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Theorem caovimo 5964
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 5781 . . . . . . 7  |-  ( ( A F w )  =  B  ->  (
( A F w ) F v )  =  ( B F v ) )
21adantl 275 . . . . . 6  |-  ( ( w  e.  S  /\  ( A F w )  =  B )  -> 
( ( A F w ) F v )  =  ( B F v ) )
323ad2ant2 1003 . . . . 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 964 . . . . . . . . 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 275 . . . . . . . . . . . . 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 981 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  A  e.  S )
8 simp2 982 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  w  e.  S )
9 simp3 983 . . . . . . . . . . . . 13  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  v  e.  S )
106, 7, 8, 9caovassd 5930 . . . . . . . . . . . 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 275 . . . . . . . . . . . . 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 5952 . . . . . . . . . . . 12  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  ( A F ( w F v ) )  =  ( w F ( A F v ) ) )
1410, 13eqtrd 2172 . . . . . . . . . . 11  |-  ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S )  ->  ( ( A F w ) F v )  =  ( w F ( A F v ) ) )
1514adantr 274 . . . . . . . . . 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 5782 . . . . . . . . . . . 12  |-  ( ( A F v )  =  B  ->  (
w F ( A F v ) )  =  ( w F B ) )
17 oveq1 5781 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  (
x F B )  =  ( w F B ) )
18 id 19 . . . . . . . . . . . . . 14  |-  ( x  =  w  ->  x  =  w )
1917, 18eqeq12d 2154 . . . . . . . . . . . . 13  |-  ( x  =  w  ->  (
( x F B )  =  x  <->  ( w F B )  =  w ) )
20 caovimo.id . . . . . . . . . . . . 13  |-  ( x  e.  S  ->  (
x F B )  =  x )
2119, 20vtoclga 2752 . . . . . . . . . . . 12  |-  ( w  e.  S  ->  (
w F B )  =  w )
2216, 21sylan9eqr 2194 . . . . . . . . . . 11  |-  ( ( w  e.  S  /\  ( A F v )  =  B )  -> 
( w F ( A F v ) )  =  w )
23223ad2antl2 1144 . . . . . . . . . 10  |-  ( ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S
)  /\  ( A F v )  =  B )  ->  (
w F ( A F v ) )  =  w )
2415, 23eqtrd 2172 . . . . . . . . 9  |-  ( ( ( A  e.  S  /\  w  e.  S  /\  v  e.  S
)  /\  ( A F v )  =  B )  ->  (
( A F w ) F v )  =  w )
254, 24sylanbr 283 . . . . . . . 8  |-  ( ( ( ( A  e.  S  /\  w  e.  S )  /\  v  e.  S )  /\  ( A F v )  =  B )  ->  (
( A F w ) F v )  =  w )
2625anasss 396 . . . . . . 7  |-  ( ( ( A  e.  S  /\  w  e.  S
)  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  -> 
( ( A F w ) F v )  =  w )
27263impa 1176 . . . . . 6  |-  ( ( A  e.  S  /\  w  e.  S  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  ->  ( ( A F w ) F v )  =  w )
28273adant2r 1211 . . . . 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 275 . . . . . . . . 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 5927 . . . . . . . 8  |-  ( v  e.  S  ->  ( B F v )  =  ( v F B ) )
34 oveq1 5781 . . . . . . . . . 10  |-  ( x  =  v  ->  (
x F B )  =  ( v F B ) )
35 id 19 . . . . . . . . . 10  |-  ( x  =  v  ->  x  =  v )
3634, 35eqeq12d 2154 . . . . . . . . 9  |-  ( x  =  v  ->  (
( x F B )  =  x  <->  ( v F B )  =  v ) )
3736, 20vtoclga 2752 . . . . . . . 8  |-  ( v  e.  S  ->  (
v F B )  =  v )
3833, 37eqtrd 2172 . . . . . . 7  |-  ( v  e.  S  ->  ( B F v )  =  v )
3938adantr 274 . . . . . 6  |-  ( ( v  e.  S  /\  ( A F v )  =  B )  -> 
( B F v )  =  v )
40393ad2ant3 1004 . . . . 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 2180 . . . 4  |-  ( ( A  e.  S  /\  ( w  e.  S  /\  ( A F w )  =  B )  /\  ( v  e.  S  /\  ( A F v )  =  B ) )  ->  w  =  v )
42413expib 1184 . . 3  |-  ( A  e.  S  ->  (
( ( w  e.  S  /\  ( A F w )  =  B )  /\  (
v  e.  S  /\  ( A F v )  =  B ) )  ->  w  =  v ) )
4342alrimivv 1847 . 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 2202 . . . 4  |-  ( w  =  v  ->  (
w  e.  S  <->  v  e.  S ) )
45 oveq2 5782 . . . . 5  |-  ( w  =  v  ->  ( A F w )  =  ( A F v ) )
4645eqeq1d 2148 . . . 4  |-  ( w  =  v  ->  (
( A F w )  =  B  <->  ( A F v )  =  B ) )
4744, 46anbi12d 464 . . 3  |-  ( w  =  v  ->  (
( w  e.  S  /\  ( A F w )  =  B )  <-> 
( v  e.  S  /\  ( A F v )  =  B ) ) )
4847mo4 2060 . 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 133 1  |-  ( A  e.  S  ->  E* w ( w  e.  S  /\  ( A F w )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962   A.wal 1329    = wceq 1331    e. wcel 1480   E*wmo 2000  (class class class)co 5774
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  recmulnqg  7199
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