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Theorem caovmo 6073
Description: Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 4-Mar-1996.)
Hypotheses
Ref Expression
caovmo.2  |-  B  e.  S
caovmo.dom  |-  dom  F  =  ( S  X.  S )
caovmo.3  |-  -.  (/)  e.  S
caovmo.com  |-  ( x F y )  =  ( y F x )
caovmo.ass  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
caovmo.id  |-  ( x  e.  S  ->  (
x F B )  =  x )
Assertion
Ref Expression
caovmo  |-  E* w
( A F w )  =  B
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, F, y, z    x, S, y, z    w, A, x, y    w, B, z   
w, F    w, S

Proof of Theorem caovmo
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5881 . . . . . 6  |-  ( u  =  A  ->  (
u F w )  =  ( A F w ) )
21eqeq1d 2304 . . . . 5  |-  ( u  =  A  ->  (
( u F w )  =  B  <->  ( A F w )  =  B ) )
32mobidv 2191 . . . 4  |-  ( u  =  A  ->  ( E* w ( u F w )  =  B  <->  E* w ( A F w )  =  B ) )
4 oveq2 5882 . . . . . . 7  |-  ( w  =  v  ->  (
u F w )  =  ( u F v ) )
54eqeq1d 2304 . . . . . 6  |-  ( w  =  v  ->  (
( u F w )  =  B  <->  ( u F v )  =  B ) )
65mo4 2189 . . . . 5  |-  ( E* w ( u F w )  =  B  <->  A. w A. v ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  w  =  v ) )
7 simpr 447 . . . . . . . . 9  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( u F v )  =  B )
87oveq2d 5890 . . . . . . . 8  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( w F ( u F v ) )  =  ( w F B ) )
9 simpl 443 . . . . . . . . . 10  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( u F w )  =  B )
109oveq1d 5889 . . . . . . . . 9  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( ( u F w ) F v )  =  ( B F v ) )
11 vex 2804 . . . . . . . . . . 11  |-  u  e. 
_V
12 vex 2804 . . . . . . . . . . 11  |-  w  e. 
_V
13 vex 2804 . . . . . . . . . . 11  |-  v  e. 
_V
14 caovmo.ass . . . . . . . . . . 11  |-  ( ( x F y ) F z )  =  ( x F ( y F z ) )
1511, 12, 13, 14caovass 6036 . . . . . . . . . 10  |-  ( ( u F w ) F v )  =  ( u F ( w F v ) )
16 caovmo.com . . . . . . . . . . 11  |-  ( x F y )  =  ( y F x )
1711, 12, 13, 16, 14caov12 6064 . . . . . . . . . 10  |-  ( u F ( w F v ) )  =  ( w F ( u F v ) )
1815, 17eqtri 2316 . . . . . . . . 9  |-  ( ( u F w ) F v )  =  ( w F ( u F v ) )
19 caovmo.2 . . . . . . . . . . 11  |-  B  e.  S
2019elexi 2810 . . . . . . . . . 10  |-  B  e. 
