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Theorem lidrideqd 12692
Description: If there is a left and right identity element for any binary operation (group operation)  .+, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
Hypotheses
Ref Expression
lidrideqd.l  |-  ( ph  ->  L  e.  B )
lidrideqd.r  |-  ( ph  ->  R  e.  B )
lidrideqd.li  |-  ( ph  ->  A. x  e.  B  ( L  .+  x )  =  x )
lidrideqd.ri  |-  ( ph  ->  A. x  e.  B  ( x  .+  R )  =  x )
Assertion
Ref Expression
lidrideqd  |-  ( ph  ->  L  =  R )
Distinct variable groups:    x, B    x, L    x, R    x,  .+
Allowed substitution hint:    ph( x)

Proof of Theorem lidrideqd
StepHypRef Expression
1 oveq1 5876 . . . 4  |-  ( x  =  L  ->  (
x  .+  R )  =  ( L  .+  R ) )
2 id 19 . . . 4  |-  ( x  =  L  ->  x  =  L )
31, 2eqeq12d 2192 . . 3  |-  ( x  =  L  ->  (
( x  .+  R
)  =  x  <->  ( L  .+  R )  =  L ) )
4 lidrideqd.ri . . 3  |-  ( ph  ->  A. x  e.  B  ( x  .+  R )  =  x )
5 lidrideqd.l . . 3  |-  ( ph  ->  L  e.  B )
63, 4, 5rspcdva 2846 . 2  |-  ( ph  ->  ( L  .+  R
)  =  L )
7 oveq2 5877 . . . 4  |-  ( x  =  R  ->  ( L  .+  x )  =  ( L  .+  R
) )
8 id 19 . . . 4  |-  ( x  =  R  ->  x  =  R )
97, 8eqeq12d 2192 . . 3  |-  ( x  =  R  ->  (
( L  .+  x
)  =  x  <->  ( L  .+  R )  =  R ) )
10 lidrideqd.li . . 3  |-  ( ph  ->  A. x  e.  B  ( L  .+  x )  =  x )
11 lidrideqd.r . . 3  |-  ( ph  ->  R  e.  B )
129, 10, 11rspcdva 2846 . 2  |-  ( ph  ->  ( L  .+  R
)  =  R )
136, 12eqtr3d 2212 1  |-  ( ph  ->  L  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   A.wral 2455  (class class class)co 5869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872
This theorem is referenced by:  lidrididd  12693
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