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Theorem mgmidmo 15760
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 457 . . . . 5  |-  ( ( ( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  -> 
( u  .+  x
)  =  x )
21ralimi 2860 . . . 4  |-  ( A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  ->  A. x  e.  B  ( u  .+  x )  =  x )
3 simpr 461 . . . . 5  |-  ( ( ( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  -> 
( x  .+  w
)  =  x )
43ralimi 2860 . . . 4  |-  ( A. x  e.  B  (
( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  ->  A. x  e.  B  ( x  .+  w )  =  x )
5 oveq1 6302 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  .+  w )  =  ( u  .+  w ) )
6 id 22 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
75, 6eqeq12d 2489 . . . . . . . 8  |-  ( x  =  u  ->  (
( x  .+  w
)  =  x  <->  ( u  .+  w )  =  u ) )
87rspcva 3217 . . . . . . 7  |-  ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  -> 
( u  .+  w
)  =  u )
9 oveq2 6303 . . . . . . . . 9  |-  ( x  =  w  ->  (
u  .+  x )  =  ( u  .+  w ) )
10 id 22 . . . . . . . . 9  |-  ( x  =  w  ->  x  =  w )
119, 10eqeq12d 2489 . . . . . . . 8  |-  ( x  =  w  ->  (
( u  .+  x
)  =  x  <->  ( u  .+  w )  =  w ) )
1211rspcva 3217 . . . . . . 7  |-  ( ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x )  -> 
( u  .+  w
)  =  w )
138, 12sylan9req 2529 . . . . . 6  |-  ( ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  /\  ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x ) )  ->  u  =  w )
1413an42s 825 . . . . 5  |-  ( ( ( u  e.  B  /\  w  e.  B
)  /\  ( A. x  e.  B  (
u  .+  x )  =  x  /\  A. x  e.  B  ( x  .+  w )  =  x ) )  ->  u  =  w )
1514ex 434 . . . 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 656 . . 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 2894 . 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 6302 . . . . . 6  |-  ( u  =  w  ->  (
u  .+  x )  =  ( w  .+  x ) )
1918eqeq1d 2469 . . . . 5  |-  ( u  =  w  ->  (
( u  .+  x
)  =  x  <->  ( w  .+  x )  =  x ) )
20 oveq2 6303 . . . . . 6  |-  ( u  =  w  ->  (
x  .+  u )  =  ( x  .+  w ) )
2120eqeq1d 2469 . . . . 5  |-  ( u  =  w  ->  (
( x  .+  u
)  =  x  <->  ( x  .+  w )  =  x ) )
2219, 21anbi12d 710 . . . 4  |-  ( u  =  w  ->  (
( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  ( ( w 
.+  x )  =  x  /\  ( x 
.+  w )  =  x ) ) )
2322ralbidv 2906 . . 3  |-  ( u  =  w  ->  ( A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  A. x  e.  B  ( ( w  .+  x )  =  x  /\  ( x  .+  w )  =  x ) ) )
2423rmo4 3301 . 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 209 1  |-  E* u  e.  B  A. x  e.  B  ( (
u  .+  x )  =  x  /\  (
x  .+  u )  =  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E*wrmo 2820  (class class class)co 6295
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-ral 2822  df-rex 2823  df-rmo 2825  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298
This theorem is referenced by:  ismgmid  15765  mndideu  15807
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