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Theorem ismgmid 14712
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
mndidcl.b  |-  B  =  ( Base `  G
)
mndidcl.o  |-  .0.  =  ( 0g `  G )
mgmidcl.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 21 . . . 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 14695 . . . . . 6  |-  E* e  e.  B A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x )
43a1i 11 . . . . 5  |-  ( ph  ->  E* e  e.  B A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
5 reu5 2923 . . . . 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 ) ) )
62, 4, 5sylanbrc 647 . . . 4  |-  ( ph  ->  E! e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
7 oveq1 6090 . . . . . . . 8  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
87eqeq1d 2446 . . . . . . 7  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
9 oveq2 6091 . . . . . . . 8  |-  ( e  =  U  ->  (
x  .+  e )  =  ( x  .+  U ) )
109eqeq1d 2446 . . . . . . 7  |-  ( e  =  U  ->  (
( x  .+  e
)  =  x  <->  ( x  .+  U )  =  x ) )
118, 10anbi12d 693 . . . . . 6  |-  ( e  =  U  ->  (
( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  ( ( U 
.+  x )  =  x  /\  ( x 
.+  U )  =  x ) ) )
1211ralbidv 2727 . . . . 5  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
1312riota2 6574 . . . 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 ) )
141, 6, 13syl2anr 466 . . 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 ) )
1514pm5.32da 624 . 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 ) ) )
16 riotacl 6566 . . . . 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
)
176, 16syl 16 . . . 4  |-  ( ph  ->  ( iota_ e  e.  B A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  e.  B
)
18 eleq1 2498 . . . 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
) )
1917, 18syl5ibcom 213 . . 3  |-  ( ph  ->  ( ( iota_ e  e.  B A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  ->  U  e.  B ) )
2019pm4.71rd 618 . 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 ) ) )
21 riotaiota 6557 . . . . 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 ) )  =  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) ) )
226, 21syl 16 . . . 4  |-  ( ph  ->  ( 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 ) ) ) )
23 mndidcl.b . . . . 5  |-  B  =  ( Base `  G
)
24 mgmidcl.p . . . . 5  |-  .+  =  ( +g  `  G )
25 mndidcl.o . . . . 5  |-  .0.  =  ( 0g `  G )
2623, 24, 25grpidval 14709 . . . 4  |-  .0.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
2722, 26syl6eqr 2488 . . 3  |-  ( ph  ->  ( iota_ e  e.  B A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  =  .0.  )
2827eqeq1d 2446 . 2  |-  ( ph  ->  ( ( iota_ e  e.  B A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  <->  .0.  =  U
) )
2915, 20, 283bitr2d 274 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    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   E!wreu 2709   E*wrmo 2710   iotacio 5418   ` cfv 5456  (class class class)co 6083   iota_crio 6544   Basecbs 13471   +g cplusg 13531   0gc0g 13725
This theorem is referenced by:  mgmidcl  14713  mgmlrid  14714  ismgmid2  14715  prds0g  14731  gsumvallem1  14773  isrngid  15691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-riota 6551  df-0g 13729
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