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Theorem mgmidmo 14370
Description: A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)
Assertion
Ref Expression
mgmidmo  |-  E* u  e.  B A. x  e.  B  ( ( u 
.+  x )  =  x  /\  ( x 
.+  u )  =  x )
Distinct variable groups:    x, u, B    u,  .+ , x

Proof of Theorem mgmidmo
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . . . 5  |-  ( ( ( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  -> 
( u  .+  x
)  =  x )
21ralimi 2618 . . . 4  |-  ( A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  ->  A. x  e.  B  ( u  .+  x )  =  x )
3 simpr 447 . . . . 5  |-  ( ( ( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  -> 
( x  .+  w
)  =  x )
43ralimi 2618 . . . 4  |-  ( A. x  e.  B  (
( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  ->  A. x  e.  B  ( x  .+  w )  =  x )
5 oveq1 5865 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  .+  w )  =  ( u  .+  w ) )
6 id 19 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
75, 6eqeq12d 2297 . . . . . . . 8  |-  ( x  =  u  ->  (
( x  .+  w
)  =  x  <->  ( u  .+  w )  =  u ) )
87rspcva 2882 . . . . . . 7  |-  ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  -> 
( u  .+  w
)  =  u )
9 oveq2 5866 . . . . . . . . 9  |-  ( x  =  w  ->  (
u  .+  x )  =  ( u  .+  w ) )
10 id 19 . . . . . . . . 9  |-  ( x  =  w  ->  x  =  w )
119, 10eqeq12d 2297 . . . . . . . 8  |-  ( x  =  w  ->  (
( u  .+  x
)  =  x  <->  ( u  .+  w )  =  w ) )
1211rspcva 2882 . . . . . . 7  |-  ( ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x )  -> 
( u  .+  w
)  =  w )
138, 12sylan9req 2336 . . . . . 6  |-  ( ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  /\  ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x ) )  ->  u  =  w )
1413an42s 800 . . . . 5  |-  ( ( ( u  e.  B  /\  w  e.  B
)  /\  ( A. x  e.  B  (
u  .+  x )  =  x  /\  A. x  e.  B  ( x  .+  w )  =  x ) )  ->  u  =  w )
1514ex 423 . . . 4  |-  ( ( u  e.  B  /\  w  e.  B )  ->  ( ( A. x  e.  B  ( u  .+  x )  =  x  /\  A. x  e.  B  ( x  .+  w )  =  x )  ->  u  =  w ) )
162, 4, 15syl2ani 637 . . 3  |-  ( ( u  e.  B  /\  w  e.  B )  ->  ( ( A. x  e.  B  ( (
u  .+  x )  =  x  /\  (
x  .+  u )  =  x )  /\  A. x  e.  B  (
( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x ) )  ->  u  =  w ) )
1716rgen2a 2609 . 2  |-  A. u  e.  B  A. w  e.  B  ( ( A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  /\  A. x  e.  B  ( (
w  .+  x )  =  x  /\  (
x  .+  w )  =  x ) )  ->  u  =  w )
18 oveq1 5865 . . . . . 6  |-  ( u  =  w  ->  (
u  .+  x )  =  ( w  .+  x ) )
1918eqeq1d 2291 . . . . 5  |-  ( u  =  w  ->  (
( u  .+  x
)  =  x  <->  ( w  .+  x )  =  x ) )
20 oveq2 5866 . . . . . 6  |-  ( u  =  w  ->  (
x  .+  u )  =  ( x  .+  w ) )
2120eqeq1d 2291 . . . . 5  |-  ( u  =  w  ->  (
( x  .+  u
)  =  x  <->  ( x  .+  w )  =  x ) )
2219, 21anbi12d 691 . . . 4  |-  ( u  =  w  ->  (
( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  ( ( w 
.+  x )  =  x  /\  ( x 
.+  w )  =  x ) ) )
2322ralbidv 2563 . . 3  |-  ( u  =  w  ->  ( A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  A. x  e.  B  ( ( w  .+  x )  =  x  /\  ( x  .+  w )  =  x ) ) )
2423rmo4 2958 . 2  |-  ( E* u  e.  B A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  <->  A. u  e.  B  A. w  e.  B  ( ( A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  /\  A. x  e.  B  ( (
w  .+  x )  =  x  /\  (
x  .+  w )  =  x ) )  ->  u  =  w )
)
2517, 24mpbir 200 1  |-  E* u  e.  B A. x  e.  B  ( ( u 
.+  x )  =  x  /\  ( x 
.+  u )  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E*wrmo 2546  (class class class)co 5858
This theorem is referenced by:  mndideu  14375  ismgmid  14387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rmo 2551  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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