MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismgmid2 Structured version   Unicode version

Theorem ismgmid2 16465
Description: Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
ismgmid.b  |-  B  =  ( Base `  G
)
ismgmid.o  |-  .0.  =  ( 0g `  G )
ismgmid.p  |-  .+  =  ( +g  `  G )
ismgmid2.u  |-  ( ph  ->  U  e.  B )
ismgmid2.l  |-  ( (
ph  /\  x  e.  B )  ->  ( U  .+  x )  =  x )
ismgmid2.r  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  U )  =  x )
Assertion
Ref Expression
ismgmid2  |-  ( ph  ->  U  =  .0.  )
Distinct variable groups:    x,  .+    x,  .0.    x, B    x, G    x, U    ph, x

Proof of Theorem ismgmid2
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 ismgmid2.u . . 3  |-  ( ph  ->  U  e.  B )
2 ismgmid2.l . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( U  .+  x )  =  x )
3 ismgmid2.r . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  U )  =  x )
42, 3jca 534 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( U  .+  x
)  =  x  /\  ( x  .+  U )  =  x ) )
54ralrimiva 2846 . . 3  |-  ( ph  ->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) )
6 ismgmid.b . . . 4  |-  B  =  ( Base `  G
)
7 ismgmid.o . . . 4  |-  .0.  =  ( 0g `  G )
8 ismgmid.p . . . 4  |-  .+  =  ( +g  `  G )
9 oveq1 6312 . . . . . . . . 9  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
109eqeq1d 2431 . . . . . . . 8  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
11 oveq2 6313 . . . . . . . . 9  |-  ( e  =  U  ->  (
x  .+  e )  =  ( x  .+  U ) )
1211eqeq1d 2431 . . . . . . . 8  |-  ( e  =  U  ->  (
( x  .+  e
)  =  x  <->  ( x  .+  U )  =  x ) )
1310, 12anbi12d 715 . . . . . . 7  |-  ( e  =  U  ->  (
( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  ( ( U 
.+  x )  =  x  /\  ( x 
.+  U )  =  x ) ) )
1413ralbidv 2871 . . . . . 6  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
1514rspcev 3188 . . . . 5  |-  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x
)  =  x  /\  ( x  .+  U )  =  x ) )  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
161, 5, 15syl2anc 665 . . . 4  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
176, 7, 8, 16ismgmid 16462 . . 3  |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x 
.+  U )  =  x ) )  <->  .0.  =  U ) )
181, 5, 17mpbi2and 929 . 2  |-  ( ph  ->  .0.  =  U )
1918eqcomd 2437 1  |-  ( ph  ->  U  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15153   0gc0g 15301
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-riota 6267  df-ov 6308  df-0g 15303
This theorem is referenced by:  grpidd  16466  submnd0  16521  mnd1id  16534  frmd0  16599  mhmid  16762  ringidss  17746  xrs10  18946
  Copyright terms: Public domain W3C validator