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Theorem ismgmid 12688
Description: The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
ismgmid.b  |-  B  =  ( Base `  G
)
ismgmid.o  |-  .0.  =  ( 0g `  G )
ismgmid.p  |-  .+  =  ( +g  `  G )
mgmidcl.e  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
Assertion
Ref Expression
ismgmid  |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x 
.+  U )  =  x ) )  <->  .0.  =  U ) )
Distinct variable groups:    x, e,  .+    .0. , e, x    B, e, x    e, G, x    U, e, x
Allowed substitution hints:    ph( x, e)

Proof of Theorem ismgmid
StepHypRef Expression
1 id 19 . . . 4  |-  ( U  e.  B  ->  U  e.  B )
2 mgmidcl.e . . . . 5  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
3 mgmidmo 12683 . . . . 5  |-  E* e  e.  B  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x )
4 reu5 2689 . . . . 5  |-  ( E! e  e.  B  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x )  <->  ( E. e  e.  B  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x )  /\  E* e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) )
52, 3, 4sylanblrc 416 . . . 4  |-  ( ph  ->  E! e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
6 oveq1 5876 . . . . . . 7  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
76eqeq1d 2186 . . . . . 6  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
87ovanraleqv 5893 . . . . 5  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
98riota2 5847 . . . 4  |-  ( ( U  e.  B  /\  E! e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  ->  ( A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x )  <->  ( iota_ e  e.  B  A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U ) )
101, 5, 9syl2anr 290 . . 3  |-  ( (
ph  /\  U  e.  B )  ->  ( A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x )  <->  ( iota_ e  e.  B  A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U ) )
1110pm5.32da 452 . 2  |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x 
.+  U )  =  x ) )  <->  ( U  e.  B  /\  ( iota_ e  e.  B  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) )  =  U ) ) )
12 riotacl 5839 . . . . 5  |-  ( E! e  e.  B  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x )  -> 
( iota_ e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  e.  B
)
135, 12syl 14 . . . 4  |-  ( ph  ->  ( iota_ e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  e.  B
)
14 eleq1 2240 . . . 4  |-  ( (
iota_ e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  =  U  ->  ( ( iota_ e  e.  B  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  e.  B  <->  U  e.  B
) )
1513, 14syl5ibcom 155 . . 3  |-  ( ph  ->  ( ( iota_ e  e.  B  A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  ->  U  e.  B ) )
1615pm4.71rd 394 . 2  |-  ( ph  ->  ( ( iota_ e  e.  B  A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  <->  ( U  e.  B  /\  ( iota_ e  e.  B  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  =  U ) ) )
17 df-riota 5825 . . . 4  |-  ( iota_ e  e.  B  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
18 rexm 3522 . . . . . . 7  |-  ( E. e  e.  B  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x )  ->  E. e  e  e.  B )
192, 18syl 14 . . . . . 6  |-  ( ph  ->  E. e  e  e.  B )
20 ismgmid.b . . . . . . . 8  |-  B  =  ( Base `  G
)
2120basmex 12503 . . . . . . 7  |-  ( e  e.  B  ->  G  e.  _V )
2221exlimiv 1598 . . . . . 6  |-  ( E. e  e  e.  B  ->  G  e.  _V )
2319, 22syl 14 . . . . 5  |-  ( ph  ->  G  e.  _V )
24 ismgmid.p . . . . . 6  |-  .+  =  ( +g  `  G )
25 ismgmid.o . . . . . 6  |-  .0.  =  ( 0g `  G )
2620, 24, 25grpidvalg 12684 . . . . 5  |-  ( G  e.  _V  ->  .0.  =  ( iota e
( e  e.  B  /\  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) ) ) )
2723, 26syl 14 . . . 4  |-  ( ph  ->  .0.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) ) ) )
2817, 27eqtr4id 2229 . . 3  |-  ( ph  ->  ( iota_ e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  =  .0.  )
2928eqeq1d 2186 . 2  |-  ( ph  ->  ( ( iota_ e  e.  B  A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  <->  .0.  =  U
) )
3011, 16, 293bitr2d 216 1  |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x 
.+  U )  =  x ) )  <->  .0.  =  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   E!wreu 2457   E*wrmo 2458   _Vcvv 2737   iotacio 5172   ` cfv 5212   iota_crio 5824  (class class class)co 5869   Basecbs 12445   +g cplusg 12518   0gc0g 12653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-ndx 12448  df-slot 12449  df-base 12451  df-0g 12655
This theorem is referenced by:  mgmidcl  12689  mgmlrid  12690  ismgmid2  12691  mgmidsssn0  12695  issrgid  12990  isringid  13034
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