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Theorem grpridd 6510
Description: Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grprinvlem.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
grprinvlem.o  |-  ( ph  ->  O  e.  B )
grprinvlem.i  |-  ( (
ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )
grprinvlem.a  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
grprinvlem.n  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  O )
Assertion
Ref Expression
grpridd  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  O )  =  x )
Distinct variable groups:    x, y,
z, B    x, O, y, z    ph, x, y, z    x,  .+ , y, z

Proof of Theorem grpridd
Dummy variables  u  n  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.n . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  O )
2 oveq1 6302 . . . . . 6  |-  ( y  =  n  ->  (
y  .+  x )  =  ( n  .+  x ) )
32eqeq1d 2469 . . . . 5  |-  ( y  =  n  ->  (
( y  .+  x
)  =  O  <->  ( n  .+  x )  =  O ) )
43cbvrexv 3094 . . . 4  |-  ( E. y  e.  B  ( y  .+  x )  =  O  <->  E. n  e.  B  ( n  .+  x )  =  O )
51, 4sylib 196 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  E. n  e.  B  ( n  .+  x )  =  O )
6 grprinvlem.a . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
76caovassg 6468 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
87adantlr 714 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  B  /\  ( n  e.  B  /\  ( n  .+  x
)  =  O ) ) )  /\  (
u  e.  B  /\  v  e.  B  /\  w  e.  B )
)  ->  ( (
u  .+  v )  .+  w )  =  ( u  .+  ( v 
.+  w ) ) )
9 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  x  e.  B )
10 simprrl 763 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  n  e.  B )
118, 9, 10, 9caovassd 6469 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( (
x  .+  n )  .+  x )  =  ( x  .+  ( n 
.+  x ) ) )
12 grprinvlem.c . . . . . . 7  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
13 grprinvlem.o . . . . . . 7  |-  ( ph  ->  O  e.  B )
14 grprinvlem.i . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )
15 simprrr 764 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( n  .+  x )  =  O )
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 6509 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( x  .+  n )  =  O )
1716oveq1d 6310 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( (
x  .+  n )  .+  x )  =  ( O  .+  x ) )
1815oveq2d 6311 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( x  .+  ( n  .+  x
) )  =  ( x  .+  O ) )
1911, 17, 183eqtr3d 2516 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( O  .+  x )  =  ( x  .+  O ) )
2019anassrs 648 . . 3  |-  ( ( ( ph  /\  x  e.  B )  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) )  ->  ( O  .+  x )  =  ( x  .+  O ) )
215, 20rexlimddv 2963 . 2  |-  ( (
ph  /\  x  e.  B )  ->  ( O  .+  x )  =  ( x  .+  O
) )
2221, 14eqtr3d 2510 1  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  O )  =  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818  (class class class)co 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298
This theorem is referenced by:  isgrpde  15946
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