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Theorem mgmidmo 12626
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 108 . . . . 5  |-  ( ( ( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  -> 
( u  .+  x
)  =  x )
21ralimi 2533 . . . 4  |-  ( A. x  e.  B  (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  ->  A. x  e.  B  ( u  .+  x )  =  x )
3 simpr 109 . . . . 5  |-  ( ( ( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  -> 
( x  .+  w
)  =  x )
43ralimi 2533 . . . 4  |-  ( A. x  e.  B  (
( w  .+  x
)  =  x  /\  ( x  .+  w )  =  x )  ->  A. x  e.  B  ( x  .+  w )  =  x )
5 oveq1 5860 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  .+  w )  =  ( u  .+  w ) )
6 id 19 . . . . . . . . 9  |-  ( x  =  u  ->  x  =  u )
75, 6eqeq12d 2185 . . . . . . . 8  |-  ( x  =  u  ->  (
( x  .+  w
)  =  x  <->  ( u  .+  w )  =  u ) )
87rspcva 2832 . . . . . . 7  |-  ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  -> 
( u  .+  w
)  =  u )
9 oveq2 5861 . . . . . . . . 9  |-  ( x  =  w  ->  (
u  .+  x )  =  ( u  .+  w ) )
10 id 19 . . . . . . . . 9  |-  ( x  =  w  ->  x  =  w )
119, 10eqeq12d 2185 . . . . . . . 8  |-  ( x  =  w  ->  (
( u  .+  x
)  =  x  <->  ( u  .+  w )  =  w ) )
1211rspcva 2832 . . . . . . 7  |-  ( ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x )  -> 
( u  .+  w
)  =  w )
138, 12sylan9req 2224 . . . . . 6  |-  ( ( ( u  e.  B  /\  A. x  e.  B  ( x  .+  w )  =  x )  /\  ( w  e.  B  /\  A. x  e.  B  ( u  .+  x )  =  x ) )  ->  u  =  w )
1413an42s 584 . . . . 5  |-  ( ( ( u  e.  B  /\  w  e.  B
)  /\  ( A. x  e.  B  (
u  .+  x )  =  x  /\  A. x  e.  B  ( x  .+  w )  =  x ) )  ->  u  =  w )
1514ex 114 . . . 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 406 . . 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 ) )
1716rgen2 2556 . 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 5860 . . . . 5  |-  ( u  =  w  ->  (
u  .+  x )  =  ( w  .+  x ) )
1918eqeq1d 2179 . . . 4  |-  ( u  =  w  ->  (
( u  .+  x
)  =  x  <->  ( w  .+  x )  =  x ) )
2019ovanraleqv 5877 . . 3  |-  ( u  =  w  ->  ( A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  A. x  e.  B  ( ( w  .+  x )  =  x  /\  ( x  .+  w )  =  x ) ) )
2120rmo4 2923 . 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 )
)
2217, 21mpbir 145 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 103    = wceq 1348    e. wcel 2141   A.wral 2448   E*wrmo 2451  (class class class)co 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rmo 2456  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856
This theorem is referenced by:  ismgmid  12631  mndideu  12662
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