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Theorem ismgmid2 15755
Description: Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
mndidcl.b  |-  B  =  ( Base `  G
)
mndidcl.o  |-  .0.  =  ( 0g `  G )
ismgmid2.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 532 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( U  .+  x
)  =  x  /\  ( x  .+  U )  =  x ) )
54ralrimiva 2878 . . 3  |-  ( ph  ->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) )
6 mndidcl.b . . . 4  |-  B  =  ( Base `  G
)
7 mndidcl.o . . . 4  |-  .0.  =  ( 0g `  G )
8 ismgmid2.p . . . 4  |-  .+  =  ( +g  `  G )
9 oveq1 6291 . . . . . . . . 9  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
109eqeq1d 2469 . . . . . . . 8  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
11 oveq2 6292 . . . . . . . . 9  |-  ( e  =  U  ->  (
x  .+  e )  =  ( x  .+  U ) )
1211eqeq1d 2469 . . . . . . . 8  |-  ( e  =  U  ->  (
( x  .+  e
)  =  x  <->  ( x  .+  U )  =  x ) )
1310, 12anbi12d 710 . . . . . . 7  |-  ( e  =  U  ->  (
( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  ( ( U 
.+  x )  =  x  /\  ( x 
.+  U )  =  x ) ) )
1413ralbidv 2903 . . . . . 6  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
1514rspcev 3214 . . . . 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 661 . . . 4  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
176, 7, 8, 16ismgmid 15752 . . 3  |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x 
.+  U )  =  x ) )  <->  .0.  =  U ) )
181, 5, 17mpbi2and 919 . 2  |-  ( ph  ->  .0.  =  U )
1918eqcomd 2475 1  |-  ( ph  ->  U  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555   0gc0g 14695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-riota 6245  df-ov 6287  df-0g 14697
This theorem is referenced by:  grpidd  15760  submnd0  15769  mnd1id  15781  frmd0  15860  rngidss  17026  xrs10  18253
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