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Theorem grprinvd 6146
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 1152 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  .+  y
)  e.  B )
76caovclg 6099 . . . 4  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B ) )  -> 
( u  .+  v
)  e.  B )
87adantlr 695 . . 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 6100 . 2  |-  ( (
ph  /\  ps )  ->  ( X  .+  N
)  e.  B )
124caovassg 6105 . . . . 5  |-  ( (
ph  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
1312adantlr 695 . . . 4  |-  ( ( ( ph  /\  ps )  /\  ( u  e.  B  /\  v  e.  B  /\  w  e.  B ) )  -> 
( ( u  .+  v )  .+  w
)  =  ( u 
.+  ( v  .+  w ) ) )
1413, 9, 10, 11caovassd 6106 . . 3  |-  ( (
ph  /\  ps )  ->  ( ( X  .+  N )  .+  ( X  .+  N ) )  =  ( X  .+  ( N  .+  ( X 
.+  N ) ) ) )
15 grprinvd.e . . . . . 6  |-  ( (
ph  /\  ps )  ->  ( N  .+  X
)  =  O )
1615oveq1d 5960 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( N  .+  X )  .+  N
)  =  ( O 
.+  N ) )
1713, 10, 9, 10caovassd 6106 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ( N  .+  X )  .+  N
)  =  ( N 
.+  ( X  .+  N ) ) )
183ralrimiva 2702 . . . . . . . 8  |-  ( ph  ->  A. x  e.  B  ( O  .+  x )  =  x )
19 oveq2 5953 . . . . . . . . . 10  |-  ( x  =  y  ->  ( O  .+  x )  =  ( O  .+  y
) )
20 id 19 . . . . . . . . . 10  |-  ( x  =  y  ->  x  =  y )
2119, 20eqeq12d 2372 . . . . . . . . 9  |-  ( x  =  y  ->  (
( O  .+  x
)  =  x  <->  ( O  .+  y )  =  y ) )
2221cbvralv 2840 . . . . . . . 8  |-  ( A. x  e.  B  ( O  .+  x )  =  x  <->  A. y  e.  B  ( O  .+  y )  =  y )
2318, 22sylib 188 . . . . . . 7  |-  ( ph  ->  A. y  e.  B  ( O  .+  y )  =  y )
2423adantr 451 . . . . . 6  |-  ( (
ph  /\  ps )  ->  A. y  e.  B  ( O  .+  y )  =  y )
25 oveq2 5953 . . . . . . . 8  |-  ( y  =  N  ->  ( O  .+  y )  =  ( O  .+  N
) )
26 id 19 . . . . . . . 8  |-  ( y  =  N  ->  y  =  N )
2725, 26eqeq12d 2372 . . . . . . 7  |-  ( y  =  N  ->  (
( O  .+  y
)  =  y  <->  ( O  .+  N )  =  N ) )
2827rspcv 2956 . . . . . 6  |-  ( N  e.  B  ->  ( A. y  e.  B  ( O  .+  y )  =  y  ->  ( O  .+  N )  =  N ) )
2910, 24, 28sylc 56 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( O  .+  N
)  =  N )
3016, 17, 293eqtr3d 2398 . . . 4  |-  ( (
ph  /\  ps )  ->  ( N  .+  ( X  .+  N ) )  =  N )
3130oveq2d 5961 . . 3  |-  ( (
ph  /\  ps )  ->  ( X  .+  ( N  .+  ( X  .+  N ) ) )  =  ( X  .+  N ) )
3214, 31eqtrd 2390 . 2  |-  ( (
ph  /\  ps )  ->  ( ( X  .+  N )  .+  ( X  .+  N ) )  =  ( X  .+  N ) )
331, 2, 3, 4, 5, 11, 32grprinvlem 6145 1  |-  ( (
ph  /\  ps )  ->  ( X  .+  N
)  =  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620  (class class class)co 5945
This theorem is referenced by:  grpridd  6147  grprcan  14614  grprinv  14628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-ov 5948
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