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Theorem mgmidmo 16445
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 458 . . . . 5  |-  ( ( ( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  -> 
( u  .+  x
)  =  x )
21ralimi 2758 . . . 4  |-  ( A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  ->  A. x  e.  B  ( u  .+  x )  =  x )
3 simpr 462 . . . . 5  |-  ( ( ( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  -> 
( x  .+  w
)  =  x )
43ralimi 2758 . . . 4  |-  ( A. x  e.  B  (
( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  ->  A. x  e.  B  ( x  .+  w )  =  x )
5 oveq1 6256 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  .+  w )  =  ( u  .+  w ) )
6 id 22 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
75, 6eqeq12d 2443 . . . . . . . 8  |-  ( x  =  u  ->  (
( x  .+  w
)  =  x  <->  ( u  .+  w )  =  u ) )
87rspcva 3123 . . . . . . 7  |-  ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  -> 
( u  .+  w
)  =  u )
9 oveq2 6257 . . . . . . . . 9  |-  ( x  =  w  ->  (
u  .+  x )  =  ( u  .+  w ) )
10 id 22 . . . . . . . . 9  |-  ( x  =  w  ->  x  =  w )
119, 10eqeq12d 2443 . . . . . . . 8  |-  ( x  =  w  ->  (
( u  .+  x
)  =  x  <->  ( u  .+  w )  =  w ) )
1211rspcva 3123 . . . . . . 7  |-  ( ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x )  -> 
( u  .+  w
)  =  w )
138, 12sylan9req 2483 . . . . . 6  |-  ( ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  /\  ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x ) )  ->  u  =  w )
1413an42s 834 . . . . 5  |-  ( ( ( u  e.  B  /\  w  e.  B
)  /\  ( A. x  e.  B  (
u  .+  x )  =  x  /\  A. x  e.  B  ( x  .+  w )  =  x ) )  ->  u  =  w )
1514ex 435 . . . 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 660 . . 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 2792 . 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 6256 . . . . . 6  |-  ( u  =  w  ->  (
u  .+  x )  =  ( w  .+  x ) )
1918eqeq1d 2430 . . . . 5  |-  ( u  =  w  ->  (
( u  .+  x
)  =  x  <->  ( w  .+  x )  =  x ) )
20 oveq2 6257 . . . . . 6  |-  ( u  =  w  ->  (
x  .+  u )  =  ( x  .+  w ) )
2120eqeq1d 2430 . . . . 5  |-  ( u  =  w  ->  (
( x  .+  u
)  =  x  <->  ( x  .+  w )  =  x ) )
2219, 21anbi12d 715 . . . 4  |-  ( u  =  w  ->  (
( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  ( ( w 
.+  x )  =  x  /\  ( x 
.+  w )  =  x ) ) )
2322ralbidv 2804 . . 3  |-  ( u  =  w  ->  ( A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  A. x  e.  B  ( ( w  .+  x )  =  x  /\  ( x  .+  w )  =  x ) ) )
2423rmo4 3206 . 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 212 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 370    = wceq 1437    e. wcel 1872   A.wral 2714   E*wrmo 2717  (class class class)co 6249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-rmo 2722  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-iota 5508  df-fv 5552  df-ov 6252
This theorem is referenced by:  ismgmid  16450  mndideu  16493
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