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Theorem caovimo 5820
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 5641 . . . . . . 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 965 . . . . 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 926 . . . . . . . . 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 943 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  A  e.  S )
8 simp2 944 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  w  e.  S )
9 simp3 945 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  v  e.  S )
106, 7, 8, 9caovassd 5786 . . . . . . . . . . . 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 5808 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  ( A F ( w F v ) )  =  ( w F ( A F v ) ) )
1410, 13eqtrd 2120 . . . . . . . . . . 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 5642 . . . . . . . . . . . 12  |-  ( ( A F v )  =  B  ->  (
w F ( A F v ) )  =  ( w F B ) )
17 oveq1 5641 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x F B )  =  ( w F B ) )
18 id 19 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  x  =  w )
1917, 18eqeq12d 2102 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  (
( x F B )  =  x  <->  ( w F B )  =  w ) )
20 caovimo.id . . . . . . . . . . . . 13  |-  ( x  e.  S  ->  (
x F B )  =  x )
2119, 20vtoclga 2685 . . . . . . . . . . . 12  |-  ( w  e.  S  ->  (
w F B )  =  w )
2216, 21sylan9eqr 2142 . . . . . . . . . . 11  |-  ( ( w  e.  S  /\  ( A F v )  =  B )  -> 
( w F ( A F v ) )  =  w )
23223ad2antl2 1106 . . . . . . . . . 10  |-  ( ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S
)  /\  ( A F v )  =  B )  ->  (
w F ( A F v ) )  =  w )
2415, 23eqtrd 2120 . . . . . . . . 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 1138 . . . . . 6  |-  ( ( A  e.  S  /\  w  e.  S  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  ->  ( ( A F w ) F v )  =  w )
28273adant2r 1169 . . . . 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 5783 . . . . . . . 8  |-  ( v  e.  S  ->  ( B F v )  =  ( v F B ) )
34 oveq1 5641 . . . . . . . . . 10  |-  ( x  =  v  ->  (
x F B )  =  ( v F B ) )
35 id 19 . . . . . . . . . 10  |-  ( x  =  v  ->  x  =  v )
3634, 35eqeq12d 2102 . . . . . . . . 9  |-  ( x  =  v  ->  (
( x F B )  =  x  <->  ( v F B )  =  v ) )
3736, 20vtoclga 2685 . . . . . . . 8  |-  ( v  e.  S  ->  (
v F B )  =  v )
3833, 37eqtrd 2120 . . . . . . 7  |-  ( v  e.  S  ->  ( B F v )  =  v )
3938adantr 270 . . . . . 6  |-  ( ( v  e.  S  /\  ( A F v )  =  B )  -> 
( B F v )  =  v )
40393ad2ant3 966 . . . . 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 2128 . . . 4  |-  ( ( A  e.  S  /\  ( w  e.  S  /\  ( A F w )  =  B )  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  ->  w  =  v )
42413expib 1146 . . 3  |-  ( A  e.  S  ->  (
( ( w  e.  S  /\  ( A F w )  =  B )  /\  (
v  e.  S  /\  ( A F v )  =  B ) )  ->  w  =  v ) )
4342alrimivv 1803 . 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 2150 . . . 4  |-  ( w  =  v  ->  (
w  e.  S  <->  v  e.  S ) )
45 oveq2 5642 . . . . 5  |-  ( w  =  v  ->  ( A F w )  =  ( A F v ) )
4645eqeq1d 2096 . . . 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 2009 . 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 924   A.wal 1287    = wceq 1289    e. wcel 1438   E*wmo 1949  (class class class)co 5634
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637
This theorem is referenced by:  recmulnqg  6929
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