Theorem ismgmid 17264

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 17259	. . . . . 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 3159	. . . . 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 698	. . . 4 |- ( ph -> E! e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
7		oveq1 6657	. . . . . . . 8 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
8	7	eqeq1d 2624	. . . . . . 7 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
9		oveq2 6658	. . . . . . . 8 |- ( e = U -> ( x .+ e ) = ( x .+ U ) )
10	9	eqeq1d 2624	. . . . . . 7 |- ( e = U -> ( ( x .+ e ) = x <-> ( x .+ U ) = x ) )
11	8, 10	anbi12d 747	. . . . . 6 |- ( e = U -> ( ( ( e .+ x ) = x /\ ( x .+ e ) = x ) <-> ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) )
12	11	ralbidv 2986	. . . . 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 6633	. . . 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 495	. . 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 673	. 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 6625	. . . . 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 2689	. . . 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 235	. . 3 |- ( ph -> ( ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = U -> U e. B ) )
20	19	pm4.71rd 667	. 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 6611	. . . . 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 17260	. . . . 5 |- .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
26	21, 25	eqtr4i 2647	. . . 4 |- ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = .0.
27	26	eqeq1i 2627	. . 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 296	1 |- ( ph -> ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) <-> .0. = U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   E*wrmo 2915   iotacio 5849   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102

This theorem is referenced by:  mgmidcl  17265  mgmlrid  17266  ismgmid2  17267  mgmidsssn0  17269  prds0g  17324  issrgid  18523  isringid  18573  signsw0g  30633