Theorem ismgmid 16008

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 16003	. . . . . 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 2998	. . . . 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 662	. . . 4 |- ( ph -> E! e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
7		oveq1 6203	. . . . . . . 8 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
8	7	eqeq1d 2384	. . . . . . 7 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
9		oveq2 6204	. . . . . . . 8 |- ( e = U -> ( x .+ e ) = ( x .+ U ) )
10	9	eqeq1d 2384	. . . . . . 7 |- ( e = U -> ( ( x .+ e ) = x <-> ( x .+ U ) = x ) )
11	8, 10	anbi12d 708	. . . . . 6 |- ( e = U -> ( ( ( e .+ x ) = x /\ ( x .+ e ) = x ) <-> ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) )
12	11	ralbidv 2821	. . . . 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 6180	. . . 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 476	. . 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 639	. 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 6172	. . . . 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 16	. . . 4 |- ( ph -> ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) e. B )
18		eleq1 2454	. . . 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 220	. . 3 |- ( ph -> ( ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = U -> U e. B ) )
20	19	pm4.71rd 633	. 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 6158	. . . . 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 16004	. . . . 5 |- .0. = ( iota e ( e e. B /\ A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) )
26	21, 25	eqtr4i 2414	. . . 4 |- ( iota_ e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) ) = .0.
27	26	eqeq1i 2389	. . 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 281	1 |- ( ph -> ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) <-> .0. = U ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1399   e. wcel 1826   A.wral 2732   E.wrex 2733   E!wreu 2734   E*wrmo 2735   iotacio 5458   ` cfv 5496   iota_crio 6157  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   0gc0g 14847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-riota 6158  df-ov 6199  df-0g 14849

This theorem is referenced by:  mgmidcl  16009  mgmlrid  16010  ismgmid2  16011  mgmidsssn0  16013  prds0g  16071  issrgid  17287  isringid  17337  signsw0g  28696