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Theorem ismgmid2 14705
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 519 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( U  .+  x
)  =  x  /\  ( x  .+  U )  =  x ) )
54ralrimiva 2781 . . 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 6080 . . . . . . . . 9  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
109eqeq1d 2443 . . . . . . . 8  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
11 oveq2 6081 . . . . . . . . 9  |-  ( e  =  U  ->  (
x  .+  e )  =  ( x  .+  U ) )
1211eqeq1d 2443 . . . . . . . 8  |-  ( e  =  U  ->  (
( x  .+  e
)  =  x  <->  ( x  .+  U )  =  x ) )
1310, 12anbi12d 692 . . . . . . 7  |-  ( e  =  U  ->  (
( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  ( ( U 
.+  x )  =  x  /\  ( x 
.+  U )  =  x ) ) )
1413ralbidv 2717 . . . . . 6  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
1514rspcev 3044 . . . . 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 643 . . . 4  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
176, 7, 8, 16ismgmid 14702 . . 3  |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x 
.+  U )  =  x ) )  <->  .0.  =  U ) )
181, 5, 17mpbi2and 888 . 2  |-  ( ph  ->  .0.  =  U )
1918eqcomd 2440 1  |-  ( ph  ->  U  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715
This theorem is referenced by:  grpidd  14710  submnd0  14717  frmd0  14797  rngidss  15682  xrs10  16729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719
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