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Theorem lidrideqd 12818
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 5895 . . . 4  |-  ( x  =  L  ->  (
x  .+  R )  =  ( L  .+  R ) )
2 id 19 . . . 4  |-  ( x  =  L  ->  x  =  L )
31, 2eqeq12d 2202 . . 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 2858 . 2  |-  ( ph  ->  ( L  .+  R
)  =  L )
7 oveq2 5896 . . . 4  |-  ( x  =  R  ->  ( L  .+  x )  =  ( L  .+  R
) )
8 id 19 . . . 4  |-  ( x  =  R  ->  x  =  R )
97, 8eqeq12d 2202 . . 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 2858 . 2  |-  ( ph  ->  ( L  .+  R
)  =  R )
136, 12eqtr3d 2222 1  |-  ( ph  ->  L  =  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    e. wcel 2158   A.wral 2465  (class class class)co 5888
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891
This theorem is referenced by:  lidrididd  12819
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