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Theorem ridlideq 24702
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 5800 . . . . . . . 8  |-  ( x  =  U  ->  ( U G x )  =  ( U G U ) )
2 id 21 . . . . . . . 8  |-  ( x  =  U  ->  x  =  U )
31, 2eqeq12d 2272 . . . . . . 7  |-  ( x  =  U  ->  (
( U G x )  =  x  <->  ( U G U )  =  U ) )
4 oveq1 5799 . . . . . . . 8  |-  ( x  =  U  ->  (
x G V )  =  ( U G V ) )
54, 2eqeq12d 2272 . . . . . . 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 2855 . . . . 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 5800 . . . . . . . 8  |-  ( x  =  V  ->  ( U G x )  =  ( U G V ) )
9 id 21 . . . . . . . 8  |-  ( x  =  V  ->  x  =  V )
108, 9eqeq12d 2272 . . . . . . 7  |-  ( x  =  V  ->  (
( U G x )  =  x  <->  ( U G V )  =  V ) )
11 oveq1 5799 . . . . . . . 8  |-  ( x  =  V  ->  (
x G V )  =  ( V G V ) )
1211, 9eqeq12d 2272 . . . . . . 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 2855 . . . . 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 2275 . . . . . . . . 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 2261 . . . . . . 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 2518  (class class class)co 5792
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 1927  ax-ext 2239
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689  df-ov 5795
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