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Theorem ridlideq 24747
Description: If a magma has a left identity element and a right identity element, they are equal. (Contributed by FL, 25-Sep-2011.)
Assertion
Ref Expression
ridlideq  |-  ( ( U  e.  X  /\  V  e.  X )  ->  ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x )  ->  U  =  V ) )
Distinct variable groups:    x, G    x, U    x, V    x, X

Proof of Theorem ridlideq
StepHypRef Expression
1 oveq2 5866 . . . . . . . 8  |-  ( x  =  U  ->  ( U G x )  =  ( U G U ) )
2 id 19 . . . . . . . 8  |-  ( x  =  U  ->  x  =  U )
31, 2eqeq12d 2297 . . . . . . 7  |-  ( x  =  U  ->  (
( U G x )  =  x  <->  ( U G U )  =  U ) )
4 oveq1 5865 . . . . . . . 8  |-  ( x  =  U  ->  (
x G V )  =  ( U G V ) )
54, 2eqeq12d 2297 . . . . . . 7  |-  ( x  =  U  ->  (
( x G V )  =  x  <->  ( U G V )  =  U ) )
63, 5anbi12d 691 . . . . . 6  |-  ( x  =  U  ->  (
( ( U G x )  =  x  /\  ( x G V )  =  x )  <->  ( ( U G U )  =  U  /\  ( U G V )  =  U ) ) )
76rspcv 2880 . . . . 5  |-  ( U  e.  X  ->  ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x )  ->  ( ( U G U )  =  U  /\  ( U G V )  =  U ) ) )
8 oveq2 5866 . . . . . . . 8  |-  ( x  =  V  ->  ( U G x )  =  ( U G V ) )
9 id 19 . . . . . . . 8  |-  ( x  =  V  ->  x  =  V )
108, 9eqeq12d 2297 . . . . . . 7  |-  ( x  =  V  ->  (
( U G x )  =  x  <->  ( U G V )  =  V ) )
11 oveq1 5865 . . . . . . . 8  |-  ( x  =  V  ->  (
x G V )  =  ( V G V ) )
1211, 9eqeq12d 2297 . . . . . . 7  |-  ( x  =  V  ->  (
( x G V )  =  x  <->  ( V G V )  =  V ) )
1310, 12anbi12d 691 . . . . . 6  |-  ( x  =  V  ->  (
( ( U G x )  =  x  /\  ( x G V )  =  x )  <->  ( ( U G V )  =  V  /\  ( V G V )  =  V ) ) )
1413rspcv 2880 . . . . 5  |-  ( V  e.  X  ->  ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x )  ->  ( ( U G V )  =  V  /\  ( V G V )  =  V ) ) )
157, 14im2anan9 808 . . . 4  |-  ( ( U  e.  X  /\  V  e.  X )  ->  ( ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x )  /\  A. x  e.  X  (
( U G x )  =  x  /\  ( x G V )  =  x ) )  ->  ( (
( U G U )  =  U  /\  ( U G V )  =  U )  /\  ( ( U G V )  =  V  /\  ( V G V )  =  V ) ) ) )
16 eqtr 2300 . . . . . . . . 9  |-  ( ( U  =  ( U G V )  /\  ( U G V )  =  V )  ->  U  =  V )
1716ex 423 . . . . . . . 8  |-  ( U  =  ( U G V )  ->  (
( U G V )  =  V  ->  U  =  V )
)
1817eqcoms 2286 . . . . . . 7  |-  ( ( U G V )  =  U  ->  (
( U G V )  =  V  ->  U  =  V )
)
1918adantrd 454 . . . . . 6  |-  ( ( U G V )  =  U  ->  (
( ( U G V )  =  V  /\  ( V G V )  =  V )  ->  U  =  V ) )
2019adantl 452 . . . . 5  |-  ( ( ( U G U )  =  U  /\  ( U G V )  =  U )  -> 
( ( ( U G V )  =  V  /\  ( V G V )  =  V )  ->  U  =  V ) )
2120imp 418 . . . 4  |-  ( ( ( ( U G U )  =  U  /\  ( U G V )  =  U )  /\  ( ( U G V )  =  V  /\  ( V G V )  =  V ) )  ->  U  =  V )
2215, 21syl6com 31 . . 3  |-  ( ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x )  /\  A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x ) )  -> 
( ( U  e.  X  /\  V  e.  X )  ->  U  =  V ) )
2322anidms 626 . 2  |-  ( A. x  e.  X  (
( U G x )  =  x  /\  ( x G V )  =  x )  ->  ( ( U  e.  X  /\  V  e.  X )  ->  U  =  V ) )
2423com12 27 1  |-  ( ( U  e.  X  /\  V  e.  X )  ->  ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x )  ->  U  =  V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543  (class class class)co 5858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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