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Theorem grprinvd 12804
Description: Deduce right inverse 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 )
grprinvd.x  |-  ( (
ph  /\  ps )  ->  X  e.  B )
grprinvd.n  |-  ( (
ph  /\  ps )  ->  N  e.  B )
grprinvd.e  |-  ( (
ph  /\  ps )  ->  ( N  .+  X
)  =  O )
Assertion
Ref Expression
grprinvd  |-  ( (
ph  /\  ps )  ->  ( X  .+  N
)  =  O )
Distinct variable groups:    x, y, z, B    x, O, y, z    ph, x, y, z   
y, N, z    x,  .+ , y, z    y, X, z    ps, y
Allowed substitution hints:    ps( x, z)    N( x)    X( x)

Proof of Theorem grprinvd
Dummy variables  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.c . 2  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
2 grprinvlem.o . 2  |-  ( ph  ->  O  e.  B )
3 grprinvlem.i . 2  |-  ( (
ph  /\  x  e.  B )  ->  ( O  .+  x )  =  x )
4 grprinvlem.a . 2  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
5 grprinvlem.n . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  O )
613expb 1204 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  .+  y
)  e.  B )
76caovclg 6026 . . . 4  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u  .+  v
)  e.  B )
87adantlr 477 . . 3  |-  ( ( ( ph  /\  ps )  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u  .+  v
)  e.  B )
9 grprinvd.x . . 3  |-  ( (
ph  /\  ps )  ->  X  e.  B )
10 grprinvd.n . . 3  |-  ( (
ph  /\  ps )  ->  N  e.  B )
118, 9, 10caovcld 6027 . 2  |-  ( (
ph  /\  ps )  ->  ( X  .+  N
)  e.  B )
124caovassg 6032 . . . . 5  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
1312adantlr 477 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
1413, 9, 10, 11caovassd 6033 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( X  .+  N )  .+  ( X  .+  N ) )  =  ( X  .+  ( N  .+  ( X 
.+  N ) ) ) )
15 grprinvd.e . . . . . 6  |-  ( (
ph  /\  ps )  ->  ( N  .+  X
)  =  O )
1615oveq1d 5889 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( N  .+  X )  .+  N
)  =  ( O 
.+  N ) )
1713, 10, 9, 10caovassd 6033 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( N  .+  X )  .+  N
)  =  ( N 
.+  ( X  .+  N ) ) )
18 oveq2 5882 . . . . . . 7  |-  ( y  =  N  ->  ( O  .+  y )  =  ( O  .+  N
) )
19 id 19 . . . . . . 7  |-  ( y  =  N  ->  y  =  N )
2018, 19eqeq12d 2192 . . . . . 6  |-  ( y  =  N  ->  (
( O  .+  y
)  =  y  <->  ( O  .+  N )  =  N ) )
213ralrimiva 2550 . . . . . . . 8  |-  ( ph  ->  A. x  e.  B  ( O  .+  x )  =  x )
22 oveq2 5882 . . . . . . . . . 10  |-  ( x  =  y  ->  ( O  .+  x )  =  ( O  .+  y
) )
23 id 19 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
2422, 23eqeq12d 2192 . . . . . . . . 9  |-  ( x  =  y  ->  (
( O  .+  x
)  =  x  <->  ( O  .+  y )  =  y ) )
2524cbvralvw 2707 . . . . . . . 8  |-  ( A. x  e.  B  ( O  .+  x )  =  x  <->  A. y  e.  B  ( O  .+  y )  =  y )
2621, 25sylib 122 . . . . . . 7  |-  ( ph  ->  A. y  e.  B  ( O  .+  y )  =  y )
2726adantr 276 . . . . . 6  |-  ( (
ph  /\  ps )  ->  A. y  e.  B  ( O  .+  y )  =  y )
2820, 27, 10rspcdva 2846 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( O  .+  N
)  =  N )
2916, 17, 283eqtr3d 2218 . . . 4  |-  ( (
ph  /\  ps )  ->  ( N  .+  ( X  .+  N ) )  =  N )
3029oveq2d 5890 . . 3  |-  ( (
ph  /\  ps )  ->  ( X  .+  ( N  .+  ( X  .+  N ) ) )  =  ( X  .+  N ) )
3114, 30eqtrd 2210 . 2  |-  ( (
ph  /\  ps )  ->  ( ( X  .+  N )  .+  ( X  .+  N ) )  =  ( X  .+  N ) )
321, 2, 3, 4, 5, 11, 31grprinvlem 12803 1  |-  ( (
ph  /\  ps )  ->  ( X  .+  N
)  =  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456  (class class class)co 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224  df-ov 5877
This theorem is referenced by:  grpridd  12805  grprcan  12909  grprinv  12922
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