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Theorem grpridd 5725
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 5547 . . . . . 6  |-  ( y  =  n  ->  (
y  .+  x )  =  ( n  .+  x ) )
32eqeq1d 2064 . . . . 5  |-  ( y  =  n  ->  (
( y  .+  x
)  =  O  <->  ( n  .+  x )  =  O ) )
43cbvrexv 2551 . . . 4  |-  ( E. y  e.  B  ( y  .+  x )  =  O  <->  E. n  e.  B  ( n  .+  x )  =  O )
51, 4sylib 131 . . 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 5687 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
87adantlr 454 . . . . . 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 491 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  x  e.  B )
10 simprrl 499 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  n  e.  B )
118, 9, 10, 9caovassd 5688 . . . . 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 500 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( n  .+  x )  =  O )
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 5724 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( x  .+  n )  =  O )
1716oveq1d 5555 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( (
x  .+  n )  .+  x )  =  ( O  .+  x ) )
1815oveq2d 5556 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( x  .+  ( n  .+  x
) )  =  ( x  .+  O ) )
1911, 17, 183eqtr3d 2096 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) ) )  ->  ( O  .+  x )  =  ( x  .+  O ) )
2019anassrs 386 . . 3  |-  ( ( ( ph  /\  x  e.  B )  /\  (
n  e.  B  /\  ( n  .+  x )  =  O ) )  ->  ( O  .+  x )  =  ( x  .+  O ) )
215, 20rexlimddv 2454 . 2  |-  ( (
ph  /\  x  e.  B )  ->  ( O  .+  x )  =  ( x  .+  O
) )
2221, 14eqtr3d 2090 1  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  O )  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543
This theorem is referenced by: (None)
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