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Theorem ismgmid 16008
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 22 . . . 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 16003 . . . . . 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 2998 . . . . 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 662 . . . 4  |-  ( ph  ->  E! e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )
7 oveq1 6203 . . . . . . . 8  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
87eqeq1d 2384 . . . . . . 7  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
9 oveq2 6204 . . . . . . . 8  |-  ( e  =  U  ->  (
x  .+  e )  =  ( x  .+  U ) )
109eqeq1d 2384 . . . . . . 7  |-  ( e  =  U  ->  (
( x  .+  e
)  =  x  <->  ( x  .+  U )  =  x ) )
118, 10anbi12d 708 . . . . . 6  |-  ( e  =  U  ->  (
( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  ( ( U 
.+  x )  =  x  /\  ( x 
.+  U )  =  x ) ) )
1211ralbidv 2821 . . . . 5  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
1312riota2 6180 . . . 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 476 . . 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 639 . 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 6172 . . . . 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 2454 . . . 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 220 . . 3  |-  ( ph  ->  ( ( iota_ e  e.  B  A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  ->  U  e.  B ) )
2019pm4.71rd 633 . 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 df-riota 6158 . . . . 5  |-  ( 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 ) ) )
22 ismgmid.b . . . . . 6  |-  B  =  ( Base `  G
)
23 ismgmid.p . . . . . 6  |-  .+  =  ( +g  `  G )
24 ismgmid.o . . . . . 6  |-  .0.  =  ( 0g `  G )
2522, 23, 24grpidval 16004 . . . . 5  |-  .0.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  (
( e  .+  x
)  =  x  /\  ( x  .+  e )  =  x ) ) )
2621, 25eqtr4i 2414 . . . 4  |-  ( iota_ e  e.  B  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x ) )  =  .0.
2726eqeq1i 2389 . . 3  |-  ( (
iota_ e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  =  U  <-> 
.0.  =  U )
2827a1i 11 . 2  |-  ( ph  ->  ( ( iota_ e  e.  B  A. x  e.  B  ( ( e 
.+  x )  =  x  /\  ( x 
.+  e )  =  x ) )  =  U  <->  .0.  =  U
) )
2915, 20, 283bitr2d 281 1  |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x 
.+  U )  =  x ) )  <->  .0.  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   E.wrex 2733   E!wreu 2734   E*wrmo 2735   iotacio 5458   ` cfv 5496   iota_crio 6157  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   0gc0g 14847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-riota 6158  df-ov 6199  df-0g 14849
This theorem is referenced by:  mgmidcl  16009  mgmlrid  16010  ismgmid2  16011  mgmidsssn0  16013  prds0g  16071  issrgid  17287  isringid  17337  signsw0g  28696
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