MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismgmOLD Structured version   Unicode version

Theorem ismgmOLD 25448
Description: Obsolete version of ismgm 15999 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 5719 . . . . 5  |-  ( g  =  G  ->  (
g : ( t  X.  t ) --> t  <-> 
G : ( t  X.  t ) --> t ) )
21exbidv 1715 . . . 4  |-  ( g  =  G  ->  ( E. t  g :
( t  X.  t
) --> t  <->  E. t  G : ( t  X.  t ) --> t ) )
3 df-mgmOLD 25447 . . . 4  |-  Magma  =  {
g  |  E. t 
g : ( t  X.  t ) --> t }
42, 3elab2g 3248 . . 3  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t  G : ( t  X.  t ) --> t ) )
5 f00 5773 . . . . . . . 8  |-  ( G : ( (/)  X.  (/) ) --> (/)  <->  ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) ) )
6 dmeq 5213 . . . . . . . . . 10  |-  ( G  =  (/)  ->  dom  G  =  dom  (/) )
7 dmeq 5213 . . . . . . . . . . 11  |-  ( dom 
G  =  dom  (/)  ->  dom  dom 
G  =  dom  dom  (/) )
8 dm0 5226 . . . . . . . . . . . . 13  |-  dom  (/)  =  (/)
98dmeqi 5214 . . . . . . . . . . . 12  |-  dom  dom  (/)  =  dom  (/)
109, 8eqtri 2486 . . . . . . . . . . 11  |-  dom  dom  (/)  =  (/)
117, 10syl6req 2515 . . . . . . . . . 10  |-  ( dom 
G  =  dom  (/)  ->  (/)  =  dom  dom 
G )
126, 11syl 16 . . . . . . . . 9  |-  ( G  =  (/)  ->  (/)  =  dom  dom 
G )
1312adantr 465 . . . . . . . 8  |-  ( ( G  =  (/)  /\  ( (/) 
X.  (/) )  =  (/) )  ->  (/)  =  dom  dom  G )
145, 13sylbi 195 . . . . . . 7  |-  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G )
15 xpeq12 5027 . . . . . . . . . 10  |-  ( ( t  =  (/)  /\  t  =  (/) )  ->  (
t  X.  t )  =  ( (/)  X.  (/) ) )
1615anidms 645 . . . . . . . . 9  |-  ( t  =  (/)  ->  ( t  X.  t )  =  ( (/)  X.  (/) ) )
17 feq23 5722 . . . . . . . . 9  |-  ( ( ( t  X.  t
)  =  ( (/)  X.  (/) )  /\  t  =  (/) )  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
1816, 17mpancom 669 . . . . . . . 8  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  <->  G :
( (/)  X.  (/) ) --> (/) ) )
19 eqeq1 2461 . . . . . . . 8  |-  ( t  =  (/)  ->  ( t  =  dom  dom  G  <->  (/)  =  dom  dom  G )
)
2018, 19imbi12d 320 . . . . . . 7  |-  ( t  =  (/)  ->  ( ( G : ( t  X.  t ) --> t  ->  t  =  dom  dom 
G )  <->  ( G : ( (/)  X.  (/) ) --> (/)  -> 
(/)  =  dom  dom  G ) ) )
2114, 20mpbiri 233 . . . . . 6  |-  ( t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
22 fdm 5741 . . . . . . . 8  |-  ( G : ( t  X.  t ) --> t  ->  dom  G  =  ( t  X.  t ) )
23 dmeq 5213 . . . . . . . 8  |-  ( dom 
G  =  ( t  X.  t )  ->  dom  dom  G  =  dom  ( t  X.  t
) )
24 df-ne 2654 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  <->  -.  t  =  (/) )
25 dmxp 5231 . . . . . . . . . . . 12  |-  ( t  =/=  (/)  ->  dom  ( t  X.  t )  =  t )
2624, 25sylbir 213 . . . . . . . . . . 11  |-  ( -.  t  =  (/)  ->  dom  ( t  X.  t
)  =  t )
2726eqeq1d 2459 . . . . . . . . . 10  |-  ( -.  t  =  (/)  ->  ( dom  ( t  X.  t
)  =  dom  dom  G  <-> 
t  =  dom  dom  G ) )
2827biimpcd 224 . . . . . . . . 9  |-  ( dom  ( t  X.  t
)  =  dom  dom  G  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G ) )
2928eqcoms 2469 . . . . . . . 8  |-  ( dom 
dom  G  =  dom  ( t  X.  t
)  ->  ( -.  t  =  (/)  ->  t  =  dom  dom  G )
)
3022, 23, 293syl 20 . . . . . . 7  |-  ( G : ( t  X.  t ) --> t  -> 
( -.  t  =  (/)  ->  t  =  dom  dom 
G ) )
3130com12 31 . . . . . 6  |-  ( -.  t  =  (/)  ->  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G ) )
3221, 31pm2.61i 164 . . . . 5  |-  ( G : ( t  X.  t ) --> t  -> 
t  =  dom  dom  G )
3332pm4.71ri 633 . . . 4  |-  ( G : ( t  X.  t ) --> t  <->  ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
3433exbii 1668 . . 3  |-  ( E. t  G : ( t  X.  t ) --> t  <->  E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t ) )
354, 34syl6bb 261 . 2  |-  ( G  e.  A  ->  ( G  e.  Magma  <->  E. t
( t  =  dom  dom 
G  /\  G :
( t  X.  t
) --> t ) ) )
36 dmexg 6730 . . 3  |-  ( G  e.  A  ->  dom  G  e.  _V )
37 dmexg 6730 . . 3  |-  ( dom 
G  e.  _V  ->  dom 
dom  G  e.  _V )
38 xpeq12 5027 . . . . . . 7  |-  ( ( t  =  dom  dom  G  /\  t  =  dom  dom 
G )  ->  (
t  X.  t )  =  ( dom  dom  G  X.  dom  dom  G
) )
3938anidms 645 . . . . . 6  |-  ( t  =  dom  dom  G  ->  ( t  X.  t
)  =  ( dom 
dom  G  X.  dom  dom  G ) )
40 feq23 5722 . . . . . 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 669 . . . . 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 2470 . . . . . . 7  |-  dom  dom  G  =  X
4443, 43xpeq12i 5030 . . . . . 6  |-  ( dom 
dom  G  X.  dom  dom  G )  =  ( X  X.  X )
4544, 43feq23i 5731 . . . . 5  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) --> dom  dom  G  <->  G :
( X  X.  X
) --> X )
4641, 45syl6bb 261 . . . 4  |-  ( t  =  dom  dom  G  ->  ( G : ( t  X.  t ) --> t  <->  G : ( X  X.  X ) --> X ) )
4746ceqsexgv 3232 . . 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 20 . 2  |-  ( G  e.  A  ->  ( E. t ( t  =  dom  dom  G  /\  G : ( t  X.  t ) --> t )  <-> 
G : ( X  X.  X ) --> X ) )
4935, 48bitrd 253 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 184    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   _Vcvv 3109   (/)c0 3793    X. cxp 5006   dom cdm 5008   -->wf 5590   Magmacmagm 25446
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-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-fun 5596  df-fn 5597  df-f 5598  df-mgmOLD 25447
This theorem is referenced by:  clmgmOLD  25449  opidonOLD  25450  issmgrpOLD  25462
  Copyright terms: Public domain W3C validator