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Theorem ismgmid 12960
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 12955 . . . . 5  |-  E* e  e.  B  A. x  e.  B  ( (
e  .+  x )  =  x  /\  (
x  .+  e )  =  x )
4 reu5 2711 . . . . 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 5925 . . . . . . 7  |-  ( e  =  U  ->  (
e  .+  x )  =  ( U  .+  x ) )
76eqeq1d 2202 . . . . . 6  |-  ( e  =  U  ->  (
( e  .+  x
)  =  x  <->  ( U  .+  x )  =  x ) )
87ovanraleqv 5942 . . . . 5  |-  ( e  =  U  ->  ( A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x )  <->  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x ) ) )
98riota2 5896 . . . 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 5888 . . . . 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 2256 . . . 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 5873 . . . 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 3546 . . . . . . 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 12677 . . . . . . 7  |-  ( e  e.  B  ->  G  e.  _V )
2221exlimiv 1609 . . . . . 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 12956 . . . . 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 2245 . . 3  |-  ( ph  ->  ( iota_ e  e.  B  A. x  e.  B  ( ( e  .+  x )  =  x  /\  ( x  .+  e )  =  x ) )  =  .0.  )
2928eqeq1d 2202 . 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 1364   E.wex 1503    e. wcel 2164   A.wral 2472   E.wrex 2473   E!wreu 2474   E*wrmo 2475   _Vcvv 2760   iotacio 5213   ` cfv 5254   iota_crio 5872  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869
This theorem is referenced by:  mgmidcl  12961  mgmlrid  12962  ismgmid2  12963  mgmidsssn0  12967  issrgid  13477  isringid  13521
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