Theorem ismgmid 16500

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 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 16495	. . . . . 6 |- E* e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x )
4	3	a1i 11	. . . . 5 |- ( ph -> E* e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
5		reu5 3007	. . . . 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 ) ) )
6	2, 4, 5	sylanbrc 669	. . . 4 |- ( ph -> E! e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
7		oveq1 6295	. . . . . . . 8 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
8	7	eqeq1d 2452	. . . . . . 7 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
9		oveq2 6296	. . . . . . . 8 |- ( e = U -> ( x .+ e ) = ( x .+ U ) )
10	9	eqeq1d 2452	. . . . . . 7 |- ( e = U -> ( ( x .+ e ) = x <-> ( x .+ U ) = x ) )
11	8, 10	anbi12d 716	. . . . . 6 |- ( e = U -> ( ( ( e .+ x ) = x /\ ( x .+ e ) = x ) <-> ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) )
12	11	ralbidv 2826	. . . . 5 |- ( e = U -> ( A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) <-> A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) )
13	12	riota2 6272	. . . 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 ) )
14	1, 6, 13	syl2anr 481	. . 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 ) )
15	14	pm5.32da 646	. 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 6264	. . . . 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 )
17	6, 16	syl 17	. . . 4 |- ( ph -> ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) e. B )
18		eleq1 2516	. . . 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 ) )
19	17, 18	syl5ibcom 224	. . 3 |- ( ph -> ( ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = U -> U e. B ) )
20	19	pm4.71rd 640	. 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 6250	. . . . 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 )
25	22, 23, 24	grpidval 16496	. . . . 5 |- .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
26	21, 25	eqtr4i 2475	. . . 4 |- ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = .0.
27	26	eqeq1i 2455	. . 3 |- ( ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = U <-> .0. = U )
28	27	a1i 11	. 2 |- ( ph -> ( ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = U <-> .0. = U ) )
29	15, 20, 28	3bitr2d 285	1 |- ( ph -> ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) <-> .0. = U ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   A.wral 2736   E.wrex 2737   E!wreu 2738   E*wrmo 2739   iotacio 5543   ` cfv 5581   iota_crio 6249  (class class class)co 6288   Basecbs 15114   +g cplusg 15183   0gc0g 15331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-riota 6250  df-ov 6291  df-0g 15333

This theorem is referenced by:  mgmidcl  16501  mgmlrid  16502  ismgmid2  16503  mgmidsssn0  16505  prds0g  16563  issrgid  17749  isringid  17799  signsw0g  29438