Theorem ismgmid2 17267

Description: Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
ismgmid.b	|- B = ( Base ` G )
ismgmid.o	|- .0. = ( 0g ` G )
ismgmid.p	|- .+ = ( +g ` G )
ismgmid2.u	|- ( ph -> U e. B )
ismgmid2.l	|- ( ( ph /\ x e. B ) -> ( U .+ x ) = x )
ismgmid2.r	|- ( ( ph /\ x e. B ) -> ( x .+ U ) = x )

Assertion

Ref	Expression
ismgmid2	|- ( ph -> U = .0. )

Distinct variable groups:   x, .+   x, .0.   x, B   x, G   x, U   ph, x

Proof of Theorem ismgmid2

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismgmid2.u	. . 3 |- ( ph -> U e. B )
2		ismgmid2.l	. . . . 5 |- ( ( ph /\ x e. B ) -> ( U .+ x ) = x )
3		ismgmid2.r	. . . . 5 |- ( ( ph /\ x e. B ) -> ( x .+ U ) = x )
4	2, 3	jca 554	. . . 4 |- ( ( ph /\ x e. B ) -> ( ( U .+ x ) = x /\ ( x .+ U ) = x ) )
5	4	ralrimiva 2966	. . 3 |- ( ph -> A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) )
6		ismgmid.b	. . . 4 |- B = ( Base ` G )
7		ismgmid.o	. . . 4 |- .0. = ( 0g ` G )
8		ismgmid.p	. . . 4 |- .+ = ( +g ` G )
9		oveq1 6657	. . . . . . . . 9 |- ( e = U -> ( e .+ x ) = ( U .+ x ) )
10	9	eqeq1d 2624	. . . . . . . 8 |- ( e = U -> ( ( e .+ x ) = x <-> ( U .+ x ) = x ) )
11		oveq2 6658	. . . . . . . . 9 |- ( e = U -> ( x .+ e ) = ( x .+ U ) )
12	11	eqeq1d 2624	. . . . . . . 8 |- ( e = U -> ( ( x .+ e ) = x <-> ( x .+ U ) = x ) )
13	10, 12	anbi12d 747	. . . . . . 7 |- ( e = U -> ( ( ( e .+ x ) = x /\ ( x .+ e ) = x ) <-> ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) )
14	13	ralbidv 2986	. . . . . 6 |- ( e = U -> ( A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) <-> A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) )
15	14	rspcev 3309	. . . . 5 |- ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) -> E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
16	1, 5, 15	syl2anc 693	. . . 4 |- ( ph -> E. e e. B A. x e. B ( ( e .+ x ) = x /\ ( x .+ e ) = x ) )
17	6, 7, 8, 16	ismgmid 17264	. . 3 |- ( ph -> ( ( U e. B /\ A. x e. B ( ( U .+ x ) = x /\ ( x .+ U ) = x ) ) <-> .0. = U ) )
18	1, 5, 17	mpbi2and 956	. 2 |- ( ph -> .0. = U )
19	18	eqcomd 2628	1 |- ( ph -> U = .0. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ` cfv 5888  (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:  grpidd  17268  submnd0  17320  mnd1id  17332  frmd0  17397  mhmid  17536  cnaddid  18273  ringidss  18577  xrs10  19785