t  X.  t )  =  ( dom  dom  G  X.  dom  dom  G
) )
3938anidms 649 . . . . . 6  |-  ( t  =  dom  dom  G  ->  ( t  X.  t
)  =  ( dom 
dom  G  X.  dom  dom  G ) )
40 feq23 5727 . . . . . 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 673 . . . . 5  |-  ( t  =  dom  dom  G  ->  ( G : ( t  X.  t ) --> t  <->  G : ( dom 
dom  G  X.  dom  dom  G ) --> dom  dom  G ) )
42 ismgmOLD.1 . . . . . . . 8  |-  X  =  dom  dom  G
4342eqcomi 2435 . . . . . . 7  |-  dom  dom  G  =  X
4443, 43xpeq12i 4871 . . . . . 6  |-  ( dom 
dom  G  X.  dom  dom  G )  =  ( X  X.  X )
4544, 43feq23i 5736 . . . . 5  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) --> dom  dom  G  <->  G :
( X  X.  X
) --> X )
4641, 45syl6bb 264 . . . 4  |-  ( t  =  dom  dom  G  ->  ( G : ( t  X.  t ) --> t  <->  G : ( X  X.  X ) --> X ) )
4746ceqsexgv 3204 . . 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 256 1  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G :
( X  X.  X
) --> X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1868    =/= wne 2618   _Vcvv 3081   (/)c0 3761    X. cxp 4847   dom cdm 4849   -->wf 5593   Magmacmagm 26031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-fun 5599  df-fn 5600  df-f 5601  df-mgmOLD 26032
This theorem is referenced by:  clmgmOLD  26034  opidonOLD  26035  issmgrpOLD  26047
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