Theorem ismgmid 13459

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 13454	. . . . 5 |- E* e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x )
4		reu5 2751	. . . . 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 6024	. . . . . . 7 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
7	6	eqeq1d 2240	. . . . . 6 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
8	7	ovanraleqv 6041	. . . . 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 5994	. . . 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 5986	. . . . 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 2294	. . . 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 5970	. . . 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 3594	. . . . . . 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 13141	. . . . . . 7 |- ( e e. B -> G e. _V )
22	21	exlimiv 1646	. . . . . 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 13455	. . . . 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 2283	. . 3 |- ( ph -> ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = .0. )
29	28	eqeq1d 2240	. 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 1397   E.wex 1540   e. wcel 2202   A.wral 2510   E.wrex 2511   E!wreu 2512   E*wrmo 2513   _Vcvv 2802   iotacio 5284   ` cfv 5326   iota_crio 5969  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-0g 13340

This theorem is referenced by:  mgmidcl  13460  mgmlrid  13461  ismgmid2  13462  mgmidsssn0  13466  prds0g  13531  issrgid  13993  isringid  14037