ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovimo Unicode version

Theorem caovimo 5725
Description: Uniqueness of inverse element in commutative, associative operation with identity. The identity element is  B. (Contributed by Jim Kingdon, 18-Sep-2019.)
Hypotheses
Ref Expression
caovimo.idel  |-  B  e.  S
caovimo.com  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  =  ( y F x ) )
caovimo.ass  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )
caovimo.id  |-  ( x  e.  S  ->  (
x F B )  =  x )
Assertion
Ref Expression
caovimo  |-  ( A  e.  S  ->  E* w ( w  e.  S  /\  ( A F w )  =  B ) )
Distinct variable groups:    w, A, x, y, z    w, B, x, y    w, F, x, y, z    w, S, x, y, z
Allowed substitution hint:    B( z)

Proof of Theorem caovimo
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 oveq1 5550 . . . . . . 7  |-  ( ( A F w )  =  B  ->  (
( A F w ) F v )  =  ( B F v ) )
21adantl 271 . . . . . 6  |-  ( ( w  e.  S  /\  ( A F w )  =  B )  -> 
( ( A F w ) F v )  =  ( B F v ) )
323ad2ant2 961 . . . . 5  |-  ( ( A  e.  S  /\  ( w  e.  S  /\  ( A F w )  =  B )  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  -> 
( ( A F w ) F v )  =  ( B F v ) )
4 df-3an 922 . . . . . . . . 9  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  <->  ( ( A  e.  S  /\  w  e.  S
)  /\  v  e.  S ) )
5 caovimo.ass . . . . . . . . . . . . . 14  |-  ( ( x  e.  S  /\  y  e.  S  /\  z  e.  S )  ->  ( ( x F y ) F z )  =  ( x F ( y F z ) ) )
65adantl 271 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S
)  /\  ( x  e.  S  /\  y  e.  S  /\  z  e.  S ) )  -> 
( ( x F y ) F z )  =  ( x F ( y F z ) ) )
7 simp1 939 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  A  e.  S )
8 simp2 940 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  w  e.  S )
9 simp3 941 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  v  e.  S )
106, 7, 8, 9caovassd 5691 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  ( ( A F w ) F v )  =  ( A F ( w F v ) ) )
11 caovimo.com . . . . . . . . . . . . . 14  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  =  ( y F x ) )
1211adantl 271 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S
)  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  =  ( y F x ) )
137, 8, 9, 12, 6caov12d 5713 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  ( A F ( w F v ) )  =  ( w F ( A F v ) ) )
1410, 13eqtrd 2114 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  ( ( A F w ) F v )  =  ( w F ( A F v ) ) )
1514adantr 270 . . . . . . . . . 10  |-  ( ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S
)  /\  ( A F v )  =  B )  ->  (
( A F w ) F v )  =  ( w F ( A F v ) ) )
16 oveq2 5551 . . . . . . . . . . . 12  |-  ( ( A F v )  =  B  ->  (
w F ( A F v ) )  =  ( w F B ) )
17 oveq1 5550 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x F B )  =  ( w F B ) )
18 id 19 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  x  =  w )
1917, 18eqeq12d 2096 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  (
( x F B )  =  x  <->  ( w F B )  =  w ) )
20 caovimo.id . . . . . . . . . . . . 13  |-  ( x  e.  S  ->  (
x F B )  =  x )
2119, 20vtoclga 2665 . . . . . . . . . . . 12  |-  ( w  e.  S  ->  (
w F B )  =  w )
2216, 21sylan9eqr 2136 . . . . . . . . . . 11  |-  ( ( w  e.  S  /\  ( A F v )  =  B )  -> 
