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Theorem grprinvd 12640
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 1199 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  .+  y
)  e.  B )
76caovclg 6005 . . . 4  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u  .+  v
)  e.  B )
87adantlr 474 . . 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 6006 . 2  |-  ( (
ph  /\  ps )  ->  ( X  .+  N
)  e.  B )
124caovassg 6011 . . . . 5  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
1312adantlr 474 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
1413, 9, 10, 11caovassd 6012 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( X  .+  N )  .+  ( X  .+  N ) )  =  ( X  .+  ( N  .+  ( X 
.+  N ) ) ) )
15 grprinvd.e . . . . . 6  |-  ( (
ph  /\  ps )  ->  ( N  .+  X
)  =  O )
1615oveq1d 5868 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( N  .+  X )  .+  N
)  =  ( O 
.+  N ) )
1713, 10, 9, 10caovassd 6012 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( N  .+  X )  .+  N
)  =  ( N 
.+  ( X  .+  N ) ) )
18 oveq2 5861 . . . . . . 7  |-  ( y  =  N  ->  ( O  .+  y )  =  ( O  .+  N
) )
19 id 19 . . . . . . 7  |-  ( y  =  N  ->  y  =  N )
2018, 19eqeq12d 2185 . . . . . 6  |-  ( y  =  N  ->  (
( O  .+  y
)  =  y  <->  ( O  .+  N )  =  N ) )
213ralrimiva 2543 . . . . . . . 8  |-  ( ph  ->  A. x  e.  B  ( O  .+  x )  =  x )
22 oveq2 5861 . . . . . . . . . 10  |-  ( x  =  y  ->  ( O  .+  x )  =  ( O  .+  y
) )
23 id 19 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
2422, 23eqeq12d 2185 . . . . . . . . 9  |-  ( x  =  y  ->  (
( O  .+  x
)  =  x  <->  ( O  .+  y )  =  y ) )
2524cbvralvw 2700 . . . . . . . 8  |-  ( A. x  e.  B  ( O  .+  x )  =  x  <->  A. y  e.  B  ( O  .+  y )  =  y )
2621, 25sylib 121 . . . . . . 7  |-  ( ph  ->  A. y  e.  B  ( O  .+  y )  =  y )
2726adantr 274 . . . . . 6  |-  ( (
ph  /\  ps )  ->  A. y  e.  B  ( O  .+  y )  =  y )
2820, 27, 10rspcdva 2839 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( O  .+  N
)  =  N )
2916, 17, 283eqtr3d 2211 . . . 4  |-  ( (
ph  /\  ps )  ->  ( N  .+  ( X  .+  N ) )  =  N )
3029oveq2d 5869 . . 3  |-  ( (
ph  /\  ps )  ->  ( X  .+  ( N  .+  ( X  .+  N ) ) )  =  ( X  .+  N ) )
3114, 30eqtrd 2203 . 2  |-  ( (
ph  /\  ps )  ->  ( ( X  .+  N )  .+  ( X  .+  N ) )  =  ( X  .+  N ) )
321, 2, 3, 4, 5, 11, 31grprinvlem 12639 1  |-  ( (
ph  /\  ps )  ->  ( X  .+  N
)  =  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 973    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449  (class class class)co 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856
This theorem is referenced by:  grpridd  12641  grprcan  12740  grprinv  12753
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