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Theorem ismgmid2 14406
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 518 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( U  .+  x
)  =  x  /\  ( x  .+  U )  =  x ) )
54ralrimiva 2639 . . 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 5881 . . . . . . . . 9  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
109eqeq1d 2304 . . . . . . . 8  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
11 oveq2 5882 . . . . . . . . 9  |-  ( e  =  U  ->  (
x  .+  e )  =  ( x  .+  U ) )
1211eqeq1d 2304 . . . . . . . 8  |-  ( e  =  U  ->  (
( x  .+  e
)  =  x  <->  ( x  .+  U )  =  x ) )
1310, 12anbi12d 691 . . . . . . 7  |-  ( e  =  U  ->  (
( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  ( ( U 
.+  x )  =  x  /\  ( x 
.+  U )  =  x ) ) )
1413ralbidv 2576 . . . . . 6  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
1514rspcev 2897 . . . . 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 642 . . . 4  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
176, 7, 8, 16ismgmid 14403 . . 3  |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x 
.+  U )  =  x ) )  <->  .0.  =  U ) )
181, 5, 17mpbi2and 887 . 2  |-  ( ph  ->  .0.  =  U )
1918eqcomd 2301 1  |-  ( ph  ->  U  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416
This theorem is referenced by:  grpidd  14411  submnd0  14418  frmd0  14498  rngidss  15383  xrs10  16426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420
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