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Theorem grpridd 6102
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 5907 . . . . . 6  |-  ( y  =  n  ->  (
y  .+  x )  =  ( n  .+  x ) )
32eqeq1d 2324 . . . . 5  |-  ( y  =  n  ->  (
( y  .+  x
)  =  O  <->  ( n  .+  x )  =  O ) )
43cbvrexv 2799 . . . 4  |-  ( E. y  e.  B  ( y  .+  x )  =  O  <->  E. n  e.  B  ( n  .+  x )  =  O )
51, 4sylib 188 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  E. n  e.  B  ( n  .+  x )  =  O )
6 grprinvlem.a . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
76caovassg 6060 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
87adantlr 695 . . . . . . . 8  |-  ( ( ( 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 732 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  x  e.  B )
10 simprrl 740 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  n  e.  B )
118, 9, 10, 9caovassd 6061 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( (
x  .+  n )  .+  x )  =  ( x  .+  ( n 
.+  x ) ) )
12 grprinvlem.c . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
13 grprinvlem.o . . . . . . . . 9  |-  ( ph  ->  O  e.  B )
14 grprinvlem.i . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )
15 simprrr 741 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( n  .+  x )  =  O )
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 6101 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( x  .+  n )  =  O )
1716oveq1d 5915 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( (
x  .+  n )  .+  x )  =  ( O  .+  x ) )
1815oveq2d 5916 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( x  .+  ( n  .+  x
) )  =  ( x  .+  O ) )
1911, 17, 183eqtr3d 2356 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( O  .+  x )  =  ( x  .+  O ) )
2019anassrs 629 . . . . 5  |-  ( ( ( ph  /\  x  e.  B )  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) )  ->  ( O  .+  x )  =  ( x  .+  O ) )
2120expr 598 . . . 4  |-  ( ( ( ph  /\  x  e.  B )  /\  n  e.  B )  ->  (
( n  .+  x
)  =  O  -> 
( O  .+  x
)  =  ( x 
.+  O ) ) )
2221rexlimdva 2701 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( E. n  e.  B  ( n  .+  x )  =  O  ->  ( O  .+  x )  =  ( x  .+  O
) ) )
235, 22mpd 14 . 2  |-  ( (
ph  /\  x  e.  B )  ->  ( O  .+  x )  =  ( x  .+  O
) )
2423, 14eqtr3d 2350 1  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  O )  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   E.wrex 2578  (class class class)co 5900
This theorem is referenced by:  isgrpde  14555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903
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