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Theorem mgmidmo 14695
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 445 . . . . 5  |-  ( ( ( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  -> 
( u  .+  x
)  =  x )
21ralimi 2783 . . . 4  |-  ( A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  ->  A. x  e.  B  ( u  .+  x )  =  x )
3 simpr 449 . . . . 5  |-  ( ( ( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  -> 
( x  .+  w
)  =  x )
43ralimi 2783 . . . 4  |-  ( A. x  e.  B  (
( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  ->  A. x  e.  B  ( x  .+  w )  =  x )
5 oveq1 6090 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  .+  w )  =  ( u  .+  w ) )
6 id 21 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
75, 6eqeq12d 2452 . . . . . . . 8  |-  ( x  =  u  ->  (
( x  .+  w
)  =  x  <->  ( u  .+  w )  =  u ) )
87rspcva 3052 . . . . . . 7  |-  ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  -> 
( u  .+  w
)  =  u )
9 oveq2 6091 . . . . . . . . 9  |-  ( x  =  w  ->  (
u  .+  x )  =  ( u  .+  w ) )
10 id 21 . . . . . . . . 9  |-  ( x  =  w  ->  x  =  w )
119, 10eqeq12d 2452 . . . . . . . 8  |-  ( x  =  w  ->  (
( u  .+  x
)  =  x  <->  ( u  .+  w )  =  w ) )
1211rspcva 3052 . . . . . . 7  |-  ( ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x )  -> 
( u  .+  w
)  =  w )
138, 12sylan9req 2491 . . . . . 6  |-  ( ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  /\  ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x ) )  ->  u  =  w )
1413an42s 802 . . . . 5  |-  ( ( ( u  e.  B  /\  w  e.  B
)  /\  ( A. x  e.  B  (
u  .+  x )  =  x  /\  A. x  e.  B  ( x  .+  w )  =  x ) )  ->  u  =  w )
1514ex 425 . . . 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 639 . . 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 2774 . 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 6090 . . . . . 6  |-  ( u  =  w  ->  (
u  .+  x )  =  ( w  .+  x ) )
1918eqeq1d 2446 . . . . 5  |-  ( u  =  w  ->  (
( u  .+  x
)  =  x  <->  ( w  .+  x )  =  x ) )
20 oveq2 6091 . . . . . 6  |-  ( u  =  w  ->  (
x  .+  u )  =  ( x  .+  w ) )
2120eqeq1d 2446 . . . . 5  |-  ( u  =  w  ->  (
( x  .+  u
)  =  x  <->  ( x  .+  w )  =  x ) )
2219, 21anbi12d 693 . . . 4  |-  ( u  =  w  ->  (
( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  ( ( w 
.+  x )  =  x  /\  ( x 
.+  w )  =  x ) ) )
2322ralbidv 2727 . . 3  |-  ( u  =  w  ->  ( A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  A. x  e.  B  ( ( w  .+  x )  =  x  /\  ( x  .+  w )  =  x ) ) )
2423rmo4 3129 . 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 202 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 360    = wceq 1653    e. wcel 1726   A.wral 2707   E*wrmo 2710  (class class class)co 6083
This theorem is referenced by:  mndideu  14700  ismgmid  14712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rmo 2715  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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