_V
2120, 13, 16caovcom 6033 . . . . . . . . 9  |-  ( B F v )  =  ( v F B )
2210, 18, 213eqtr3g 2351 . . . . . . . 8  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( w F ( u F v ) )  =  ( v F B ) )
238, 22eqtr3d 2330 . . . . . . 7  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( w F B )  =  ( v F B ) )
249, 19syl6eqel 2384 . . . . . . . . . 10  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( u F w )  e.  S
)
25 caovmo.dom . . . . . . . . . . 11  |-  dom  F  =  ( S  X.  S )
26 caovmo.3 . . . . . . . . . . 11  |-  -.  (/)  e.  S
2725, 26ndmovrcl 6022 . . . . . . . . . 10  |-  ( ( u F w )  e.  S  ->  (
u  e.  S  /\  w  e.  S )
)
2824, 27syl 15 . . . . . . . . 9  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( u  e.  S  /\  w  e.  S ) )
2928simprd 449 . . . . . . . 8  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  w  e.  S
)
30 oveq1 5881 . . . . . . . . . 10  |-  ( x  =  w  ->  (
x F B )  =  ( w F B ) )
31 id 19 . . . . . . . . . 10  |-  ( x  =  w  ->  x  =  w )
3230, 31eqeq12d 2310 . . . . . . . . 9  |-  ( x  =  w  ->  (
( x F B )  =  x  <->  ( w F B )  =  w ) )
33 caovmo.id . . . . . . . . 9  |-  ( x  e.  S  ->  (
x F B )  =  x )
3432, 33vtoclga 2862 . . . . . . . 8  |-  ( w  e.  S  ->  (
w F B )  =  w )
3529, 34syl 15 . . . . . . 7  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( w F B )  =  w )
367, 19syl6eqel 2384 . . . . . . . . . 10  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( u F v )  e.  S
)
3725, 26ndmovrcl 6022 . . . . . . . . . 10  |-  ( ( u F v )  e.  S  ->  (
u  e.  S  /\  v  e.  S )
)
3836, 37syl 15 . . . . . . . . 9  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( u  e.  S  /\  v  e.  S ) )
3938simprd 449 . . . . . . . 8  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  v  e.  S
)
40 oveq1 5881 . . . . . . . . . 10  |-  ( x  =  v  ->  (
x F B )  =  ( v F B ) )
41 id 19 . . . . . . . . . 10  |-  ( x  =  v  ->  x  =  v )
4240, 41eqeq12d 2310 . . . . . . . . 9  |-  ( x  =  v  ->  (
( x F B )  =  x  <->  ( v F B )  =  v ) )
4342, 33vtoclga 2862 . . . . . . . 8  |-  ( v  e.  S  ->  (
v F B )  =  v )
4439, 43syl 15 . . . . . . 7  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  ( v F B )  =  v )
4523, 35, 443eqtr3d 2336 . . . . . 6  |-  ( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  w  =  v )
4645ax-gen 1536 . . . . 5  |-  A. v
( ( ( u F w )  =  B  /\  ( u F v )  =  B )  ->  w  =  v )
476, 46mpgbir 1540 . . . 4  |-  E* w
( u F w )  =  B
483, 47vtoclg 2856 . . 3  |-  ( A  e.  S  ->  E* w ( A F w )  =  B )
49 moanimv 2214 . . 3  |-  ( E* w ( A  e.  S  /\  ( A F w )  =  B )  <->  ( A  e.  S  ->  E* w
( A F w )  =  B ) )
5048, 49mpbir 200 . 2  |-  E* w
( A  e.  S  /\  ( A F w )  =  B )
51 eleq1 2356 . . . . . . 7  |-  ( ( A F w )  =  B  ->  (
( A F w )  e.  S  <->  B  e.  S ) )
5219, 51mpbiri 224 . . . . . 6  |-  ( ( A F w )  =  B  ->  ( A F w )  e.  S )
5325, 26ndmovrcl 6022 . . . . . 6  |-  ( ( A F w )  e.  S  ->  ( A  e.  S  /\  w  e.  S )
)
5452, 53syl 15 . . . . 5  |-  ( ( A F w )  =  B  ->  ( A  e.  S  /\  w  e.  S )
)
5554simpld 445 . . . 4  |-  ( ( A F w )  =  B  ->  A  e.  S )
5655ancri 535 . . 3  |-  ( ( A F w )  =  B  ->  ( A  e.  S  /\  ( A F w )  =  B ) )
5756moimi 2203 . 2  |-  ( E* w ( A  e.  S  /\  ( A F w )  =  B )  ->  E* w ( A F w )  =  B )
5850, 57ax-mp 8 1  |-  E* w
( A F w )  =  B
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   E*wmo 2157   (/)c0 3468    X. cxp 4703   dom cdm 4705  (class class class)co 5874
This theorem is referenced by:  recmulnq  8604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-iota 5235  df-fv 5279  df-ov 5877
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