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Theorem rzrlzreq 25336
Description: If a magma has a left zero element and a right zero element, they are equal. (Contributed by FL, 25-Dec-2011.)
Assertion
Ref Expression
rzrlzreq  |-  ( ( U  e.  X  /\  V  e.  X )  ->  ( A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V )  ->  U  =  V ) )
Distinct variable groups:    x, G    x, U    x, V    x, X

Proof of Theorem rzrlzreq
StepHypRef Expression
1 oveq2 5866 . . . . . . . . 9  |-  ( x  =  U  ->  ( U G x )  =  ( U G U ) )
21eqeq1d 2291 . . . . . . . 8  |-  ( x  =  U  ->  (
( U G x )  =  U  <->  ( U G U )  =  U ) )
3 oveq1 5865 . . . . . . . . 9  |-  ( x  =  U  ->  (
x G V )  =  ( U G V ) )
43eqeq1d 2291 . . . . . . . 8  |-  ( x  =  U  ->  (
( x G V )  =  V  <->  ( U G V )  =  V ) )
52, 4anbi12d 691 . . . . . . 7  |-  ( x  =  U  ->  (
( ( U G x )  =  U  /\  ( x G V )  =  V )  <->  ( ( U G U )  =  U  /\  ( U G V )  =  V ) ) )
65rspcva 2882 . . . . . 6  |-  ( ( U  e.  X  /\  A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V ) )  ->  ( ( U G U )  =  U  /\  ( U G V )  =  V ) )
7 oveq2 5866 . . . . . . . . 9  |-  ( x  =  V  ->  ( U G x )  =  ( U G V ) )
87eqeq1d 2291 . . . . . . . 8  |-  ( x  =  V  ->  (
( U G x )  =  U  <->  ( U G V )  =  U ) )
9 oveq1 5865 . . . . . . . . 9  |-  ( x  =  V  ->  (
x G V )  =  ( V G V ) )
109eqeq1d 2291 . . . . . . . 8  |-  ( x  =  V  ->  (
( x G V )  =  V  <->  ( V G V )  =  V ) )
118, 10anbi12d 691 . . . . . . 7  |-  ( x  =  V  ->  (
( ( U G x )  =  U  /\  ( x G V )  =  V )  <->  ( ( U G V )  =  U  /\  ( V G V )  =  V ) ) )
1211rspcva 2882 . . . . . 6  |-  ( ( V  e.  X  /\  A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V ) )  ->  ( ( U G V )  =  U  /\  ( V G V )  =  V ) )
13 eqtr 2300 . . . . . . . . . . . 12  |-  ( ( U  =  ( U G V )  /\  ( U G V )  =  V )  ->  U  =  V )
1413ex 423 . . . . . . . . . . 11  |-  ( U  =  ( U G V )  ->  (
( U G V )  =  V  ->  U  =  V )
)
1514eqcoms 2286 . . . . . . . . . 10  |-  ( ( U G V )  =  U  ->  (
( U G V )  =  V  ->  U  =  V )
)
1615adantr 451 . . . . . . . . 9  |-  ( ( ( U G V )  =  U  /\  ( V G V )  =  V )  -> 
( ( U G V )  =  V  ->  U  =  V ) )
1716com12 27 . . . . . . . 8  |-  ( ( U G V )  =  V  ->  (
( ( U G V )  =  U  /\  ( V G V )  =  V )  ->  U  =  V ) )
1817adantl 452 . . . . . . 7  |-  ( ( ( U G U )  =  U  /\  ( U G V )  =  V )  -> 
( ( ( U G V )  =  U  /\  ( V G V )  =  V )  ->  U  =  V ) )
1918imp 418 . . . . . 6  |-  ( ( ( ( U G U )  =  U  /\  ( U G V )  =  V )  /\  ( ( U G V )  =  U  /\  ( V G V )  =  V ) )  ->  U  =  V )
206, 12, 19syl2an 463 . . . . 5  |-  ( ( ( U  e.  X  /\  A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V ) )  /\  ( V  e.  X  /\  A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V ) ) )  ->  U  =  V )
2120an4s 799 . . . 4  |-  ( ( ( U  e.  X  /\  V  e.  X
)  /\  ( A. x  e.  X  (
( U G x )  =  U  /\  ( x G V )  =  V )  /\  A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V ) ) )  ->  U  =  V )
2221expcom 424 . . 3  |-  ( ( A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V )  /\  A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V ) )  -> 
( ( U  e.  X  /\  V  e.  X )  ->  U  =  V ) )
2322anidms 626 . 2  |-  ( A. x  e.  X  (
( U G x )  =  U  /\  ( x G V )  =  V )  ->  ( ( U  e.  X  /\  V  e.  X )  ->  U  =  V ) )
2423com12 27 1  |-  ( ( U  e.  X  /\  V  e.  X )  ->  ( A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V )  ->  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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