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Theorem ismgmid 14387
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 19 . . . . 5  |-  ( U  e.  B  ->  U  e.  B )
2 mgmidcl.e . . . . . 6  |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
3 mgmidmo 14370 . . . . . . 7  |-  E* e  e.  B A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x )
43a1i 10 . . . . . 6  |-  ( ph  ->  E* e  e.  B A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
5 reu5 2753 . . . . . 6  |-  ( 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 645 . . . . 5  |-  ( ph  ->  E! e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
7 oveq1 5865 . . . . . . . . 9  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
87eqeq1d 2291 . . . . . . . 8  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
9 oveq2 5866 . . . . . . . . 9  |-  ( e  =  U  ->  (
x  .+  e )  =  ( x  .+  U ) )
109eqeq1d 2291 . . . . . . . 8  |-  ( e  =  U  ->  (
( x  .+  e
)  =  x  <->  ( x  .+  U )  =  x ) )
118, 10anbi12d 691 . . . . . . 7  |-  ( e  =  U  ->  (
( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  ( ( U 
.+  x )  =  x  /\  ( x 
.+  U )  =  x ) ) )
1211ralbidv 2563 . . . . . 6  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
1312riota2 6327 . . . . 5  |-  ( ( 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 464 . . . 4  |-  ( (
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 622 . . 3  |-  ( 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 6319 . . . . . 6  |-  ( 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 15 . . . . 5  |-  ( ph  ->  ( iota_ e  e.  B A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  e.  B
)
18 eleq1 2343 . . . . 5  |-  ( (
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 211 . . . 4  |-  ( ph  ->  ( ( iota_ e  e.  B A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  ->  U  e.  B ) )
2019pm4.71rd 616 . . 3  |-  ( 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 ) ) )
2115, 20bitr4d 247 . 2  |-  ( 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 ) )
22 riotaiota 6310 . . . . 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 ) ) ) )
236, 22syl 15 . . . 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 ) ) ) )
24 mndidcl.b . . . . 5  |-  B  =  ( Base `  G
)
25 mgmidcl.p . . . . 5  |-  .+  =  ( +g  `  G )
26 mndidcl.o . . . . 5  |-  .0.  =  ( 0g `  G )
2724, 25, 26grpidval 14384 . . . 4  |-  .0.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
2823, 27syl6eqr 2333 . . 3  |-  ( ph  ->  ( iota_ e  e.  B A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  =  .0.  )
2928eqeq1d 2291 . 2  |-  ( ph  ->  ( ( iota_ e  e.  B A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  <->  .0.  =  U
) )
3021, 29bitrd 244 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545   E*wrmo 2546   iotacio 5217   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208   0gc0g 13400
This theorem is referenced by:  mgmidcl  14388  mgmlrid  14389  ismgmid2  14390  prds0g  14406  gsumvallem1  14448  isrngid  15366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404
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