Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ridlideq Unicode version

Theorem ridlideq 24501
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 5718 . . . . . . . 8  |-  ( x  =  U  ->  ( U G x )  =  ( U G U ) )
2 id 21 . . . . . . . 8  |-  ( x  =  U  ->  x  =  U )
31, 2eqeq12d 2267 . . . . . . 7  |-  ( x  =  U  ->  (
( U G x )  =  x  <->  ( U G U )  =  U ) )
4 oveq1 5717 . . . . . . . 8  |-  ( x  =  U  ->  (
x G V )  =  ( U G V ) )
54, 2eqeq12d 2267 . . . . . . 7  |-  ( x  =  U  ->  (
( x G V )  =  x  <->  ( U G V )  =  U ) )
63, 5anbi12d 694 . . . . . 6  |-  ( x  =  U  ->  (
( ( U G x )  =  x  /\  ( x G V )  =  x )  <->  ( ( U G U )  =  U  /\  ( U G V )  =  U ) ) )
76rcla4v 2817 . . . . 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 5718 . . . . . . . 8  |-  ( x  =  V  ->  ( U G x )  =  ( U G V ) )
9 id 21 . . . . . . . 8  |-  ( x  =  V  ->  x  =  V )
108, 9eqeq12d 2267 . . . . . . 7  |-  ( x  =  V  ->  (
( U G x )  =  x  <->  ( U G V )  =  V ) )
11 oveq1 5717 . . . . . . . 8  |-  ( x  =  V  ->  (
x G V )  =  ( V G V ) )
1211, 9eqeq12d 2267 . . . . . . 7  |-  ( x  =  V  ->  (
( x G V )  =  x  <->  ( V G V )  =  V ) )
1310, 12anbi12d 694 . . . . . 6  |-  ( x  =  V  ->  (
( ( U G x )  =  x  /\  ( x G V )  =  x )  <->  ( ( U G V )  =  V  /\  ( V G V )  =  V ) ) )
1413rcla4v 2817 . . . . 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 811 . . . 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 2270 . . . . . . . . 9  |-  ( ( U  =  ( U G V )  /\  ( U G V )  =  V )  ->  U  =  V )
1716ex 425 . . . . . . . 8  |-  ( U  =  ( U G V )  ->  (
( U G V )  =  V  ->  U  =  V )
)
1817eqcoms 2256 . . . . . . 7  |-  ( ( U G V )  =  U  ->  (
( U G V )  =  V  ->  U  =  V )
)
1918adantrd 456 . . . . . 6  |-  ( ( U G V )  =  U  ->  (
( ( U G V )  =  V  /\  ( V G V )  =  V )  ->  U  =  V ) )
2019adantl 454 . . . . 5  |-  ( ( ( U G U )  =  U  /\  ( U G V )  =  U )  -> 
( ( ( U G V )  =  V  /\  ( V G V )  =  V )  ->  U  =  V ) )
2120imp 420 . . . 4  |-  ( ( ( ( U G U )  =  U  /\  ( U G V )  =  U )  /\  ( ( U G V )  =  V  /\  ( V G V )  =  V ) )  ->  U  =  V )
2215, 21syl6com 33 . . 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 629 . 2  |-  ( A. x  e.  X  (
( U G x )  =  x  /\  ( x G V )  =  x )  ->  ( ( U  e.  X  /\  V  e.  X )  ->  U  =  V ) )
2423com12 29 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509  (class class class)co 5710
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ov 5713
  Copyright terms: Public domain W3C validator