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Theorem ridlideq 25438
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 5882 . . . . . . . 8  |-  ( x  =  U  ->  ( U G x )  =  ( U G U ) )
2 id 19 . . . . . . . 8  |-  ( x  =  U  ->  x  =  U )
31, 2eqeq12d 2310 . . . . . . 7  |-  ( x  =  U  ->  (
( U G x )  =  x  <->  ( U G U )  =  U ) )
4 oveq1 5881 . . . . . . . 8  |-  ( x  =  U  ->  (
x G V )  =  ( U G V ) )
54, 2eqeq12d 2310 . . . . . . 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 2893 . . . . 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 5882 . . . . . . . 8  |-  ( x  =  V  ->  ( U G x )  =  ( U G V ) )
9 id 19 . . . . . . . 8  |-  ( x  =  V  ->  x  =  V )
108, 9eqeq12d 2310 . . . . . . 7  |-  ( x  =  V  ->  (
( U G x )  =  x  <->  ( U G V )  =  V ) )
11 oveq1 5881 . . . . . . . 8  |-  ( x  =  V  ->  (
x G V )  =  ( V G V ) )
1211, 9eqeq12d 2310 . . . . . . 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 2893 . . . . 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 2313 . . . . . . . . 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 2299 . . . . . . 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 1632    e. wcel 1696   A.wral 2556  (class class class)co 5874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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