( w F ( A F v ) )  =  w )
23223ad2antl2 1102 . . . . . . . . . 10  |-  ( ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S
)  /\  ( A F v )  =  B )  ->  (
w F ( A F v ) )  =  w )
2415, 23eqtrd 2114 . . . . . . . . 9  |-  ( ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S
)  /\  ( A F v )  =  B )  ->  (
( A F w ) F v )  =  w )
254, 24sylanbr 279 . . . . . . . 8  |-  ( ( ( ( A  e.  S  /\  w  e.  S )  /\  v  e.  S )  /\  ( A F v )  =  B )  ->  (
( A F w ) F v )  =  w )
2625anasss 391 . . . . . . 7  |-  ( ( ( A  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  -> 
( ( A F w ) F v )  =  w )
27263impa 1134 . . . . . 6  |-  ( ( A  e.  S  /\  w  e.  S  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  ->  ( ( A F w ) F v )  =  w )
28273adant2r 1165 . . . . 5  |-  ( ( A  e.  S  /\  ( w  e.  S  /\  ( A F w )  =  B )  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  -> 
( ( A F w ) F v )  =  w )
2911adantl 271 . . . . . . . . 9  |-  ( ( v  e.  S  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x F y )  =  ( y F x ) )
30 caovimo.idel . . . . . . . . . 10  |-  B  e.  S
3130a1i 9 . . . . . . . . 9  |-  ( v  e.  S  ->  B  e.  S )
32 id 19 . . . . . . . . 9  |-  ( v  e.  S  ->  v  e.  S )
3329, 31, 32caovcomd 5688 . . . . . . . 8  |-  ( v  e.  S  ->  ( B F v )  =  ( v F B ) )
34 oveq1 5550 . . . . . . . . . 10  |-  ( x  =  v  ->  (
x F B )  =  ( v F B ) )
35 id 19 . . . . . . . . . 10  |-  ( x  =  v  ->  x  =  v )
3634, 35eqeq12d 2096 . . . . . . . . 9  |-  ( x  =  v  ->  (
( x F B )  =  x  <->  ( v F B )  =  v ) )
3736, 20vtoclga 2665 . . . . . . . 8  |-  ( v  e.  S  ->  (
v F B )  =  v )
3833, 37eqtrd 2114 . . . . . . 7  |-  ( v  e.  S  ->  ( B F v )  =  v )
3938adantr 270 . . . . . 6  |-  ( ( v  e.  S  /\  ( A F v )  =  B )  -> 
( B F v )  =  v )
40393ad2ant3 962 . . . . 5  |-  ( ( A  e.  S  /\  ( w  e.  S  /\  ( A F w )  =  B )  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  -> 
( B F v )  =  v )
413, 28, 403eqtr3d 2122 . . . 4  |-  ( ( A  e.  S  /\  ( w  e.  S  /\  ( A F w )  =  B )  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  ->  w  =  v )
42413expib 1142 . . 3  |-  ( A  e.  S  ->  (
( ( w  e.  S  /\  ( A F w )  =  B )  /\  (
v  e.  S  /\  ( A F v )  =  B ) )  ->  w  =  v ) )
4342alrimivv 1797 . 2  |-  ( A  e.  S  ->  A. w A. v ( ( ( w  e.  S  /\  ( A F w )  =  B )  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  ->  w  =  v ) )
44 eleq1 2142 . . . 4  |-  ( w  =  v  ->  (
w  e.  S  <->  v  e.  S ) )
45 oveq2 5551 . . . . 5  |-  ( w  =  v  ->  ( A F w )  =  ( A F v ) )
4645eqeq1d 2090 . . . 4  |-  ( w  =  v  ->  (
( A F w )  =  B  <->  ( A F v )  =  B ) )
4744, 46anbi12d 457 . . 3  |-  ( w  =  v  ->  (
( w  e.  S  /\  ( A F w )  =  B )  <-> 
( v  e.  S  /\  ( A F v )  =  B ) ) )
4847mo4 2003 . 2  |-  ( E* w ( w  e.  S  /\  ( A F w )  =  B )  <->  A. w A. v ( ( ( w  e.  S  /\  ( A F w )  =  B )  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  ->  w  =  v ) )
4943, 48sylibr 132 1  |-  ( A  e.  S  ->  E* w ( w  e.  S  /\  ( A F w )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920   A.wal 1283    = wceq 1285    e. wcel 1434   E*wmo 1943  (class class class)co 5543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  recmulnqg  6643
  Copyright terms: Public domain W3C validator