Theorem ismgmid 13640

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

Step	Hyp	Ref	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 13635	. . . . 5 |- E* e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x )
4		reu5 2764	. . . . 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 ) ) )
5	2, 3, 4	sylanblrc 416	. . . 4 |- ( ph -> E! e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
6		oveq1 6065	. . . . . . 7 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
7	6	eqeq1d 2243	. . . . . 6 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
8	7	ovanraleqv 6082	. . . . 5 |- ( e = U -> ( A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) <-> A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) )
9	8	riota2 6035	. . . 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 ) )
10	1, 5, 9	syl2anr 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 ) )
11	10	pm5.32da 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 6027	. . . . 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 )
13	5, 12	syl 14	. . . 4 |- ( ph -> ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) e. B )
14		eleq1 2297	. . . 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 ) )
15	13, 14	syl5ibcom 155	. . 3 |- ( ph -> ( ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = U -> U e. B ) )
16	15	pm4.71rd 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 6011	. . . 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 3613	. . . . . . 7 |- ( E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) -> E. e e e. B )
19	2, 18	syl 14	. . . . . 6 |- ( ph -> E. e e e. B )
20		ismgmid.b	. . . . . . . 8 |- B = ( Base ` G )
21	20	basmex 13356	. . . . . . 7 |- ( e e. B -> G e. _V )
22	21	exlimiv 1647	. . . . . 6 |- ( E. e e e. B -> G e. _V )
23	19, 22	syl 14	. . . . 5 |- ( ph -> G e. _V )
24		ismgmid.p	. . . . . 6 |- .+ = ( +g ` G )
25		ismgmid.o	. . . . . 6 |- .0. = ( 0g ` G )
26	20, 24, 25	grpidvalg 13636	. . . . 5 |- ( G e. _V -> .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) )
27	23, 26	syl 14	. . . 4 |- ( ph -> .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) ) )
28	17, 27	eqtr4id 2286	. . 3 |- ( ph -> ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = .0. )
29	28	eqeq1d 2243	. 2 |- ( ph -> ( ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = U <-> .0. = U ) )
30	11, 16, 29	3bitr2d 216	1 |- ( ph -> ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) <-> .0. = U ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   E!wreu 2524   E*wrmo 2525   _Vcvv 2815   iotacio 5315   ` cfv 5357   iota_crio 6010  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-0g 13555

This theorem is referenced by:  mgmidcl  13641  mgmlrid  13642  ismgmid2  13643  mgmidsssn0  13647  prds0g  14137  issrgid  14224  isringid  14268